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Theorem pw2f1o 6969
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 2285 . . . 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 6968 . . . . 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 2793 . . . 4  |-  y  e. 
_V
1211cnvex 5211 . . 3  |-  `' y  e.  _V
13 imaexg 5028 . . 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 6968 . 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 6069 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 1625    e. wcel 1686    =/= wne 2448   _Vcvv 2790   ifcif 3567   ~Pcpw 3627   {csn 3642   {cpr 3643    e. cmpt 4079   `'ccnv 4690   "cima 4694   -1-1-onto->wf1o 5256  (class class class)co 5860    ^m cmap 6774
This theorem is referenced by:  pw2eng  6970  indf1o  23609
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-rab 2554  df-v 2792  df-sbc 2994  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-op 3651  df-uni 3830  df-br 4026  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-ov 5863  df-oprab 5864  df-mpt2 5865  df-map 6776
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