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Theorem List for Metamath Proof Explorer - 27301-27400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremmdandyvr12 27301 Given the equivalences set in the hypotheses, there exist a proof where ch, th, ta, et match ze, si accordingly (Contributed by Jarvin Udandy, 7-Sep-2016.)
 |-  ( ph 
 <->  ze )   &    |-  ( ps  <->  si )   &    |-  ( ch  <->  ph )   &    |-  ( th  <->  ph )   &    |-  ( ta  <->  ps )   &    |-  ( et  <->  ps )   =>    |-  ( ( ( ( ch  <->  ze )  /\  ( th 
 <->  ze ) )  /\  ( ta  <->  si ) )  /\  ( et  <->  si ) )
 
Theoremmdandyvr13 27302 Given the equivalences set in the hypotheses, there exist a proof where ch, th, ta, et match ze, si accordingly (Contributed by Jarvin Udandy, 7-Sep-2016.)
 |-  ( ph 
 <->  ze )   &    |-  ( ps  <->  si )   &    |-  ( ch  <->  ps )   &    |-  ( th  <->  ph )   &    |-  ( ta  <->  ps )   &    |-  ( et  <->  ps )   =>    |-  ( ( ( ( ch  <->  si )  /\  ( th 
 <->  ze ) )  /\  ( ta  <->  si ) )  /\  ( et  <->  si ) )
 
Theoremmdandyvr14 27303 Given the equivalences set in the hypotheses, there exist a proof where ch, th, ta, et match ze, si accordingly (Contributed by Jarvin Udandy, 7-Sep-2016.)
 |-  ( ph 
 <->  ze )   &    |-  ( ps  <->  si )   &    |-  ( ch  <->  ph )   &    |-  ( th  <->  ps )   &    |-  ( ta  <->  ps )   &    |-  ( et  <->  ps )   =>    |-  ( ( ( ( ch  <->  ze )  /\  ( th 
 <-> 
 si ) )  /\  ( ta  <->  si ) )  /\  ( et  <->  si ) )
 
Theoremmdandyvr15 27304 Given the equivalences set in the hypotheses, there exist a proof where ch, th, ta, et match ze, si accordingly (Contributed by Jarvin Udandy, 7-Sep-2016.)
 |-  ( ph 
 <->  ze )   &    |-  ( ps  <->  si )   &    |-  ( ch  <->  ps )   &    |-  ( th  <->  ps )   &    |-  ( ta  <->  ps )   &    |-  ( et  <->  ps )   =>    |-  ( ( ( ( ch  <->  si )  /\  ( th 
 <-> 
 si ) )  /\  ( ta  <->  si ) )  /\  ( et  <->  si ) )
 
Theoremmdandyvrx0 27305 Given the exclusivities set in the hypotheses, there exist a proof where ch, th, ta, et exclude ze, si accordingly (Contributed by Jarvin Udandy, 7-Sep-2016.)
 |-  ( ph \/_ ze )   &    |-  ( ps \/_ si )   &    |-  ( ch 
 <-> 
 ph )   &    |-  ( th  <->  ph )   &    |-  ( ta  <->  ph )   &    |-  ( et  <->  ph )   =>    |-  ( ( ( ( ch \/_ ze )  /\  ( th \/_ ze ) )  /\  ( ta
 \/_ ze ) )  /\  ( et \/_ ze )
 )
 
Theoremmdandyvrx1 27306 Given the exclusivities set in the hypotheses, there exist a proof where ch, th, ta, et exclude ze, si accordingly (Contributed by Jarvin Udandy, 7-Sep-2016.)
 |-  ( ph \/_ ze )   &    |-  ( ps \/_ si )   &    |-  ( ch 
 <->  ps )   &    |-  ( th  <->  ph )   &    |-  ( ta  <->  ph )   &    |-  ( et  <->  ph )   =>    |-  ( ( ( ( ch \/_ si )  /\  ( th \/_ ze ) )  /\  ( ta
 \/_ ze ) )  /\  ( et \/_ ze )
 )
 
Theoremmdandyvrx2 27307 Given the exclusivities set in the hypotheses, there exist a proof where ch, th, ta, et exclude ze, si accordingly (Contributed by Jarvin Udandy, 7-Sep-2016.)
 |-  ( ph \/_ ze )   &    |-  ( ps \/_ si )   &    |-  ( ch 
 <-> 
 ph )   &    |-  ( th  <->  ps )   &    |-  ( ta  <->  ph )   &    |-  ( et  <->  ph )   =>    |-  ( ( ( ( ch \/_ ze )  /\  ( th \/_ si )
 )  /\  ( ta \/_
 ze ) )  /\  ( et \/_ ze )
 )
 
Theoremmdandyvrx3 27308 Given the exclusivities set in the hypotheses, there exist a proof where ch, th, ta, et exclude ze, si accordingly (Contributed by Jarvin Udandy, 7-Sep-2016.)
 |-  ( ph \/_ ze )   &    |-  ( ps \/_ si )   &    |-  ( ch 
 <->  ps )   &    |-  ( th  <->  ps )   &    |-  ( ta  <->  ph )   &    |-  ( et  <->  ph )   =>    |-  ( ( ( ( ch \/_ si )  /\  ( th \/_ si )
 )  /\  ( ta \/_
 ze ) )  /\  ( et \/_ ze )
 )
 
Theoremmdandyvrx4 27309 Given the exclusivities set in the hypotheses, there exist a proof where ch, th, ta, et exclude ze, si accordingly (Contributed by Jarvin Udandy, 7-Sep-2016.)
 |-  ( ph \/_ ze )   &    |-  ( ps \/_ si )   &    |-  ( ch 
 <-> 
 ph )   &    |-  ( th  <->  ph )   &    |-  ( ta  <->  ps )   &    |-  ( et  <->  ph )   =>    |-  ( ( ( ( ch \/_ ze )  /\  ( th \/_ ze ) )  /\  ( ta
 \/_ si ) )  /\  ( et \/_ ze )
 )
 
