Theorem isf34lem5 8559

Description: Lemma for isfin3-4 8563. (Contributed by Stefan O'Rear, 7-Nov-2014.) (Revised by Mario Carneiro, 17-May-2015.)

Hypothesis

Ref	Expression
compss.a	|- F = ( x e. ~P A |-> ( A \ x ) )

Assertion

Ref	Expression
isf34lem5	|- ( ( A e. V /\ ( X C_ ~P A /\ X =/= (/) ) ) -> ( F ` |^| X ) = U. ( F " X ) )

Distinct variable groups:   x, A   x, V

Allowed substitution hints:   F( x)   X( x)

Proof of Theorem isf34lem5

Step	Hyp	Ref	Expression
1		imassrn 5192	. . . . . . 7 |- ( F " X ) C_ ran F
2		compss.a	. . . . . . . . . 10 |- F = ( x e. ~P A |-> ( A \ x ) )
3	2	isf34lem2 8554	. . . . . . . . 9 |- ( A e. V -> F : ~P A --> ~P A )
4	3	adantr 465	. . . . . . . 8 |- ( ( A e. V /\ ( X C_ ~P A /\ X =/= (/) ) ) -> F : ~P A --> ~P A )
5		frn 5577	. . . . . . . 8 |- ( F : ~P A --> ~P A -> ran F C_ ~P A )
6	4, 5	syl 16	. . . . . . 7 |- ( ( A e. V /\ ( X C_ ~P A /\ X =/= (/) ) ) -> ran F C_ ~P A )
7	1, 6	syl5ss 3379	. . . . . 6 |- ( ( A e. V /\ ( X C_ ~P A /\ X =/= (/) ) ) -> ( F " X ) C_ ~P A )
8		simprl 755	. . . . . . . . . 10 |- ( ( A e. V /\ ( X C_ ~P A /\ X =/= (/) ) ) -> X C_ ~P A )
9		fdm 5575	. . . . . . . . . . 11 |- ( F : ~P A --> ~P A -> dom F = ~P A )
10	4, 9	syl 16	. . . . . . . . . 10 |- ( ( A e. V /\ ( X C_ ~P A /\ X =/= (/) ) ) -> dom F = ~P A )
11	8, 10	sseqtr4d 3405	. . . . . . . . 9 |- ( ( A e. V /\ ( X C_ ~P A /\ X =/= (/) ) ) -> X C_ dom F )
12		dfss1 3567	. . . . . . . . 9 |- ( X C_ dom F <-> ( dom F i^i X ) = X )
13	11, 12	sylib 196	. . . . . . . 8 |- ( ( A e. V /\ ( X C_ ~P A /\ X =/= (/) ) ) -> ( dom F i^i X ) = X )
14		simprr 756	. . . . . . . 8 |- ( ( A e. V /\ ( X C_ ~P A /\ X =/= (/) ) ) -> X =/= (/) )
15	13, 14	eqnetrd 2638	. . . . . . 7 |- ( ( A e. V /\ ( X C_ ~P A /\ X =/= (/) ) ) -> ( dom F i^i X ) =/= (/) )
16		imadisj 5200	. . . . . . . 8 |- ( ( F " X ) = (/) <-> ( dom F i^i X ) = (/) )
17	16	necon3bii 2652	. . . . . . 7 |- ( ( F " X ) =/= (/) <-> ( dom F i^i X ) =/= (/) )
18	15, 17	sylibr 212	. . . . . 6 |- ( ( A e. V /\ ( X C_ ~P A /\ X =/= (/) ) ) -> ( F " X ) =/= (/) )
19	7, 18	jca 532	. . . . 5 |- ( ( A e. V /\ ( X C_ ~P A /\ X =/= (/) ) ) -> ( ( F " X ) C_ ~P A /\ ( F " X ) =/= (/) ) )
20	2	isf34lem4 8558	. . . . 5 |- ( ( A e. V /\ ( ( F " X ) C_ ~P A /\ ( F " X ) =/= (/) ) ) -> ( F ` U. ( F " X ) ) = |^| ( F " ( F " X ) ) )
21	19, 20	syldan 470	. . . 4 |- ( ( A e. V /\ ( X C_ ~P A /\ X =/= (/) ) ) -> ( F ` U. ( F " X ) ) = |^| ( F " ( F " X ) ) )
22	2	isf34lem3 8556	. . . . . 6 |- ( ( A e. V /\ X C_ ~P A ) -> ( F " ( F " X ) ) = X )
23	22	adantrr 716	. . . . 5 |- ( ( A e. V /\ ( X C_ ~P A /\ X =/= (/) ) ) -> ( F " ( F " X ) ) = X )
