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Theorem pw2f1o 7529
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 2454 . . . 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 7528 . . . . 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 484 . . . 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 684 . . 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 459 . 2  |-  ( (
ph  /\  x  e.  ~P A )  ->  (
z  e.  A  |->  if ( z  e.  x ,  C ,  B ) )  e.  ( { B ,  C }  ^m  A ) )
11 vex 3081 . . . 4  |-  y  e. 
_V
1211cnvex 6638 . . 3  |-  `' y  e.  _V
13 imaexg 6628 . . 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 7528 . 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 6423 1  |-  ( ph  ->  F : ~P A -1-1-onto-> ( { B ,  C }  ^m  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758    =/= wne 2648   _Vcvv 3078   ifcif 3902   ~Pcpw 3971   {csn 3988   {cpr 3990    |-> cmpt 4461   `'ccnv 4950   "cima 4954   -1-1-onto->wf1o 5528  (class class class)co 6203    ^m cmap 7327
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-map 7329
This theorem is referenced by:  pw2eng  7530  indf1o  26645
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