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Theorem pw2f1o 6967
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
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 pw2f1o.5 . 2  |-  F  =  ( x  e.  ~P A  |->  ( z  e.  A  |->  if ( z  e.  x ,  C ,  B ) ) )
2 eqid 2283 . . . 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 6966 . . . . 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 470 . . . 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 665 . . 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 445 . 2  |-  ( (
ph  /\  x  e.  ~P A )  ->  (
z  e.  A  |->  if ( z  e.  x ,  C ,  B ) )  e.  ( { B ,  C }  ^m  A ) )
11 vex 2791 . . . 4  |-  y  e. 
_V
1211cnvex 5209 . . 3  |-  `' y  e.  _V
13 imaexg 5026 . . 3  |-  ( `' y  e.  _V  ->  ( `' y " { C } )  e.  _V )
1412, 13mp1i 11 . 2  |-  ( (
ph  /\  y  e.  ( { B ,  C }  ^m  A ) )  ->  ( `' y
" { C }
)  e.  _V )
153, 4, 5, 6pw2f1olem 6966 . 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 6067 1  |-  ( ph  ->  F : ~P A -1-1-onto-> ( { B ,  C }  ^m  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   _Vcvv 2788   ifcif 3565   ~Pcpw 3625   {csn 3640   {cpr 3641    e. cmpt 4077   `'ccnv 4688   "cima 4692   -1-1-onto->wf1o 5254  (class class class)co 5858    ^m cmap 6772
This theorem is referenced by:  pw2eng  6968  indf1o  23607
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-map 6774
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