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Theorem funfveu 5188
Description: A function has one value given an argument in its domain. (Contributed by Jim Kingdon, 29-Dec-2018.)
Assertion
Ref Expression
funfveu  |-  ( ( Fun  F  /\  A  e.  dom  F )  ->  E! y  A F
y )
Distinct variable groups:    y, A    y, F

Proof of Theorem funfveu
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 eleq1 2100 . . . . 5  |-  ( x  =  A  ->  (
x  e.  dom  F  <->  A  e.  dom  F ) )
21anbi2d 437 . . . 4  |-  ( x  =  A  ->  (
( Fun  F  /\  x  e.  dom  F )  <-> 
( Fun  F  /\  A  e.  dom  F ) ) )
3 breq1 3767 . . . . 5  |-  ( x  =  A  ->  (
x F y  <->  A F
y ) )
43eubidv 1908 . . . 4  |-  ( x  =  A  ->  ( E! y  x F
y  <->  E! y  A F y ) )
52, 4imbi12d 223 . . 3  |-  ( x  =  A  ->  (
( ( Fun  F  /\  x  e.  dom  F )  ->  E! y  x F y )  <->  ( ( Fun  F  /\  A  e. 
dom  F )  ->  E! y  A F
y ) ) )
6 dffun8 4929 . . . . 5  |-  ( Fun 
F  <->  ( Rel  F  /\  A. x  e.  dom  F E! y  x F y ) )
76simprbi 260 . . . 4  |-  ( Fun 
F  ->  A. x  e.  dom  F E! y  x F y )
87r19.21bi 2407 . . 3  |-  ( ( Fun  F  /\  x  e.  dom  F )  ->  E! y  x F
y )
95, 8vtoclg 2613 . 2  |-  ( A  e.  dom  F  -> 
( ( Fun  F  /\  A  e.  dom  F )  ->  E! y  A F y ) )
109anabsi7 515 1  |-  ( ( Fun  F  /\  A  e.  dom  F )  ->  E! y  A F
y )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 97    = wceq 1243    e. wcel 1393   E!weu 1900   A.wral 2306   class class class wbr 3764   dom cdm 4345   Rel wrel 4350   Fun wfun 4896
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-id 4030  df-cnv 4353  df-co 4354  df-dm 4355  df-fun 4904
This theorem is referenced by:  funfvex  5192
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