Theorem f1opw2 6519

Description: A one-to-one mapping induces a one-to-one mapping on power sets. This version of f1opw 6520 avoids the Axiom of Replacement. (Contributed by Mario Carneiro, 26-Jun-2015.)

Hypotheses

Ref	Expression
f1opw2.1	|- ( ph -> F : A -1-1-onto-> B )
f1opw2.2	|- ( ph -> ( `' F " a ) e. _V )
f1opw2.3	|- ( ph -> ( F " b ) e. _V )

Assertion

Ref	Expression
f1opw2	|- ( ph -> ( b e. ~P A |-> ( F " b ) ) : ~P A -1-1-onto-> ~P B )

Distinct variable groups:   a, b, A   B, a, b   F, a, b   ph, a, b

Proof of Theorem f1opw2

Step	Hyp	Ref	Expression
1		eqid 2450	. 2 |- ( b e. ~P A |-> ( F " b ) ) = ( b e. ~P A |-> ( F " b ) )
2		imassrn 5178	. . . . 5 |- ( F " b ) C_ ran F
3		f1opw2.1	. . . . . . 7 |- ( ph -> F : A -1-1-onto-> B )
4		f1ofo 5819	. . . . . . 7 |- ( F : A -1-1-onto-> B -> F : A -onto-> B )
5	3, 4	syl 17	. . . . . 6 |- ( ph -> F : A -onto-> B )
6		forn 5794	. . . . . 6 |- ( F : A -onto-> B -> ran F = B )
7	5, 6	syl 17	. . . . 5 |- ( ph -> ran F = B )
8	2, 7	syl5sseq 3479	. . . 4 |- ( ph -> ( F " b ) C_ B )
9		f1opw2.3	. . . . 5 |- ( ph -> ( F " b ) e. _V )
10		elpwg 3958	. . . . 5 |- ( ( F " b ) e. _V -> ( ( F " b ) e. ~P B <-> ( F " b ) C_ B ) )
11	9, 10	syl 17	. . . 4 |- ( ph -> ( ( F " b ) e. ~P B <-> ( F " b ) C_ B ) )
12	8, 11	mpbird 236	. . 3 |- ( ph -> ( F " b ) e. ~P B )
13	12	adantr 467	. 2 |- ( ( ph /\ b e. ~P A ) -> ( F " b ) e. ~P B )
14		imassrn 5178	. . . . 5 |- ( `' F " a ) C_ ran `' F
15		dfdm4 5026	. . . . . 6 |- dom F = ran `' F
16		f1odm 5816	. . . . . . 7 |- ( F : A -1-1-onto-> B -> dom F = A )
17	3, 16	syl 17	. . . . . 6 |- ( ph -> dom F = A )
18	15, 17	syl5eqr 2498	. . . . 5 |- ( ph -> ran `' F = A )
19	14, 18	syl5sseq 3479	. . . 4 |- ( ph -> ( `' F " a ) C_ A )
20		f1opw2.2	. . . . 5 |- ( ph -> ( `' F " a ) e. _V )
21		elpwg 3958	. . . . 5 |- ( ( `' F " a ) e. _V -> ( ( `' F " a ) e. ~P A <-> ( `' F " a ) C_ A ) )
22	20, 21	syl 17	. . . 4 |- ( ph -> ( ( `' F " a ) e. ~P A <-> ( `' F " a ) C_ A ) )
23	19, 22	mpbird 236	. . 3 |- ( ph -> ( `' F " a ) e. ~P A )
24	23	adantr 467	. 2 |- ( ( ph /\ a e. ~P B ) -> ( `' F " a ) e. ~P A )
25		elpwi 3959	. . . . . . 7 |- ( a e. ~P B -> a C_ B )
26	25	adantl 468	. . . . . 6 |- ( ( b e. ~P A /\ a e. ~P B ) -> a C_ B )
27		foimacnv 5829	. . . . . 6 |- ( ( F : A -onto-> B /\ a C_ B ) -> ( F " ( `' F " a ) ) = a )
28	5, 26, 27	syl2an 480	. . . . 5 |- ( ( ph /\ ( b e. ~P A /\ a e. ~P B ) ) -> ( F " ( `' F " a ) ) = a )
29	28	eqcomd 2456	. . . 4 |- ( ( ph /\ ( b e. ~P A /\ a e. ~P B ) ) -> a = ( F " ( `' F " a ) ) )
30		imaeq2 5163	. . . . 5 |- ( b = ( `' F " a ) -> ( F " b ) = ( F " ( `' F " a ) ) )
31	30	eqeq2d 2460	. . . 4 |- ( b = ( `' F " a ) -> ( a = ( F " b ) <-> a = ( F " ( `' F " a ) ) ) )
32	29, 31	syl5ibrcom 226	. . 3 |- ( ( ph /\ ( b e. ~P A /\ a e. ~P B ) ) -> ( b = ( `' F " a ) -> a = ( F " b ) ) )
33		f1of1 5811	. . . . . . 7 |- ( F : A -1-1-onto-> B -> F : A -1-1-> B )
34	3, 33	syl 17	. . . . . 6 |- ( ph -> F : A -1-1-> B )
35		elpwi 3959	. . . . . . 7 |- ( b e. ~P A -> b C_ A )
36	35	adantr 467	. . . . . 6 |- ( ( b e. ~P A /\ a e. ~P B ) -> b C_ A )
37		f1imacnv 5828	. . . . . 6 |- ( ( F : A -1-1-> B /\ b C_ A ) -> ( `' F " ( F " b ) ) = b )
38	34, 36, 37	syl2an 480	. . . . 5 |- ( ( ph /\ ( b e. ~P A /\ a e. ~P B ) ) -> ( `' F " ( F " b ) ) = b )
39	38	eqcomd 2456	. . . 4 |- ( ( ph /\ ( b e. ~P A /\ a e. ~P B ) ) -> b = ( `' F " ( F " b ) ) )
40		imaeq2 5163	. . . . 5 |- ( a = ( F " b ) -> ( `' F " a ) = ( `' F " ( F " b ) ) )
41	40	eqeq2d 2460	. . . 4 |- ( a = ( F " b ) -> ( b = ( `' F " a ) <-> b = ( `' F " ( F " b ) ) ) )
42	39, 41	syl5ibrcom 226	. . 3 |- ( ( ph /\ ( b e. ~P A /\ a e. ~P B ) ) -> ( a = ( F " b ) -> b = ( `' F " a ) ) )
43	32, 42	impbid 194	. 2 |- ( ( ph /\ ( b e. ~P A /\ a e. ~P B ) ) -> ( b = ( `' F " a ) <-> a = ( F " b ) ) )
44	1, 13, 24, 43	f1o2d 6518	1 |- ( ph -> ( b e. ~P A |-> ( F " b ) ) : ~P A -1-1-onto-> ~P B )

Syntax hints:   -> wi 4   <-> wb 188   /\ wa 371   = wceq 1443   e. wcel 1886   _Vcvv 3044   C_ wss 3403   ~Pcpw 3950   |-> cmpt 4460   `'ccnv 4832   dom cdm 4833   ran crn 4834   "cima 4836   -1-1->wf1 5578   -onto->wfo 5579   -1-1-onto->wf1o 5580

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-sep 4524  ax-nul 4533  ax-pr 4638

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-ral 2741  df-rex 2742  df-rab 2745  df-v 3046  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-op 3974  df-br 4402  df-opab 4461  df-mpt 4462  df-id 4748  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588

This theorem is referenced by:  f1opw  6520