Theoremmdandyvrx5 27310 Given the exclusivities set in the hypotheses, there exist a proof where ch, th, ta, et exclude ze, si accordingly (Contributed by Jarvin Udandy, 7-Sep-2016.)
 |-  ( ph \/_ ze )   &    |-  ( ps \/_ si )   &    |-  ( ch 
 <->  ps )   &    |-  ( th  <->  ph )   &    |-  ( ta  <->  ps )   &    |-  ( et  <->  ph )   =>    |-  ( ( ( ( ch \/_ si )  /\  ( th \/_ ze ) )  /\  ( ta
 \/_ si ) )  /\  ( et \/_ ze )
 )
 
Theoremmdandyvrx6 27311 Given the exclusivities set in the hypotheses, there exist a proof where ch, th, ta, et exclude ze, si accordingly (Contributed by Jarvin Udandy, 7-Sep-2016.)
 |-  ( ph \/_ ze )   &    |-  ( ps \/_ si )   &    |-  ( ch 
 <-> 
 ph )   &    |-  ( th  <->  ps )   &    |-  ( ta  <->  ps )   &    |-  ( et  <->  ph )   =>    |-  ( ( ( ( ch \/_ ze )  /\  ( th \/_ si )
 )  /\  ( ta \/_ si ) )  /\  ( et \/_ ze ) )
 
Theoremmdandyvrx7 27312 Given the exclusivities set in the hypotheses, there exist a proof where ch, th, ta, et exclude ze, si accordingly (Contributed by Jarvin Udandy, 7-Sep-2016.)
 |-  ( ph \/_ ze )   &    |-  ( ps \/_ si )   &    |-  ( ch 
 <->  ps )   &    |-  ( th  <->  ps )   &    |-  ( ta  <->  ps )   &    |-  ( et  <->  ph )   =>    |-  ( ( ( ( ch \/_ si )  /\  ( th \/_ si )
 )  /\  ( ta \/_ si ) )  /\  ( et \/_ ze ) )
 
Theoremmdandyvrx8 27313 Given the exclusivities set in the hypotheses, there exist a proof where ch, th, ta, et exclude ze, si accordingly (Contributed by Jarvin Udandy, 7-Sep-2016.)
 |-  ( ph \/_ ze )   &    |-  ( ps \/_ si )   &    |-  ( ch 
 <-> 
 ph )   &    |-  ( th  <->  ph )   &    |-  ( ta  <->  ph )   &    |-  ( et  <->  ps )   =>    |-  ( ( ( ( ch \/_ ze )  /\  ( th \/_ ze ) )  /\  ( ta
 \/_ ze ) )  /\  ( et \/_ si )
 )
 
Theoremmdandyvrx9 27314 Given the exclusivities set in the hypotheses, there exist a proof where ch, th, ta, et exclude ze, si accordingly (Contributed by Jarvin Udandy, 7-Sep-2016.)
 |-  ( ph \/_ ze )   &    |-  ( ps \/_ si )   &    |-  ( ch 
 <->  ps )   &    |-  ( th  <->  ph )   &    |-  ( ta  <->  ph )   &    |-  ( et  <->  ps )   =>    |-  ( ( ( ( ch \/_ si )  /\  ( th \/_ ze ) )  /\  ( ta
 \/_ ze ) )  /\  ( et \/_ si )
 )
 
Theoremmdandyvrx10 27315 Given the exclusivities set in the hypotheses, there exist a proof where ch, th, ta, et exclude ze, si accordingly (Contributed by Jarvin Udandy, 7-Sep-2016.)
 |-  ( ph \/_ ze )   &    |-  ( ps \/_ si )   &    |-  ( ch 
 <-> 
 ph )   &    |-  ( th  <->  ps )   &    |-  ( ta  <->  ph )   &    |-  ( et  <->  ps )   =>    |-  ( ( ( ( ch \/_ ze )  /\  ( th \/_ si )
 )  /\  ( ta \/_
 ze ) )  /\  ( et \/_ si )
 )
 
Theoremmdandyvrx11 27316 Given the exclusivities set in the hypotheses, there exist a proof where ch, th, ta, et exclude ze, si accordingly (Contributed by Jarvin Udandy, 7-Sep-2016.)
 |-  ( ph \/_ ze )   &    |-  ( ps \/_ si )   &    |-  ( ch 
 <->  ps )   &    |-  ( th  <->  ps )   &    |-  ( ta  <->  ph )   &    |-  ( et  <->  ps )   =>    |-  ( ( ( ( ch \/_ si )  /\  ( th \/_ si )
 )  /\  ( ta \/_
 ze ) )  /\  ( et \/_ si )
 )
 
Theoremmdandyvrx12 27317 Given the exclusivities set in the hypotheses, there exist a proof where ch, th, ta, et exclude ze, si accordingly (Contributed by Jarvin Udandy, 7-Sep-2016.)
 |-  ( ph \/_ ze )   &    |-  ( ps \/_ si )   &    |-  ( ch 
 <-> 
 ph )   &    |-  ( th  <->  ph )   &    |-  ( ta  <->  ps )   &    |-  ( et  <->  ps )   =>    |-  ( ( ( ( ch \/_ ze )  /\  ( th \/_ ze ) )  /\  ( ta
 \/_ si ) )  /\  ( et \/_ si )
 )
 
Theoremmdandyvrx13 27318 Given the exclusivities set in the hypotheses, there exist a proof where ch, th, ta, et exclude ze, si accordingly (Contributed by Jarvin Udandy, 7-Sep-2016.)
 |-  ( ph \/_ ze )   &    |-  ( ps \/_ si )   &    |-  ( ch 
 <->  ps )   &    |-  ( th  <->  ph )   &    |-  ( ta  <->  ps )   &    |-  ( et  <->  ps )   =>    |-  ( ( ( ( ch \/_ si )  /\  ( th \/_ ze ) )  /\  ( ta
 \/_ si ) )  /\  ( et \/_ si )
 )
 
