Theorem pw2f1odc 6939

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 2206	. . . 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 6938	. . . . 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 2776	. . . . 5 |- y e. _V
13	12	cnvex 5226	. . . 4 |- `' y e. _V
14	13	imaex 5042	. . 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 6938	. 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 6156	1 |- ( ph -> F : ~P A -1-1-onto-> ( { B , C } ^m A ) )

Syntax hints:   -> wi 4   /\ wa 104  DECID wdc 836   = wceq 1373   e. wcel 2177   =/= wne 2377   A.wral 2485   _Vcvv 2773   ifcif 3572   ~Pcpw 3617   {csn 3634   {cpr 3635   |-> cmpt 4109   `'ccnv 4678   "cima 4682   -1-1-onto->wf1o 5275  (class class class)co 5951   ^m cmap 6742

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-sep 4166  ax-pow 4222  ax-pr 4257  ax-un 4484  ax-setind 4589

This theorem depends on definitions:  df-bi 117  df-stab 833  df-dc 837  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-ral 2490  df-rex 2491  df-rab 2494  df-v 2775  df-sbc 3000  df-csb 3095  df-dif 3169  df-un 3171  df-in 3173  df-ss 3180  df-if 3573  df-pw 3619  df-sn 3640  df-pr 3641  df-op 3643  df-uni 3853  df-br 4048  df-opab 4110  df-mpt 4111  df-id 4344  df-xp 4685  df-rel 4686  df-cnv 4687  df-co 4688  df-dm 4689  df-rn 4690  df-res 4691  df-ima 4692  df-iota 5237  df-fun 5278  df-fn 5279  df-f 5280  df-f1 5281  df-fo 5282  df-f1o 5283  df-fv 5284  df-ov 5954  df-oprab 5955  df-mpo 5956  df-map 6744

This theorem is referenced by:  exmidpw2en  7016