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Theorem pw2f1o 7176
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 2408 . . . 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 7175 . . . . 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 471 . . . 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 666 . . 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 446 . 2  |-  ( (
ph  /\  x  e.  ~P A )  ->  (
z  e.  A  |->  if ( z  e.  x ,  C ,  B ) )  e.  ( { B ,  C }  ^m  A ) )
11 vex 2923 . . . 4  |-  y  e. 
_V
1211cnvex 5369 . . 3  |-  `' y  e.  _V
13 imaexg 5180 . . 3  |-  ( `' y  e.  _V  ->  ( `' y " { C } )  e.  _V )
1412, 13mp1i 12 . 2  |-  ( (
ph  /\  y  e.  ( { B ,  C }  ^m  A ) )  ->  ( `' y
" { C }
)  e.  _V )
153, 4, 5, 6pw2f1olem 7175 . 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 6257 1  |-  ( ph  ->  F : ~P A -1-1-onto-> ( { B ,  C }  ^m  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721    =/= wne 2571   _Vcvv 2920   ifcif 3703   ~Pcpw 3763   {csn 3778   {cpr 3779    e. cmpt 4230   `'ccnv 4840   "cima 4844   -1-1-onto->wf1o 5416  (class class class)co 6044    ^m cmap 6981
This theorem is referenced by:  pw2eng  7177  indf1o  24378
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-rab 2679  df-v 2922  df-sbc 3126  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-br 4177  df-opab 4231  df-mpt 4232  df-id 4462  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-map 6983
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