Theoremmdandyvrx14 27319 Given the exclusivities set in the hypotheses, there exist a proof where ch, th, ta, et exclude ze, si accordingly (Contributed by Jarvin Udandy, 7-Sep-2016.)
 |-  ( ph \/_ ze )   &    |-  ( ps \/_ si )   &    |-  ( ch 
 <-> 
 ph )   &    |-  ( th  <->  ps )   &    |-  ( ta  <->  ps )   &    |-  ( et  <->  ps )   =>    |-  ( ( ( ( ch \/_ ze )  /\  ( th \/_ si )
 )  /\  ( ta \/_ si ) )  /\  ( et \/_ si ) )
 
Theoremmdandyvrx15 27320 Given the exclusivities set in the hypotheses, there exist a proof where ch, th, ta, et exclude ze, si accordingly (Contributed by Jarvin Udandy, 7-Sep-2016.)
 |-  ( ph \/_ ze )   &    |-  ( ps \/_ si )   &    |-  ( ch 
 <->  ps )   &    |-  ( th  <->  ps )   &    |-  ( ta  <->  ps )   &    |-  ( et  <->  ps )   =>    |-  ( ( ( ( ch \/_ si )  /\  ( th \/_ si )
 )  /\  ( ta \/_ si ) )  /\  ( et \/_ si ) )
 
TheoremH15NH16TH15IH16 27321 Given 15 hypotheses and a 16th hypothesis, there exists a proof the 15 imply the 16th. (Contributed by Jarvin Udandy, 8-Sep-2016.)
 |-  ph   &    |-  ps   &    |-  ch   &    |-  th   &    |-  ta   &    |-  et   &    |-  ze   &    |-  si   &    |-  rh   &    |-  mu   &    |-  la   &    |-  ka   &    |- jph   &    |- jps   &    |- jch   &    |- jth   =>    |-  (
 ( ( ( ( ( ( ( ( ( ( ( ( ( ( ph  /\  ps )  /\  ch )  /\  th )  /\  ta )  /\  et )  /\  ze )  /\  si )  /\  rh )  /\  mu )  /\  la )  /\  ka )  /\ jph )  /\ jps
 )  /\ jch ) 
 -> jth )
 
Theoremdandysum2p2e4 27322

CONTRADICTION PROVED AT 1 + 1 = 2 .

Given the right hypotheses we can prove a dandysum of 2+2=4. The qed step is the value '4' in Decimal BEING IMPLIED by the hypotheses.

Note: Values that when added which exceed a 4bit value are not supported.

Note: Digits begin from left (least) to right (greatest). e.g. 1000 would be '1', 0100 would be '2'. 0010 would be '4'.

How to perceive the the hypotheses' bits in order: ( th <-> F. ), ( ta <-> F. ) Would be input value X's first bit, and input value Y's first bit.

( et <-> F ), ( ze <-> F. ) would be input value X's second bit, and input value Y's second bit. (Contributed by Jarvin Udandy, 6-Sep-2016.)

 |-  ( ph 
 <->  ( th  /\  ta ) )   &    |-  ( ps  <->  ( et  /\  ze ) )   &    |-  ( ch  <->  ( si  /\  rh ) )   &    |-  ( th  <->  F.  )   &    |-  ( ta  <->  F.  )   &    |-  ( et  <->  T.  )   &    |-  ( ze 
 <->  T.  )   &    |-  ( si  <->  F.  )   &    |-  ( rh  <->  F.  )   &    |-  ( mu  <->  F.  )   &    |-  ( la  <->  F.  )   &    |-  ( ka  <->  ( ( th \/_ ta ) \/_ ( th  /\  ta ) ) )   &    |-  (jph  <->  (
 ( et \/_ ze )  \/  ph ) )   &    |-  (jps  <->  ( ( si \/_ rh )  \/  ps ) )   &    |-  (jch  <->  ( ( mu
 \/_ la )  \/  ch ) )   =>    |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (
 ph 
 <->  ( th  /\  ta ) )  /\  ( ps  <->  ( et  /\  ze )
 ) )  /\  ( ch 
 <->  ( si  /\  rh ) ) )  /\  ( th  <->  F.  ) )  /\  ( ta  <->  F.  ) )  /\  ( et  <->  T.  ) )  /\  ( ze  <->  T.  ) )  /\  ( si  <->  F.  ) )  /\  ( rh  <->  F.  ) )  /\  ( mu  <->  F.  ) )  /\  ( la  <->  F.  ) )  /\  ( ka  <->  ( ( th \/_ ta ) \/_ ( th  /\  ta ) ) ) )  /\  (jph  <->  ( ( et \/_ ze )  \/  ph ) ) ) 
 /\  (jps  <->  (
 ( si \/_ rh )  \/  ps ) ) ) 
 /\  (jch  <->  (
 ( mu \/_ la )  \/  ch ) ) ) 
 ->  ( ( ( ( ka  <->  F.  )  /\  (jph  <->  F.  ) )  /\  (jps  <->  T.  ) )  /\  (jch  <->  F.  ) ) )
 
Theoremmdandysum2p2e4 27323 CONTRADICTION PROVED AT 1 + 1 = 2 . Luckily Mario Canerio did a successful version of his own.

See Mario's Relevant Work: 1.3.14 Half-adders and full adders in propositional calculus

Given the right hypotheses we can prove a dandysum of 2+2=4. The qed step is the value '4' in Decimal BEING IMPLIED by the hypotheses.

Note: Values that when added which exceed a 4bit value are not supported.

Note: Digits begin from left (least) to right (greatest). e.g. 1000 would be '1', 0100 would be '2'. 0010 would be '4'.

How to perceive the the hypotheses' bits in order: ( th <-> F. ), ( ta <-> F. ) Would be input value X's first bit, and input value Y's first bit.

( et <-> F. ), ( ze <-> F. ) would be input value X's second bit, and input value Y's second bit.

In mdandysum2p2e4, one might imagine what jth or jta could be then do the math with their truths. Also limited to the restriction jth, jta are having opposite truths equivalent to the stated truth constants.

