Theorem pw2f1odc 7026

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 2230	. . . 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 7025	. . . . 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 2804	. . . . 5 |- y e. _V
13	12	cnvex 5277	. . . 4 |- `' y e. _V
14	13	imaex 5093	. . 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 7025	. 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 6231	1 |- ( ph -> F : ~P A -1-1-onto-> ( { B , C } ^m A ) )

Syntax hints:   -> wi 4   /\ wa 104  DECID wdc 841   = wceq 1397   e. wcel 2201   =/= wne 2401   A.wral 2509   _Vcvv 2801   ifcif 3604   ~Pcpw 3653   {csn 3670   {cpr 3671   |-> cmpt 4151   `'ccnv 4726   "cima 4730   -1-1-onto->wf1o 5327  (class class class)co 6023   ^m cmap 6822

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2203  ax-14 2204  ax-ext 2212  ax-sep 4208  ax-pow 4266  ax-pr 4301  ax-un 4532  ax-setind 4637

This theorem depends on definitions:  df-bi 117  df-stab 838  df-dc 842  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1810  df-eu 2081  df-mo 2082  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ne 2402  df-ral 2514  df-rex 2515  df-rab 2518  df-v 2803  df-sbc 3031  df-csb 3127  df-dif 3201  df-un 3203  df-in 3205  df-ss 3212  df-if 3605  df-pw 3655  df-sn 3676  df-pr 3677  df-op 3679  df-uni 3895  df-br 4090  df-opab 4152  df-mpt 4153  df-id 4392  df-xp 4733  df-rel 4734  df-cnv 4735  df-co 4736  df-dm 4737  df-rn 4738  df-res 4739  df-ima 4740  df-iota 5288  df-fun 5330  df-fn 5331  df-f 5332  df-f1 5333  df-fo 5334  df-f1o 5335  df-fv 5336  df-ov 6026  df-oprab 6027  df-mpo 6028  df-map 6824

This theorem is referenced by:  exmidpw2en  7109