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Theorem pw2f1o 6852
Description: The power set of a set is equinumerous to set exponentiation with an unordered pair base of ordinal 2. Generalized from Proposition 10.44 of [TakeutiZaring] p. 96. (Contributed by Mario Carneiro, 6-Oct-2014.)
Hypotheses
Ref Expression
pw2f1o.1  |-  ( ph  ->  A  e.  V )
pw2f1o.2  |-  ( ph  ->  B  e.  W )
pw2f1o.3  |-  ( ph  ->  C  e.  W )
pw2f1o.4  |-  ( ph  ->  B  =/=  C )
pw2f1o.5  |-  F  =  ( x  e.  ~P A  |->  ( z  e.  A  |->  if ( z  e.  x ,  C ,  B ) ) )
Assertion
Ref Expression
pw2f1o  |-  ( ph  ->  F : ~P A -1-1-onto-> ( { B ,  C }  ^m  A ) )
Distinct variable groups:    x, z, A    x, B, z    x, C, z    ph, x
Allowed substitution hints:    ph( z)    F( x, z)    V( x, z)    W( x, z)

Proof of Theorem pw2f1o
StepHypRef Expression
1 pw2f1o.5 . 2  |-  F  =  ( x  e.  ~P A  |->  ( z  e.  A  |->  if ( z  e.  x ,  C ,  B ) ) )
2 eqid 2253 . . . 4  |-  ( z  e.  A  |->  if ( z  e.  x ,  C ,  B ) )  =  ( z  e.  A  |->  if ( z  e.  x ,  C ,  B ) )
3 pw2f1o.1 . . . . . 6  |-  ( ph  ->  A  e.  V )
4 pw2f1o.2 . . . . . 6  |-  ( ph  ->  B  e.  W )
5 pw2f1o.3 . . . . . 6  |-  ( ph  ->  C  e.  W )
6 pw2f1o.4 . . . . . 6  |-  ( ph  ->  B  =/=  C )
73, 4, 5, 6pw2f1olem 6851 . . . . 5  |-  ( ph  ->  ( ( x  e. 
~P A  /\  (
z  e.  A  |->  if ( z  e.  x ,  C ,  B ) )  =  ( z  e.  A  |->  if ( z  e.  x ,  C ,  B ) ) )  <->  ( (
z  e.  A  |->  if ( z  e.  x ,  C ,  B ) )  e.  ( { B ,  C }  ^m  A )  /\  x  =  ( `' ( z  e.  A  |->  if ( z  e.  x ,  C ,  B ) ) " { C } ) ) ) )
87biimpa 472 . . . 4  |-  ( (
ph  /\  ( x  e.  ~P A  /\  (
z  e.  A  |->  if ( z  e.  x ,  C ,  B ) )  =  ( z  e.  A  |->  if ( z  e.  x ,  C ,  B ) ) ) )  -> 
( ( z  e.  A  |->  if ( z  e.  x ,  C ,  B ) )  e.  ( { B ,  C }  ^m  A )  /\  x  =  ( `' ( z  e.  A  |->  if ( z  e.  x ,  C ,  B ) ) " { C } ) ) )
92, 8mpanr2 668 . . 3  |-  ( (
ph  /\  x  e.  ~P A )  ->  (
( z  e.  A  |->  if ( z  e.  x ,  C ,  B ) )  e.  ( { B ,  C }  ^m  A )  /\  x  =  ( `' ( z  e.  A  |->  if ( z  e.  x ,  C ,  B ) ) " { C } ) ) )
109simpld 447 . 2  |-  ( (
ph  /\  x  e.  ~P A )  ->  (
z  e.  A  |->  if ( z  e.  x ,  C ,  B ) )  e.  ( { B ,  C }  ^m  A ) )
11 vex 2730 . . . 4  |-  y  e. 
_V
1211cnvex 5115 . . 3  |-  `' y  e.  _V
13 imaexg 4933 . . 3  |-  ( `' y  e.  _V  ->  ( `' y " { C } )  e.  _V )
1412, 13mp1i 13 . 2  |-  ( (
ph  /\  y  e.  ( { B ,  C }  ^m  A ) )  ->  ( `' y
" { C }
)  e.  _V )
153, 4, 5, 6pw2f1olem 6851 . 2  |-  ( ph  ->  ( ( x  e. 
~P A  /\  y  =  ( z  e.  A  |->  if ( z  e.  x ,  C ,  B ) ) )  <-> 
( y  e.  ( { B ,  C }  ^m  A )  /\  x  =  ( `' y " { C }
) ) ) )
161, 10, 14, 15f1od 5919 1  |-  ( ph  ->  F : ~P A -1-1-onto-> ( { B ,  C }  ^m  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360    = wceq 1619    e. wcel 1621    =/= wne 2412   _Vcvv 2727   ifcif 3470   ~Pcpw 3530   {csn 3544   {cpr 3545    e. cmpt 3974   `'ccnv 4579   "cima 4583   -1-1-onto->wf1o 4591  (class class class)co 5710    ^m cmap 6658
This theorem is referenced by:  pw2eng  6853
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-rab 2516  df-v 2729  df-sbc 2922  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-br 3921  df-opab 3975  df-mpt 3976  df-id 4202  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-map 6660
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