(Contributed by Jarvin Udandy, 6-Sep-2016.)

 |-  (jth  <->  F.  )   &    |-  (jta  <->  T.  )   &    |-  ( ph  <->  ( th  /\  ta ) )   &    |-  ( ps  <->  ( et  /\  ze ) )   &    |-  ( ch  <->  ( si  /\  rh ) )   &    |-  ( th  <-> jth )   &    |-  ( ta 
 <-> jth
 )   &    |-  ( et  <-> jta )   &    |-  ( ze 
 <-> jta
 )   &    |-  ( si  <-> jth )   &    |-  ( rh 
 <-> jth
 )   &    |-  ( mu  <-> jth )   &    |-  ( la 
 <-> jth
 )   &    |-  ( ka  <->  ( ( th \/_ ta ) \/_ ( th  /\  ta ) ) )   &    |-  (jph  <->  (
 ( et \/_ ze )  \/  ph ) )   &    |-  (jps  <->  ( ( si \/_ rh )  \/  ps ) )   &    |-  (jch  <->  ( ( mu
 \/_ la )  \/  ch ) )   =>    |-  ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (
 ph 
 <->  ( th  /\  ta ) )  /\  ( ps  <->  ( et  /\  ze )
 ) )  /\  ( ch 
 <->  ( si  /\  rh ) ) )  /\  ( th  <->  F.  ) )  /\  ( ta  <->  F.  ) )  /\  ( et  <->  T.  ) )  /\  ( ze  <->  T.  ) )  /\  ( si  <->  F.  ) )  /\  ( rh  <->  F.  ) )  /\  ( mu  <->  F.  ) )  /\  ( la  <->  F.  ) )  /\  ( ka  <->  ( ( th \/_ ta ) \/_ ( th  /\  ta ) ) ) )  /\  (jph  <->  ( ( et \/_ ze )  \/  ph ) ) ) 
 /\  (jps  <->  (
 ( si \/_ rh )  \/  ps ) ) ) 
 /\  (jch  <->  (
 ( mu \/_ la )  \/  ch ) ) ) 
 ->  ( ( ( ( ka  <->  F.  )  /\  (jph  <->  F.  ) )  /\  (jps  <->  T.  ) )  /\  (jch  <->  F.  ) ) )
 
18.21  Mathbox for Alexander van der Vekens
 
18.21.1  1.1 Restricted quantification (extension)
 
Theoremr19.32 27324 Theorem 19.32 of [Margaris] p. 90 with restricted quantifiers, analogous to r19.32v 2687. (Contributed by Alexander van der Vekens, 29-Jun-2017.)
 |-  F/ x ph   =>    |-  ( A. x  e.  A  ( ph  \/  ps )  <->  ( ph  \/  A. x  e.  A  ps ) )
 
Theoremrexsb 27325* An equivalent expression for restricted existence, analogous to exsb 2072. (Contributed by Alexander van der Vekens, 1-Jul-2017.)
 |-  ( E. x  e.  A  ph  <->  E. y  e.  A  A. x ( x  =  y  ->  ph ) )
 
Theoremrexrsb 27326* An equivalent expression for restricted existence, analogous to exsb 2072. (Contributed by Alexander van der Vekens, 1-Jul-2017.)
 |-  ( E. x  e.  A  ph  <->  E. y  e.  A  A. x  e.  A  ( x  =  y  ->  ph ) )
 
Theorem2rexsb 27327* An equivalent expression for double restricted existence, analogous to rexsb 27325. (Contributed by Alexander van der Vekens, 1-Jul-2017.)
 |-  ( E. x  e.  A  E. y  e.  B  ph  <->  E. z  e.  A  E. w  e.  B  A. x A. y ( ( x  =  z  /\  y  =  w )  ->  ph )
 )
 
Theorem2rexrsb 27328* An equivalent expression for double restricted existence, analogous to 2exsb 2074. (Contributed by Alexander van der Vekens, 1-Jul-2017.)
 |-  ( E. x  e.  A  E. y  e.  B  ph  <->  E. z  e.  A  E. w  e.  B  A. x  e.  A  A. y  e.  B  ( ( x  =  z  /\  y  =  w )  ->  ph )
 )
 
Theoremcbvral2 27329* Change bound variables of double restricted universal quantification, using implicit substitution, analogous to cbvral2v 2773. (Contributed by Alexander van der Vekens, 2-Jul-2017.)
 |-  F/ z ph   &    |-  F/ x ch   &    |-  F/ w ch   &    |-  F/ y ps   &    |-  ( x  =  z  ->  ( ph  <->  ch ) )   &    |-  (
 y  =  w  ->  ( ch  <->  ps ) )   =>    |-  ( A. x  e.  A  A. y  e.  B  ph  <->  A. z  e.  A  A. w  e.  B  ps )
 
Theoremcbvrex2 27330* Change bound variables of double restricted universal quantification, using implicit substitution, analogous to cbvrex2v 2774. (Contributed by Alexander van der Vekens, 2-Jul-2017.)
 |-  F/ z ph   &    |-  F/ x ch   &    |-  F/ w ch   &    |-  F/ y ps   &    |-  ( x  =  z  ->  ( ph  <->  ch ) )   &    |-  (
 y  =  w  ->  ( ch  <->  ps ) )   =>    |-  ( E. x  e.  A  E. y  e.  B  ph  <->  E. z  e.  A  E. w  e.  B  ps )
 
Theorem2ralbiim 27331 Split a biconditional and distribute 2 quantifiers, analogous to 2albiim 1600 and ralbiim 2681. (Contributed by Alexander van der Vekens, 2-Jul-2017.)
 |-  ( A. x  e.  A  A. y  e.  B  (
 ph 
 <->  ps )  <->  ( A. x  e.  A  A. y  e.  B  ( ph  ->  ps )  /\  A. x  e.  A  A. y  e.  B  ( ps  ->  ph ) ) )
 
18.21.2  1.2 The empty set (extension)
 
Theoremraaan2 27332* Rearrange restricted quantifiers with two different restricting classes, analogous to raaan 3562. It is necessary that either both restricting classes are empty or both are not empty. (Contributed by Alexander van der Vekens, 29-Jun-2017.)
 |-  F/ y ph   &    |-  F/ x ps   =>    |-  (
 ( A  =  (/)  <->  B  =  (/) )  ->  ( A. x  e.  A  A. y  e.  B  (
 ph  /\  ps )  <->  (
 A. x  e.  A  ph 
 /\  A. y  e.  B  ps ) ) )
 
