Theorem f1opw2 6510

Description: A one-to-one mapping induces a one-to-one mapping on power sets. This version of f1opw 6511 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 2467	. 2 |- ( b e. ~P A |-> ( F " b ) ) = ( b e. ~P A |-> ( F " b ) )
2		imassrn 5346	. . . . 5 |- ( F " b ) C_ ran F
3		f1opw2.1	. . . . . . 7 |- ( ph -> F : A -1-1-onto-> B )
4		f1ofo 5821	. . . . . . 7 |- ( F : A -1-1-onto-> B -> F : A -onto-> B )
5	3, 4	syl 16	. . . . . 6 |- ( ph -> F : A -onto-> B )
6		forn 5796	. . . . . 6 |- ( F : A -onto-> B -> ran F = B )
7	5, 6	syl 16	. . . . 5 |- ( ph -> ran F = B )
8	2, 7	syl5sseq 3552	. . . 4 |- ( ph -> ( F " b ) C_ B )
9		f1opw2.3	. . . . 5 |- ( ph -> ( F " b ) e. _V )
10		elpwg 4018	. . . . 5 |- ( ( F " b ) e. _V -> ( ( F " b ) e. ~P B <-> ( F " b ) C_ B ) )
11	9, 10	syl 16	. . . 4 |- ( ph -> ( ( F " b ) e. ~P B <-> ( F " b ) C_ B ) )
12	8, 11	mpbird 232	. . 3 |- ( ph -> ( F " b ) e. ~P B )
13	12	adantr 465	. 2 |- ( ( ph /\ b e. ~P A ) -> ( F " b ) e. ~P B )
14		imassrn 5346	. . . . 5 |- ( `' F " a ) C_ ran `' F
15		dfdm4 5193	. . . . . 6 |- dom F = ran `' F
16		f1odm 5818	. . . . . . 7 |- ( F : A -1-1-onto-> B -> dom F = A )
17	3, 16	syl 16	. . . . . 6 |- ( ph -> dom F = A )
18	15, 17	syl5eqr 2522	. . . . 5 |- ( ph -> ran `' F = A )
19	14, 18	syl5sseq 3552	. . . 4 |- ( ph -> ( `' F " a ) C_ A )
20		f1opw2.2	. . . . 5 |- ( ph -> ( `' F " a ) e. _V )
21		elpwg 4018	. . . . 5 |- ( ( `' F " a ) e. _V -> ( ( `' F " a ) e. ~P A <-> ( `' F " a ) C_ A ) )
22	20, 21	syl 16	. . . 4 |- ( ph -> ( ( `' F " a ) e. ~P A <-> ( `' F " a ) C_ A ) )
23	19, 22	mpbird 232	. . 3 |- ( ph -> ( `' F " a ) e. ~P A )
24	23	adantr 465	. 2 |- ( ( ph /\ a e. ~P B ) -> ( `' F " a ) e. ~P A )
25		elpwi 4019	. . . . . . 7 |- ( a e. ~P B -> a C_ B )
26	25	adantl 466	. . . . . 6 |- ( ( b e. ~P A /\ a e. ~P B ) -> a C_ B )
27		foimacnv 5831	. . . . . 6 |- ( ( F : A -onto-> B /\ a C_ B ) -> ( F " ( `' F " a ) ) = a )
28	5, 26, 27	syl2an 477	. . . . 5 |- ( ( ph /\ ( b e. ~P A /\ a e. ~P B ) ) -> ( F " ( `' F " a ) ) = a )
29	28	eqcomd 2475	. . . 4 |- ( ( ph /\ ( b e. ~P A /\ a e. ~P B ) ) -> a = ( F " ( `' F " a ) ) )
30		imaeq2 5331	. . . . 5 |- ( b = ( `' F " a ) -> ( F " b ) = ( F " ( `' F " a ) ) )
31	30	eqeq2d 2481	. . . 4 |- ( b = ( `' F " a ) -> ( a = ( F " b ) <-> a = ( F " ( `' F " a ) ) ) )
32	29, 31	syl5ibrcom 222	. . 3 |- ( ( ph /\ ( b e. ~P A /\ a e. ~P B ) ) -> ( b = ( `' F " a ) -> a = ( F " b ) ) )
33		f1of1 5813	. . . . . . 7 |- ( F : A -1-1-onto-> B -> F : A -1-1-> B )
34	3, 33	syl 16	. . . . . 6 |- ( ph -> F : A -1-1-> B )
35		elpwi 4019	. . . . . . 7 |- ( b e. ~P A -> b C_ A )
36	35	adantr 465	. . . . . 6 |- ( ( b e. ~P A /\ a e. ~P B ) -> b C_ A )
37		f1imacnv 5830	. . . . . 6 |- ( ( F : A -1-1-> B /\ b C_ A ) -> ( `' F " ( F " b ) ) = b )
38	34, 36, 37	syl2an 477	. . . . 5 |- ( ( ph /\ ( b e. ~P A /\ a e. ~P B ) ) -> ( `' F " ( F " b ) ) = b )
39	38	eqcomd 2475	. . . 4 |- ( ( ph /\ ( b e. ~P A /\ a e. ~P B ) ) -> b = ( `' F " ( F " b ) ) )
40		imaeq2 5331	. . . . 5 |- ( a = ( F " b ) -> ( `' F " a ) = ( `' F " ( F " b ) ) )
41	40	eqeq2d 2481	. . . 4 |- ( a = ( F " b ) -> ( b = ( `' F " a ) <-> b = ( `' F " ( F " b ) ) ) )
42	39, 41	syl5ibrcom 222	. . 3 |- ( ( ph /\ ( b e. ~P A /\ a e. ~P B ) ) -> ( a = ( F " b ) -> b = ( `' F " a ) ) )
43	32, 42	impbid 191	. 2 |- ( ( ph /\ ( b e. ~P A /\ a e. ~P B ) ) -> ( b = ( `' F " a ) <-> a = ( F " b ) ) )
44	1, 13, 24, 43	f1o2d 6509	1 |- ( ph -> ( b e. ~P A |-> ( F " b ) ) : ~P A -1-1-onto-> ~P B )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1379   e. wcel 1767   _Vcvv 3113   C_ wss 3476   ~Pcpw 4010   |-> cmpt 4505   `'ccnv 4998   dom cdm 4999   ran crn 5000   "cima 5002   -1-1->wf1 5583   -onto->wfo 5584   -1-1-onto->wf1o 5585

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593

This theorem is referenced by:  f1opw  6511