24	23	inteqd 4145	. . . 4 |- ( ( A e. V /\ ( X C_ ~P A /\ X =/= (/) ) ) -> |^| ( F " ( F " X ) ) = |^| X )
25	21, 24	eqtrd 2475	. . 3 |- ( ( A e. V /\ ( X C_ ~P A /\ X =/= (/) ) ) -> ( F ` U. ( F " X ) ) = |^| X )
26	25	fveq2d 5707	. 2 |- ( ( A e. V /\ ( X C_ ~P A /\ X =/= (/) ) ) -> ( F ` ( F ` U. ( F " X ) ) ) = ( F ` |^| X ) )
27	2	compsscnv 8552	. . . 4 |- `' F = F
28	27	fveq1i 5704	. . 3 |- ( `' F ` ( F ` U. ( F " X ) ) ) = ( F ` ( F ` U. ( F " X ) ) )
29	2	compssiso 8555	. . . . . 6 |- ( A e. V -> F Isom [ C.] , `' [ C.] ( ~P A , ~P A ) )
30		isof1o 6028	. . . . . 6 |- ( F Isom [ C.] , `' [ C.] ( ~P A , ~P A ) -> F : ~P A -1-1-onto-> ~P A )
31	29, 30	syl 16	. . . . 5 |- ( A e. V -> F : ~P A -1-1-onto-> ~P A )
32	31	adantr 465	. . . 4 |- ( ( A e. V /\ ( X C_ ~P A /\ X =/= (/) ) ) -> F : ~P A -1-1-onto-> ~P A )
33		sspwuni 4268	. . . . . 6 |- ( ( F " X ) C_ ~P A <-> U. ( F " X ) C_ A )
34	7, 33	sylib 196	. . . . 5 |- ( ( A e. V /\ ( X C_ ~P A /\ X =/= (/) ) ) -> U. ( F " X ) C_ A )
35		elpw2g 4467	. . . . . 6 |- ( A e. V -> ( U. ( F " X ) e. ~P A <-> U. ( F " X ) C_ A ) )
36	35	adantr 465	. . . . 5 |- ( ( A e. V /\ ( X C_ ~P A /\ X =/= (/) ) ) -> ( U. ( F " X ) e. ~P A <-> U. ( F " X ) C_ A ) )
37	34, 36	mpbird 232	. . . 4 |- ( ( A e. V /\ ( X C_ ~P A /\ X =/= (/) ) ) -> U. ( F " X ) e. ~P A )
38		f1ocnvfv1 5995	. . . 4 |- ( ( F : ~P A -1-1-onto-> ~P A /\ U. ( F " X ) e. ~P A ) -> ( `' F ` ( F ` U. ( F " X ) ) ) = U. ( F " X ) )
39	32, 37, 38	syl2anc 661	. . 3 |- ( ( A e. V /\ ( X C_ ~P A /\ X =/= (/) ) ) -> ( `' F ` ( F ` U. ( F " X ) ) ) = U. ( F " X ) )
40	28, 39	syl5eqr 2489	. 2 |- ( ( A e. V /\ ( X C_ ~P A /\ X =/= (/) ) ) -> ( F ` ( F ` U. ( F " X ) ) ) = U. ( F " X ) )
41	26, 40	eqtr3d 2477	1 |- ( ( A e. V /\ ( X C_ ~P A /\ X =/= (/) ) ) -> ( F ` |^| X ) = U. ( F " X ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1369   e. wcel 1756   =/= wne 2618   \ cdif 3337   i^i cin 3339   C_ wss 3340   (/)c0 3649   ~Pcpw 3872   U.cuni 4103   |^|cint 4140   e. cmpt 4362   `'ccnv 4851   dom cdm 4852   ran crn 4853   "cima 4855   -->wf 5426   -1-1-onto->wf1o 5429   ` cfv 5430   Isom wiso 5431   [ C.] crpss 6371

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-ral 2732  df-rex 2733  df-rab 2736  df-v 2986  df-sbc 3199  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-pss 3356  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-op 3896  df-uni 4104  df-int 4141  df-br 4305  df-opab 4363  df-mpt 4364  df-id 4648  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-isom 5439  df-rpss 6372

This theorem is referenced by:  isf34lem7  8560