18.21.3  1.3 Restricted uniqueness and "at most one" quantification
 
Theoremrmoimi 27333 Restricted "at most one" is preserved through implication (note wff reversal). (Contributed by Alexander van der Vekens, 17-Jun-2017.)
 |-  ( ph  ->  ps )   =>    |-  ( E* x  e.  A ps  ->  E* x  e.  A ph )
 
Theorem2reu5a 27334 Double restricted existential uniqueness in terms of restricted existence and restricted "at most one." (Contributed by Alexander van der Vekens, 17-Jun-2017.)
 |-  ( E! x  e.  A  E! y  e.  B  ph  <->  ( E. x  e.  A  ( E. y  e.  B  ph 
 /\  E* y  e.  B ph )  /\  E* x  e.  A ( E. y  e.  B  ph  /\  E* y  e.  B ph ) ) )
 
Theoremreuimrmo 27335 Restricted uniqueness implies restricted "at most one" through implication, analogous to euimmo 2193. (Contributed by Alexander van der Vekens, 25-Jun-2017.)
 |-  ( A. x  e.  A  ( ph  ->  ps )  ->  ( E! x  e.  A  ps  ->  E* x  e.  A ph ) )
 
Theoremrmoanim 27336* Introduction of a conjunct into restricted "at most one" quantifier, analogous to moanim 2200. (Contributed by Alexander van der Vekens, 25-Jun-2017.)
 |-  F/ x ph   =>    |-  ( E* x  e.  A ( ph  /\  ps ) 
 <->  ( ph  ->  E* x  e.  A ps ) )
 
Theoremreuan 27337* Introduction of a conjunct into restricted uniqueness quantifier, analogous to euan 2201. (Contributed by Alexander van der Vekens, 2-Jul-2017.)
 |-  F/ x ph   =>    |-  ( E! x  e.  A  ( ph  /\  ps ) 
 <->  ( ph  /\  E! x  e.  A  ps ) )
 
18.21.4  1.4 Analogs to 1.6.6 Existential uniqueness (double quantification)
 
Theorem2reurex 27338* Double restricted quantification with existential uniqueness, analogous to 2euex 2216. (Contributed by Alexander van der Vekens, 24-Jun-2017.)
 |-  ( E! x  e.  A  E. y  e.  B  ph 
 ->  E. y  e.  B  E! x  e.  A  ph )
 
Theorem2reurmo 27339* Double restricted quantification with restricted existential uniqueness and restricted "at most one.", analogous to 2eumo 2217. (Contributed by Alexander van der Vekens, 24-Jun-2017.)
 |-  ( E! x  e.  A  E* y  e.  B ph 
 ->  E* x  e.  A E! y  e.  B  ph )
 
Theorem2reu2rex 27340* Double restricted existential uniqueness, analogous to 2eu2ex 2218. (Contributed by Alexander van der Vekens, 25-Jun-2017.)
 |-  ( E! x  e.  A  E! y  e.  B  ph 
 ->  E. x  e.  A  E. y  e.  B  ph )
 
Theorem2rmoswap 27341* A condition allowing swap of restricted "at most one" and restricted existential quantifiers, analogous to 2moswap 2219. (Contributed by Alexander van der Vekens, 25-Jun-2017.)
 |-  ( A. x  e.  A  E* y  e.  B ph 
 ->  ( E* x  e.  A E. y  e.  B  ph  ->  E* y  e.  B E. x  e.  A  ph ) )
 
Theorem2rexreu 27342* Double restricted existential uniqueness implies double restricted uniqueness quantification, analogous to 2exeu 2221. (Contributed by Alexander van der Vekens, 25-Jun-2017.)
 |-  (
 ( E! x  e.  A  E. y  e.  B  ph  /\  E! y  e.  B  E. x  e.  A  ph )  ->  E! x  e.  A  E! y  e.  B  ph )
 
Theorem2reu1 27343* Double restricted existential uniqueness. This theorem shows a condition under which a "naive" definition matches the correct one, analogous to 2eu1 2224. (Contributed by Alexander van der Vekens, 25-Jun-2017.)
 |-  ( A. x  e.  A  E* y  e.  B ph 
 ->  ( E! x  e.  A  E! y  e.  B  ph  <->  ( E! x  e.  A  E. y  e.  B  ph  /\  E! y  e.  B  E. x  e.  A  ph ) ) )
 
Theorem2reu2 27344* Double restricted existential uniqueness, analogous to 2eu2 2225. (Contributed by Alexander van der Vekens, 29-Jun-2017.)
 |-  ( E! y  e.  B  E. x  e.  A  ph 
 ->  ( E! x  e.  A  E! y  e.  B  ph  <->  E! x  e.  A  E. y  e.  B  ph ) )
 
Theorem2reu3 27345* Double restricted existential uniqueness, analogous to 2eu3 2226. (Contributed by Alexander van der Vekens, 29-Jun-2017.)
 |-  ( A. x  e.  A  A. y  e.  B  ( E* x  e.  A ph 
 \/  E* y  e.  B ph )  ->  ( ( E! x  e.  A  E! y  e.  B  ph 
 /\  E! y  e.  B  E! x  e.  A  ph )  <->  ( E! x  e.  A  E. y  e.  B  ph  /\  E! y  e.  B  E. x  e.  A  ph ) ) )
 
Theorem2reu4a 27346* Definition of double restricted existential uniqueness ("exactly one  x and exactly one  y"), analogous to 2eu4 2227 with the additional requirement that the restricting classes are not empty (which is not necessary as shown in 2reu4 27347). (Contributed by Alexander van der Vekens, 1-Jul-2017.)
 |-  (
 ( A  =/=  (/)  /\  B  =/= 
 (/) )  ->  (
 ( E! x  e.  A  E. y  e.  B  ph  /\  E! y  e.  B  E. x  e.  A  ph )  <->  ( E. x  e.  A  E. y  e.  B  ph  /\  E. z  e.  A  E. w  e.  B  A. x  e.  A  A. y  e.  B  ( ph  ->  ( x  =  z  /\  y  =  w )
 ) ) ) )
 
