Theorem f1opw2 6888

Description: A one-to-one mapping induces a one-to-one mapping on power sets. This version of f1opw 6889 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 2622	. 2 |- ( b e. ~P A |-> ( F " b ) ) = ( b e. ~P A |-> ( F " b ) )
2		imassrn 5477	. . . . 5 |- ( F " b ) C_ ran F
3		f1opw2.1	. . . . . . 7 |- ( ph -> F : A -1-1-onto-> B )
4		f1ofo 6144	. . . . . . 7 |- ( F : A -1-1-onto-> B -> F : A -onto-> B )
5	3, 4	syl 17	. . . . . 6 |- ( ph -> F : A -onto-> B )
6		forn 6118	. . . . . 6 |- ( F : A -onto-> B -> ran F = B )
7	5, 6	syl 17	. . . . 5 |- ( ph -> ran F = B )
8	2, 7	syl5sseq 3653	. . . 4 |- ( ph -> ( F " b ) C_ B )
9		f1opw2.3	. . . . 5 |- ( ph -> ( F " b ) e. _V )
10		elpwg 4166	. . . . 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 247	. . 3 |- ( ph -> ( F " b ) e. ~P B )
13	12	adantr 481	. 2 |- ( ( ph /\ b e. ~P A ) -> ( F " b ) e. ~P B )
14		imassrn 5477	. . . . 5 |- ( `' F " a ) C_ ran `' F
15		dfdm4 5316	. . . . . 6 |- dom F = ran `' F
16		f1odm 6141	. . . . . . 7 |- ( F : A -1-1-onto-> B -> dom F = A )
17	3, 16	syl 17	. . . . . 6 |- ( ph -> dom F = A )
18	15, 17	syl5eqr 2670	. . . . 5 |- ( ph -> ran `' F = A )
19	14, 18	syl5sseq 3653	. . . 4 |- ( ph -> ( `' F " a ) C_ A )
20		f1opw2.2	. . . . 5 |- ( ph -> ( `' F " a ) e. _V )
21		elpwg 4166	. . . . 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 247	. . 3 |- ( ph -> ( `' F " a ) e. ~P A )
24	23	adantr 481	. 2 |- ( ( ph /\ a e. ~P B ) -> ( `' F " a ) e. ~P A )
25		elpwi 4168	. . . . . . 7 |- ( a e. ~P B -> a C_ B )
26	25	adantl 482	. . . . . 6 |- ( ( b e. ~P A /\ a e. ~P B ) -> a C_ B )
27		foimacnv 6154	. . . . . 6 |- ( ( F : A -onto-> B /\ a C_ B ) -> ( F " ( `' F " a ) ) = a )
28	5, 26, 27	syl2an 494	. . . . 5 |- ( ( ph /\ ( b e. ~P A /\ a e. ~P B ) ) -> ( F " ( `' F " a ) ) = a )
29	28	eqcomd 2628	. . . 4 |- ( ( ph /\ ( b e. ~P A /\ a e. ~P B ) ) -> a = ( F " ( `' F " a ) ) )
30		imaeq2 5462	. . . . 5 |- ( b = ( `' F " a ) -> ( F " b ) = ( F " ( `' F " a ) ) )
31	30	eqeq2d 2632	. . . 4 |- ( b = ( `' F " a ) -> ( a = ( F " b ) <-> a = ( F " ( `' F " a ) ) ) )
32	29, 31	syl5ibrcom 237	. . 3 |- ( ( ph /\ ( b e. ~P A /\ a e. ~P B ) ) -> ( b = ( `' F " a ) -> a = ( F " b ) ) )
33		f1of1 6136	. . . . . . 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 4168	. . . . . . 7 |- ( b e. ~P A -> b C_ A )
36	35	adantr 481	. . . . . 6 |- ( ( b e. ~P A /\ a e. ~P B ) -> b C_ A )
37		f1imacnv 6153	. . . . . 6 |- ( ( F : A -1-1-> B /\ b C_ A ) -> ( `' F " ( F " b ) ) = b )
38	34, 36, 37	syl2an 494	. . . . 5 |- ( ( ph /\ ( b e. ~P A /\ a e. ~P B ) ) -> ( `' F " ( F " b ) ) = b )
39	38	eqcomd 2628	. . . 4 |- ( ( ph /\ ( b e. ~P A /\ a e. ~P B ) ) -> b = ( `' F " ( F " b ) ) )
40		imaeq2 5462	. . . . 5 |- ( a = ( F " b ) -> ( `' F " a ) = ( `' F " ( F " b ) ) )
41	40	eqeq2d 2632	. . . 4 |- ( a = ( F " b ) -> ( b = ( `' F " a ) <-> b = ( `' F " ( F " b ) ) ) )
42	39, 41	syl5ibrcom 237	. . 3 |- ( ( ph /\ ( b e. ~P A /\ a e. ~P B ) ) -> ( a = ( F " b ) -> b = ( `' F " a ) ) )
43	32, 42	impbid 202	. 2 |- ( ( ph /\ ( b e. ~P A /\ a e. ~P B ) ) -> ( b = ( `' F " a ) <-> a = ( F " b ) ) )
44	1, 13, 24, 43	f1o2d 6887	1 |- ( ph -> ( b e. ~P A |-> ( F " b ) ) : ~P A -1-1-onto-> ~P B )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   C_ wss 3574   ~Pcpw 4158   |-> cmpt 4729   `'ccnv 5113   dom cdm 5114   ran crn 5115   "cima 5117   -1-1->wf1 5885   -onto->wfo 5886   -1-1-onto->wf1o 5887

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895

This theorem is referenced by:  f1opw  6889