Theorem pw2f1odc 6964

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 )
pw2f1odc.4	|- ( ph -> A. p e. A A. q e. ~P ADECID p e. q )
pw2f1o.5	|- F = ( x e. ~P A |-> ( z e. A |-> if ( z e. x , C , B ) ) )

Assertion

Ref	Expression
pw2f1odc	|- ( ph -> F : ~P A -1-1-onto-> ( { B , C } ^m A ) )

Distinct variable groups:   A, p, q, x   z, A, x   x, B, z   x, C, z   ph, x

Allowed substitution hints:   ph( z, q, p)   B( q, p)   C( q, p)   F( x, z, q, p)   V( x, z, q, p)   W( x, z, q, p)

Proof of Theorem pw2f1odc

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		pw2f1o.5	. 2 |- F = ( x e. ~P A |-> ( z e. A |-> if ( z e. x , C , B ) ) )
2		eqid 2209	. . . 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 )
7		pw2f1odc.4	. . . . . 6 |- ( ph -> A. p e. A A. q e. ~P ADECID p e. q )
8	3, 4, 5, 6, 7	pw2f1odclem 6963	. . . . 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 } ) ) ) )
9	8	biimpa 296	. . . 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 } ) ) )
10	2, 9	mpanr2 438	. . 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 } ) ) )
11	10	simpld 112	. 2 |- ( ( ph /\ x e. ~P A ) -> ( z e. A |-> if ( z e. x , C , B ) ) e. ( { B , C } ^m A ) )
12		vex 2782	. . . . 5 |- y e. _V
13	12	cnvex 5243	. . . 4 |- `' y e. _V
14	13	imaex 5059	. . 3 |- ( `' y " { C } ) e. _V
15	14	a1i 9	. 2 |- ( ( ph /\ y e. ( { B , C } ^m A ) ) -> ( `' y " { C } ) e. _V )
16	3, 4, 5, 6, 7	pw2f1odclem 6963	. 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 } ) ) ) )
17	1, 11, 15, 16	f1od 6179	1 |- ( ph -> F : ~P A -1-1-onto-> ( { B , C } ^m A ) )

Syntax hints:   -> wi 4   /\ wa 104  DECID wdc 838   = wceq 1375   e. wcel 2180   =/= wne 2380   A.wral 2488   _Vcvv 2779   ifcif 3582   ~Pcpw 3629   {csn 3646   {cpr 3647   |-> cmpt 4124   `'ccnv 4695   "cima 4699   -1-1-onto->wf1o 5293  (class class class)co 5974   ^m cmap 6765

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-13 2182  ax-14 2183  ax-ext 2191  ax-sep 4181  ax-pow 4237  ax-pr 4272  ax-un 4501  ax-setind 4606

This theorem depends on definitions:  df-bi 117  df-stab 835  df-dc 839  df-3an 985  df-tru 1378  df-fal 1381  df-nf 1487  df-sb 1789  df-eu 2060  df-mo 2061  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-ne 2381  df-ral 2493  df-rex 2494  df-rab 2497  df-v 2781  df-sbc 3009  df-csb 3105  df-dif 3179  df-un 3181  df-in 3183  df-ss 3190  df-if 3583  df-pw 3631  df-sn 3652  df-pr 3653  df-op 3655  df-uni 3868  df-br 4063  df-opab 4125  df-mpt 4126  df-id 4361  df-xp 4702  df-rel 4703  df-cnv 4704  df-co 4705  df-dm 4706  df-rn 4707  df-res 4708  df-ima 4709  df-iota 5254  df-fun 5296  df-fn 5297  df-f 5298  df-f1 5299  df-fo 5300  df-f1o 5301  df-fv 5302  df-ov 5977  df-oprab 5978  df-mpo 5979  df-map 6767

This theorem is referenced by:  exmidpw2en  7042