Theorem2reu4 27347* Definition of double restricted existential uniqueness ("exactly one  x and exactly one  y"), analogous to 2eu4 2227. (Contributed by Alexander van der Vekens, 1-Jul-2017.)
 |-  (
 ( E! x  e.  A  E. y  e.  B  ph  /\  E! y  e.  B  E. x  e.  A  ph )  <->  ( E. x  e.  A  E. y  e.  B  ph  /\  E. z  e.  A  E. w  e.  B  A. x  e.  A  A. y  e.  B  ( ph  ->  ( x  =  z  /\  y  =  w )
 ) ) )
 
Theorem2reu7 27348* Two equivalent expressions for double restricted existential uniqueness, analogous to 2eu7 2230. (Contributed by Alexander van der Vekens, 2-Jul-2017.)
 |-  (
 ( E! x  e.  A  E. y  e.  B  ph  /\  E! y  e.  B  E. x  e.  A  ph )  <->  E! x  e.  A  E! y  e.  B  ( E. x  e.  A  ph 
 /\  E. y  e.  B  ph ) )
 
Theorem2reu8 27349* Two equivalent expressions for double restricted existential uniqueness, analogous to 2eu8 2231. Curiously, we can put  E! on either of the internal conjuncts but not both. We can also commute  E! x  e.  A E! y  e.  B using 2reu7 27348. (Contributed by Alexander van der Vekens, 2-Jul-2017.)
 |-  ( E! x  e.  A  E! y  e.  B  ( E. x  e.  A  ph 
 /\  E. y  e.  B  ph )  <->  E! x  e.  A  E! y  e.  B  ( E! x  e.  A  ph 
 /\  E. y  e.  B  ph ) )
 
18.21.5  2.1a Restricted quantification (extension)
 
Theoremralbinrald 27350* Elemination of a restricted universal quantification under certain conditions. (Contributed by Alexander van der Vekens, 2-Aug-2017.)
 |-  ( ph  ->  X  e.  A )   &    |-  ( x  e.  A  ->  x  =  X )   &    |-  ( x  =  X  ->  ( ps  <->  th ) )   =>    |-  ( ph  ->  (
 A. x  e.  A  ps 
 <-> 
 th ) )
 
18.21.6  2.1b The universal class (extension)
 
Theoremnvelim 27351 If a class is the universal class it doesn't belong to any class, generalisation of nvel 4154. (Contributed by Alexander van der Vekens, 26-May-2017.)
 |-  ( A  =  _V  ->  -.  A  e.  B )
 
18.21.7  2.2 Relations (extension)
 
Theoremsbcrel 27352 Distribute proper substitution through the domain of a class. (Contributed by Alexander van der Vekens, 23-Jul-2017.)
 |-  ( A  e.  V  ->  (
 [. A  /  x ].
 Rel  R  <->  Rel  [_ A  /  x ]_ R ) )
 
Theoremcsbdmg 27353 Distribute proper substitution through the domain of a class. (Contributed by Alexander van der Vekens, 23-Jul-2017.)
 |-  ( A  e.  V  ->  [_ A  /  x ]_ dom  B  =  dom  [_  A  /  x ]_ B )
 
Theoremdmmpt2g 27354* Domain of a class given by the "maps to" notation, closed form of dmmpt2 6155. (Contributed by Alexander van der Vekens, 1-Jun-2017.)
 |-  F  =  ( x  e.  A ,  y  e.  B  |->  C )   =>    |-  ( C  e.  V  ->  dom  F  =  ( A  X.  B ) )
 
Theoremeldmressn 27355 Element of the domain of a restriction to a singleton. (Contributed by Alexander van der Vekens, 2-Jul-2017.)
 |-  ( B  e.  dom  (  F  |`  { A } )  ->  B  =  A )
 
Theoremdmressnsn 27356 The domain of a restriction to a singleton is a singleton. (Contributed by Alexander van der Vekens, 2-Jul-2017.)
 |-  ( A  e.  dom  F  ->  dom  (  F  |`  { A } )  =  { A } )
 
Theoremeldmressnsn 27357 The element of the domain of a restriction to a singleton is the element of the singleton. (Contributed by Alexander van der Vekens, 2-Jul-2017.)
 |-  ( A  e.  dom  F  ->  A  e.  dom  (  F  |` 
 { A } )
 )
 
18.21.8  2.3 Functions (extension)
 
Theoremsbcfun 27358 Distribute proper substitution through the domain of a class. (Contributed by Alexander van der Vekens, 23-Jul-2017.)
 |-  ( A  e.  V  ->  (
 [. A  /  x ].
 Fun  F  <->  Fun  [_ A  /  x ]_ F ) )
 
Theoremfvfundmfvn0 27359 If a class' value at an argument is not the empty set, the argument is contained in the domain of the class, and the class restricted to the argument is a function. (Contributed by Alexander van der Vekens, 26-May-2017.)
 |-  (
 ( F `  A )  =/=  (/)  ->  ( A  e.  dom  F  /\  Fun  ( F  |`  { A } ) ) )
 
Theoremfveqvfvv 27360 If a function's value at an argument is the universal class (which can never be the case because of fvex 5499), the function's value at this argument is any set (especially the empty set). In short "If a function's value is a proper class, it is a set", which sounds strange/contradictory, but which is a consequence of that a contradiction implies anything ( see pm2.21i 125). (Contributed by Alexander van der Vekens, 26-May-2017.)
 |-  (
 ( F `  A )  =  _V  ->  ( F `  A )  =  B )
 
Theoremfunresfunco 27361 Composition of two functions, generalization of funco 5257. (Contributed by Alexander van der Vekens, 25-Jul-2017.)
 |-  (
 ( Fun  ( F  |` 
 ran  G )  /\  Fun  G )  ->  Fun  ( F  o.  G ) )
 
Theoremfnresfnco 27362 Composition of two functions, similar to fnco 5317. (Contributed by Alexander van der Vekens, 25-Jul-2017.)
 |-  (
 ( ( F  |`  ran  G )  Fn  ran  G  /\  G  Fn  B )  ->  ( F  o.  G )  Fn  B )
 
