Theorem f1opw2 5648

Description: A one-to-one mapping induces a one-to-one mapping on power sets. This version of f1opw 5649 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. ~PA |-> ( F " b)) : ~PA -1-1-onto-> ~PB)

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 2037	. 2 |- ( b e. ~PA |-> ( F " b)) = ( b e. ~PA |-> ( F " b))
2		imassrn 4622	. . . . 5 |- ( F " b) C_ ran F
3		f1opw2.1	. . . . . . 7 |- (ph -> F :A -1-1-onto->B)
4		f1ofo 5076	. . . . . . 7 |- ( F :A -1-1-onto->B -> F :A -onto->B)
5	3, 4	syl 14	. . . . . 6 |- (ph -> F :A -onto->B)
6		forn 5052	. . . . . 6 |- ( F :A -onto->B -> ran F = B)
7	5, 6	syl 14	. . . . 5 |- (ph -> ran F = B)
8	2, 7	syl5sseq 2987	. . . 4 |- (ph -> ( F " b) C_ B)
9		f1opw2.3	. . . . 5 |- (ph -> ( F " b) e. _V)
10		elpwg 3359	. . . . 5 |- (( F " b) e. _V -> (( F " b) e. ~PB <-> ( F " b) C_ B))
11	9, 10	syl 14	. . . 4 |- (ph -> (( F " b) e. ~PB <-> ( F " b) C_ B))
12	8, 11	mpbird 156	. . 3 |- (ph -> ( F " b) e. ~PB)
13	12	adantr 261	. 2 |- ((ph /\ b e. ~PA) -> ( F " b) e. ~PB)
14		imassrn 4622	. . . . 5 |- ( `' F " a) C_ ran `' F
15		dfdm4 4470	. . . . . 6 |- dom F = ran `' F
16		f1odm 5073	. . . . . . 7 |- ( F :A -1-1-onto->B -> dom F = A)
17	3, 16	syl 14	. . . . . 6 |- (ph -> dom F = A)
18	15, 17	syl5eqr 2083	. . . . 5 |- (ph -> ran `' F = A)
19	14, 18	syl5sseq 2987	. . . 4 |- (ph -> ( `' F " a) C_ A)
20		f1opw2.2	. . . . 5 |- (ph -> ( `' F " a) e. _V)
21		elpwg 3359	. . . . 5 |- (( `' F " a) e. _V -> (( `' F " a) e. ~PA <-> ( `' F " a) C_ A))
22	20, 21	syl 14	. . . 4 |- (ph -> (( `' F " a) e. ~PA <-> ( `' F " a) C_ A))
23	19, 22	mpbird 156	. . 3 |- (ph -> ( `' F " a) e. ~PA)
24	23	adantr 261	. 2 |- ((ph /\ a e. ~PB) -> ( `' F " a) e. ~PA)
25		elpwi 3360	. . . . . . 7 |- ( a e. ~PB -> a C_ B)
26	25	adantl 262	. . . . . 6 |- (( b e. ~PA /\ a e. ~PB) -> a C_ B)
27		foimacnv 5087	. . . . . 6 |- (( F :A -onto->B /\ a C_ B) -> ( F "( `' F " a)) = a)
28	5, 26, 27	syl2an 273	. . . . 5 |- ((ph /\ ( b e. ~PA /\ a e. ~PB)) -> ( F "( `' F " a)) = a)
29	28	eqcomd 2042	. . . 4 |- ((ph /\ ( b e. ~PA /\ a e. ~PB)) -> a = ( F "( `' F " a)))
30		imaeq2 4607	. . . . 5 |- ( b = ( `' F " a) -> ( F " b) = ( F "( `' F " a)))
31	30	eqeq2d 2048	. . . 4 |- ( b = ( `' F " a) -> ( a = ( F " b) <-> a = ( F "( `' F " a))))
32	29, 31	syl5ibrcom 146	. . 3 |- ((ph /\ ( b e. ~PA /\ a e. ~PB)) -> ( b = ( `' F " a) -> a = ( F " b)))
33		f1of1 5068	. . . . . . 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 3360	. . . . . . 7 |- ( b e. ~PA -> b C_ A)
36	35	adantr 261	. . . . . 6 |- (( b e. ~PA /\ a e. ~PB) -> b C_ A)
37		f1imacnv 5086	. . . . . 6 |- (( F :A -1-1->B /\ b C_ A) -> ( `' F "( F " b)) = b)
38	34, 36, 37	syl2an 273	. . . . 5 |- ((ph /\ ( b e. ~PA /\ a e. ~PB)) -> ( `' F "( F " b)) = b)
39	38	eqcomd 2042	. . . 4 |- ((ph /\ ( b e. ~PA /\ a e. ~PB)) -> b = ( `' F "( F " b)))
40		imaeq2 4607	. . . . 5 |- ( a = ( F " b) -> ( `' F " a) = ( `' F "( F " b)))
41	40	eqeq2d 2048	. . . 4 |- ( a = ( F " b) -> ( b = ( `' F " a) <-> b = ( `' F "( F " b))))
42	39, 41	syl5ibrcom 146	. . 3 |- ((ph /\ ( b e. ~PA /\ a e. ~PB)) -> ( a = ( F " b) -> b = ( `' F " a)))
43	32, 42	impbid 120	. 2 |- ((ph /\ ( b e. ~PA /\ a e. ~PB)) -> ( b = ( `' F " a) <-> a = ( F " b)))
44	1, 13, 24, 43	f1o2d 5647	1 |- (ph -> ( b e. ~PA |-> ( F " b)) : ~PA -1-1-onto-> ~PB)

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   = wceq 1242   e. wcel 1390   _Vcvv 2551   C_ wss 2911   ~Pcpw 3351   |-> cmpt 3809   `'ccnv 4287   dom cdm 4288   ran crn 4289   "cima 4291   -1-1->wf1 4842   -onto->wfo 4843   -1-1-onto->wf1o 4844

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852

This theorem is referenced by:  f1opw  5649