Theorem pw2f1odc 7090

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 2234	. . . 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 7089	. . . . 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 2818	. . . . 5 |- y e. _V
13	12	cnvex 5303	. . . 4 |- `' y e. _V
14	13	imaex 5118	. . 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 7089	. 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 6260	1 |- ( ph -> F : ~P A -1-1-onto-> ( { B , C } ^m A ) )

Syntax hints:   -> wi 4   /\ wa 104  DECID wdc 842   = wceq 1398   e. wcel 2205   =/= wne 2414   A.wral 2522   _Vcvv 2815   ifcif 3622   ~Pcpw 3671   {csn 3691   {cpr 3692   |-> cmpt 4173   `'ccnv 4750   "cima 4754   -1-1-onto->wf1o 5353  (class class class)co 6052   ^m cmap 6884

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661

This theorem depends on definitions:  df-bi 117  df-stab 839  df-dc 843  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-rab 2531  df-v 2817  df-sbc 3045  df-csb 3141  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-if 3623  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-br 4112  df-opab 4174  df-mpt 4175  df-id 4416  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-fo 5360  df-f1o 5361  df-fv 5362  df-ov 6055  df-oprab 6056  df-mpo 6057  df-map 6886

This theorem is referenced by:  exmidpw2en  7174