Theoremfuncoressn 27363 A composition restricted to a singleton is a function under certain conditions. (Contributed by Alexander van der Vekens, 25-Jul-2017.)
 |-  (
 ( ( ( G `
  X )  e. 
 dom  F  /\  Fun  ( F  |`  { ( G `
  X ) }
 ) )  /\  ( G  Fn  A  /\  X  e.  A ) )  ->  Fun  ( ( F  o.  G )  |`  { X } ) )
 
Theoremfunressnfv 27364 A restriction to a singleton with a function value is a function under certain conditions. (Contributed by Alexander van der Vekens, 25-Jul-2017.)
 |-  (
 ( ( X  e.  dom  (  F  o.  G )  /\  Fun  ( ( F  o.  G )  |`  { X } ) ) 
 /\  ( G  Fn  A  /\  X  e.  A ) )  ->  Fun  ( F  |`  { ( G `
  X ) }
 ) )
 
18.21.9  2.4 Predicate "defined at"
 
Syntaxwdfat 27365 Extend the definition of a wff to include the "defined at" predicate. (Read: (The Function)  F is defined at (the argument)  A
 wff  Fdef@  A
 
Definitiondf-dfat 27366 Definition of the predicate that determines if some class  F is defined as function for an argument  A or, in other words, if the function value for some class  F for an argument  A is defined. We say that  F is defined at  A if a  F is a function restricted to the member  A of its domain. (Contributed by Alexander van der Vekens, 25-May-2017.)
 |-  ( Fdef@  A  <->  ( A  e.  dom 
 F  /\  Fun  ( F  |`  { A } )
 ) )
 
Theoremdfateq12d 27367 Equality deduction for "defined at". (Contributed by Alexander van der Vekens, 26-May-2017.)
 |-  ( ph  ->  F  =  G )   &    |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( Fdef@  A  <->  Gdef@  B ) )
 
Theoremnfdfat 27368 Bound-variable hypothesis builder for "defined at". To prove a deduction version of this theorem is not easily possible because many deduction versions for bound-variable hypothesis builder for constructs the definition of "defined at" is based on are not available (e.g. for Fun/Rel, dom, C_, etc.). (Contributed by Alexander van der Vekens, 26-May-2017.)
 |-  F/_ x F   &    |-  F/_ x A   =>    |- 
 F/ x  Fdef@  A
 
Theoremdfdfat2 27369* Alternate definition of the predicate "defined at" not using the  Fun predicate. (Contributed by Alexander van der Vekens, 22-Jul-2017.)
 |-  ( Fdef@  A  <->  ( A  e.  dom 
 F  /\  E! y  A F y ) )
 
18.21.10  2.5 Alternative definition of the value of a function
 
Syntaxcafv 27370 Extend the definition of a class to include the value of a function. (Read: The value of  F at  A, or " F of  A.")
 class  ( F'
 A )
 
Definitiondf-afv 27371* Alternative definition of the value of a function,  ( F' A ), also known as function application. In contrast to  ( F' A )  =  (/) (see df-fv 5229 and ndmfv 5513),  ( F' A )  =  _V if F is not defined for A! (Contributed by Alexander van der Vekens, 25-May-2017.)
 |-  ( F' A )  =  if ( Fdef@  A ,  U. { x  |  ( F " { A } )  =  { x } } ,  _V )
 
Theoremdfafv2 27372 Alternative definition of  ( F' A ) using 
( F `  A
) directly. (Contributed by Alexander van der Vekens, 22-Jul-2017.)
 |-  ( F' A )  =  if ( Fdef@  A ,  ( F `
  A ) ,  _V )
 
Theoremafveq12d 27373 Equality deduction for function value, analogous to fveq12d 5491. (Contributed by Alexander van der Vekens, 26-May-2017.)
 |-  ( ph  ->  F  =  G )   &    |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( F' A )  =  ( G' B ) )
 
Theoremafveq1 27374 Equality theorem for function value, analogous to fveq1 5484. (Contributed by Alexander van der Vekens, 22-Jul-2017.)
 |-  ( F  =  G  ->  ( F' A )  =  ( G' A ) )
 
Theoremafveq2 27375 Equality theorem for function value, analogous to fveq1 5484. (Contributed by Alexander van der Vekens, 22-Jul-2017.)
 |-  ( A  =  B  ->  ( F' A )  =  ( F' B ) )
 
Theoremnfafv 27376 Bound-variable hypothesis builder for function value, analogous to nffv 5492. To prove a deduction version of this analogous to nffvd 5494 is not easily possible because a deduction version of nfdfat 27368 cannot be shown easily. (Contributed by Alexander van der Vekens, 26-May-2017.)
 |-  F/_ x F   &    |-  F/_ x A   =>    |-  F/_ x ( F' A )
 
Theoremcsbafv12g 27377 Move class substitution in and out of a function value, analogous to csbfv12g 5495, with a direct proof proposed by Mario Carneiro, analogous to csbovg 5850. (Contributed by Alexander van der Vekens, 23-Jul-2017.)
 |-  ( A  e.  V  ->  [_ A  /  x ]_ ( F' B )  =  ( [_ A  /  x ]_ F' [_ A  /  x ]_ B ) )
 
Theoremafvfundmfveq 27378 If a class is a function restricted to a member of its domain, then the function value for this member is equal for both definitions. (Contributed by Alexander van der Vekens, 25-May-2017.)
 |-  ( Fdef@  A  ->  ( F' A )  =  ( F `
  A ) )
 
Theoremafvnfundmuv 27379 If a set is not in the domain of a class or the class is not a function restricted to the set, then the function value for this set is the universe. (Contributed by Alexander van der Vekens, 26-May-2017.)
 |-  ( -.  Fdef@  A  ->  ( F'
 A )  =  _V )
 
Theoremndmafv 27380 The value of a class outside its domain is the universe, compare with ndmfv 5513. (Contributed by Alexander van der Vekens, 25-May-2017.)
 |-  ( -.  A  e.  dom  F  ->  ( F' A )  =  _V )
 
