Theorem f1opw2 6071

Description: A one-to-one mapping induces a one-to-one mapping on power sets. This version of f1opw 6072 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 2177	. 2 |- ( b e. ~P A |-> ( F " b ) ) = ( b e. ~P A |-> ( F " b ) )
2		imassrn 4977	. . . . 5 |- ( F " b ) C_ ran F
3		f1opw2.1	. . . . . . 7 |- ( ph -> F : A -1-1-onto-> B )
4		f1ofo 5464	. . . . . . 7 |- ( F : A -1-1-onto-> B -> F : A -onto-> B )
5	3, 4	syl 14	. . . . . 6 |- ( ph -> F : A -onto-> B )
6		forn 5437	. . . . . 6 |- ( F : A -onto-> B -> ran F = B )
7	5, 6	syl 14	. . . . 5 |- ( ph -> ran F = B )
8	2, 7	sseqtrid 3205	. . . 4 |- ( ph -> ( F " b ) C_ B )
9		f1opw2.3	. . . . 5 |- ( ph -> ( F " b ) e. _V )
10		elpwg 3582	. . . . 5 |- ( ( F " b ) e. _V -> ( ( F " b ) e. ~P B <-> ( F " b ) C_ B ) )
11	9, 10	syl 14	. . . 4 |- ( ph -> ( ( F " b ) e. ~P B <-> ( F " b ) C_ B ) )
12	8, 11	mpbird 167	. . 3 |- ( ph -> ( F " b ) e. ~P B )
13	12	adantr 276	. 2 |- ( ( ph /\ b e. ~P A ) -> ( F " b ) e. ~P B )
14		imassrn 4977	. . . . 5 |- ( `' F " a ) C_ ran `' F
15		dfdm4 4815	. . . . . 6 |- dom F = ran `' F
16		f1odm 5461	. . . . . . 7 |- ( F : A -1-1-onto-> B -> dom F = A )
17	3, 16	syl 14	. . . . . 6 |- ( ph -> dom F = A )
18	15, 17	eqtr3id 2224	. . . . 5 |- ( ph -> ran `' F = A )
19	14, 18	sseqtrid 3205	. . . 4 |- ( ph -> ( `' F " a ) C_ A )
20		f1opw2.2	. . . . 5 |- ( ph -> ( `' F " a ) e. _V )
21		elpwg 3582	. . . . 5 |- ( ( `' F " a ) e. _V -> ( ( `' F " a ) e. ~P A <-> ( `' F " a ) C_ A ) )
22	20, 21	syl 14	. . . 4 |- ( ph -> ( ( `' F " a ) e. ~P A <-> ( `' F " a ) C_ A ) )
23	19, 22	mpbird 167	. . 3 |- ( ph -> ( `' F " a ) e. ~P A )
24	23	adantr 276	. 2 |- ( ( ph /\ a e. ~P B ) -> ( `' F " a ) e. ~P A )
25		elpwi 3583	. . . . . . 7 |- ( a e. ~P B -> a C_ B )
26	25	adantl 277	. . . . . 6 |- ( ( b e. ~P A /\ a e. ~P B ) -> a C_ B )
27		foimacnv 5475	. . . . . 6 |- ( ( F : A -onto-> B /\ a C_ B ) -> ( F " ( `' F " a ) ) = a )
28	5, 26, 27	syl2an 289	. . . . 5 |- ( ( ph /\ ( b e. ~P A /\ a e. ~P B ) ) -> ( F " ( `' F " a ) ) = a )
29	28	eqcomd 2183	. . . 4 |- ( ( ph /\ ( b e. ~P A /\ a e. ~P B ) ) -> a = ( F " ( `' F " a ) ) )
30		imaeq2 4962	. . . . 5 |- ( b = ( `' F " a ) -> ( F " b ) = ( F " ( `' F " a ) ) )
31	30	eqeq2d 2189	. . . 4 |- ( b = ( `' F " a ) -> ( a = ( F " b ) <-> a = ( F " ( `' F " a ) ) ) )
32	29, 31	syl5ibrcom 157	. . 3 |- ( ( ph /\ ( b e. ~P A /\ a e. ~P B ) ) -> ( b = ( `' F " a ) -> a = ( F " b ) ) )
33		f1of1 5456	. . . . . . 7 |- ( F : A -1-1-onto-> B -> F : A -1-1-> B )
34	3, 33	syl 14	. . . . . 6 |- ( ph -> F : A -1-1-> B )
35		elpwi 3583	. . . . . . 7 |- ( b e. ~P A -> b C_ A )
36	35	adantr 276	. . . . . 6 |- ( ( b e. ~P A /\ a e. ~P B ) -> b C_ A )
37		f1imacnv 5474	. . . . . 6 |- ( ( F : A -1-1-> B /\ b C_ A ) -> ( `' F " ( F " b ) ) = b )
38	34, 36, 37	syl2an 289	. . . . 5 |- ( ( ph /\ ( b e. ~P A /\ a e. ~P B ) ) -> ( `' F " ( F " b ) ) = b )
39	38	eqcomd 2183	. . . 4 |- ( ( ph /\ ( b e. ~P A /\ a e. ~P B ) ) -> b = ( `' F " ( F " b ) ) )
40		imaeq2 4962	. . . . 5 |- ( a = ( F " b ) -> ( `' F " a ) = ( `' F " ( F " b ) ) )
41	40	eqeq2d 2189	. . . 4 |- ( a = ( F " b ) -> ( b = ( `' F " a ) <-> b = ( `' F " ( F " b ) ) ) )
42	39, 41	syl5ibrcom 157	. . 3 |- ( ( ph /\ ( b e. ~P A /\ a e. ~P B ) ) -> ( a = ( F " b ) -> b = ( `' F " a ) ) )
43	32, 42	impbid 129	. 2 |- ( ( ph /\ ( b e. ~P A /\ a e. ~P B ) ) -> ( b = ( `' F " a ) <-> a = ( F " b ) ) )
44	1, 13, 24, 43	f1o2d 6070	1 |- ( ph -> ( b e. ~P A |-> ( F " b ) ) : ~P A -1-1-onto-> ~P B )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1353   e. wcel 2148   _Vcvv 2737   C_ wss 3129   ~Pcpw 3574   |-> cmpt 4061   `'ccnv 4622   dom cdm 4623   ran crn 4624   "cima 4626   -1-1->wf1 5209   -onto->wfo 5210   -1-1-onto->wf1o 5211

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-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4206

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4290  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219

This theorem is referenced by:  f1opw  6072