Theoremafvvdm 27381 If the function value of a class for an argument is a set, the argument is contained in the domain of the class. (Contributed by Alexander van der Vekens, 25-May-2017.)
 |-  (
 ( F' A )  e.  B  ->  A  e.  dom  F )
 
Theoremnfunsnafv 27382 If the restriction of a class to a singleton is not a function, its value is the universe, compare with nfunsn 5519 (Contributed by Alexander van der Vekens, 25-May-2017.)
 |-  ( -.  Fun  ( F  |`  { A } )  ->  ( F'
 A )  =  _V )
 
Theoremafvvfunressn 27383 If the function value of a class for an argument is a set, the class restricted to the singleton of the argument is a function. (Contributed by Alexander van der Vekens, 25-May-2017.)
 |-  (
 ( F' A )  e.  B  ->  Fun  ( F  |`  { A }
 ) )
 
Theoremafvprc 27384 A function's value at a proper class is the universe, compare with fvprc 5482. (Contributed by Alexander van der Vekens, 25-May-2017.)
 |-  ( -.  A  e.  _V  ->  ( F' A )  =  _V )
 
Theoremafvvv 27385 If a function's value at an argument is a set, the argument is also a set. (Contributed by Alexander van der Vekens, 25-May-2017.)
 |-  (
 ( F' A )  e.  B  ->  A  e.  _V )
 
Theoremafvpcfv0 27386 If the value of the alternative function at an argument is the universe, the function's value at this argument is the empty set. (Contributed by Alexander van der Vekens, 25-May-2017.)
 |-  (
 ( F' A )  =  _V  ->  ( F `  A )  =  (/) )
 
Theoremafvnufveq 27387 The value of the alternative function at a set as argument equals the function's value at this argument. (Contributed by Alexander van der Vekens, 25-May-2017.)
 |-  (
 ( F' A )  =/=  _V  ->  ( F' A )  =  ( F `  A ) )
 
Theoremafvvfveq 27388 The value of the alternative function at a set as argument equals the function's value at this argument. (Contributed by Alexander van der Vekens, 25-May-2017.)
 |-  (
 ( F' A )  e.  B  ->  ( F' A )  =  ( F `  A ) )
 
Theoremafv0fv0 27389 If the value of the alternative function at an argument is the empty set, the function's value at this argument is the empty set. (Contributed by Alexander van der Vekens, 25-May-2017.)
 |-  (
 ( F' A )  =  (/)  ->  ( F `  A )  =  (/) )
 
Theoremafvfvn0fveq 27390 If the function's value at an argument is not the empty set, it equals the value of the alternative function at this argument. (Contributed by Alexander van der Vekens, 25-May-2017.)
 |-  (
 ( F `  A )  =/=  (/)  ->  ( F' A )  =  ( F `
  A ) )
 
Theoremafv0nbfvbi 27391 The function's value at an argument is an element of a set if and only if the value of the alternative function at this argument is an element of that set, if the set does not contain the empty set. (Contributed by Alexander van der Vekens, 25-May-2017.)
 |-  ( (/)  e/  B  ->  ( ( F' A )  e.  B  <->  ( F `  A )  e.  B ) )
 
Theoremafvfv0bi 27392 The function's value at an argument is the empty set if and only if the value of the alternative function at this argument is either the empty set or the universe. (Contributed by Alexander van der Vekens, 25-May-2017.)
 |-  (
 ( F `  A )  =  (/)  <->  ( ( F'
 A )  =  (/)  \/  ( F' A )  =  _V ) )
 
Theoremfnbrafvb 27393 Equivalence of function value and binary relation, analogous to fnbrfvb 5524. (Contributed by Alexander van der Vekens, 25-May-2017.)
 |-  (
 ( F  Fn  A  /\  B  e.  A ) 
 ->  ( ( F' B )  =  C  <->  B F C ) )
 
Theoremfnopafvb 27394 Equivalence of function value and ordered pair membership, analogous to fnopfvb 5525. (Contributed by Alexander van der Vekens, 25-May-2017.)
 |-  (
 ( F  Fn  A  /\  B  e.  A ) 
 ->  ( ( F' B )  =  C  <->  <. B ,  C >.  e.  F ) )
 
Theoremfunbrafvb 27395 Equivalence of function value and binary relation, analogous to funbrfvb 5526. (Contributed by Alexander van der Vekens, 25-May-2017.)
 |-  (
 ( Fun  F  /\  A  e.  dom  F ) 
 ->  ( ( F' A )  =  B  <->  A F B ) )
 
Theoremfunopafvb 27396 Equivalence of function value and ordered pair membership, analogous to funopfvb 5527. (Contributed by Alexander van der Vekens, 25-May-2017.)
 |-  (
 ( Fun  F  /\  A  e.  dom  F ) 
 ->  ( ( F' A )  =  B  <->  <. A ,  B >.  e.  F ) )
 
Theoremfunbrafv 27397 The second argument of a binary relation on a function is the function's value, analogous to funbrfv 5522. (Contributed by Alexander van der Vekens, 25-May-2017.)
 |-  ( Fun  F  ->  ( A F B  ->  ( F'
 A )  =  B ) )
 
Theoremfunbrafv2b 27398 Function value in terms of a binary relation, analogous to funbrfv2b 5528. (Contributed by Alexander van der Vekens, 25-May-2017.)
 |-  ( Fun  F  ->  ( A F B  <->  ( A  e.  dom 
 F  /\  ( F' A )  =  B ) ) )
 
Theoremdfafn5a 27399* Representation of a function in terms of its values, analogous to dffn5 5529 (only one direction of implication!). (Contributed by Alexander van der Vekens, 25-May-2017.)
 |-  ( F  Fn  A  ->  F  =  ( x  e.  A  |->  ( F' x ) ) )
 
Theoremdfafn5b 27400* Representation of a function in terms of its values, analogous to dffn5 5529 (only if it is assumed that the function value for each x is a set). (Contributed by Alexander van der Vekens, 25-May-2017.)
 |-  ( A. x  e.  A  ( F' x )  e.  V  ->  ( F  Fn  A  <->  F  =  ( x  e.  A  |->  ( F' x ) ) ) )
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