Theorem isf34lem5 8759

Description: Lemma for isfin3-4 8763. (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 5348	. . . . . . 7 |- ( F " X ) C_ ran F
2		compss.a	. . . . . . . . . 10 |- F = ( x e. ~P A |-> ( A \ x ) )
3	2	isf34lem2 8754	. . . . . . . . 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 5737	. . . . . . . 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 3515	. . . . . 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 5735	. . . . . . . . . . 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 3541	. . . . . . . . 9 |- ( ( A e. V /\ ( X C_ ~P A /\ X =/= (/) ) ) -> X C_ dom F )
12		dfss1 3703	. . . . . . . . 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 2760	. . . . . . 7 |- ( ( A e. V /\ ( X C_ ~P A /\ X =/= (/) ) ) -> ( dom F i^i X ) =/= (/) )
16		imadisj 5356	. . . . . . . 8 |- ( ( F " X ) = (/) <-> ( dom F i^i X ) = (/) )
17	16	necon3bii 2735	. . . . . . 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 8758	. . . . 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 8756	. . . . . 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 4287	. . . 4 |- ( ( A e. V /\ ( X C_ ~P A /\ X =/= (/) ) ) -> |^| ( F " ( F " X ) ) = |^| X )
25	21, 24	eqtrd 2508	. . 3 |- ( ( A e. V /\ ( X C_ ~P A /\ X =/= (/) ) ) -> ( F ` U. ( F " X ) ) = |^| X )
26	25	fveq2d 5870	. 2 |- ( ( A e. V /\ ( X C_ ~P A /\ X =/= (/) ) ) -> ( F ` ( F ` U. ( F " X ) ) ) = ( F ` |^| X ) )
27	2	compsscnv 8752	. . . 4 |- `' F = F
28	27	fveq1i 5867	. . 3 |- ( `' F ` ( F ` U. ( F " X ) ) ) = ( F ` ( F ` U. ( F " X ) ) )
29	2	compssiso 8755	. . . . . 6 |- ( A e. V -> F Isom [ C.] , `' [ C.] ( ~P A , ~P A ) )
30		isof1o 6210	. . . . . 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 4411	. . . . . 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 4610	. . . . . 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 6171	. . . 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 2522	. 2 |- ( ( A e. V /\ ( X C_ ~P A /\ X =/= (/) ) ) -> ( F ` ( F ` U. ( F " X ) ) ) = U. ( F " X ) )
41	26, 40	eqtr3d 2510	1 |- ( ( A e. V /\ ( X C_ ~P A /\ X =/= (/) ) ) -> ( F ` |^| X ) = U. ( F " X ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1379   e. wcel 1767   =/= wne 2662   \ cdif 3473   i^i cin 3475   C_ wss 3476   (/)c0 3785   ~Pcpw 4010   U.cuni 4245   |^|cint 4282   |-> cmpt 4505   `'ccnv 4998   dom cdm 4999   ran crn 5000   "cima 5002   -->wf 5584   -1-1-onto->wf1o 5587   ` cfv 5588   Isom wiso 5589   [ C.] crpss 6564

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-8 1769  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-pow 4625  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-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-int 4283  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-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-isom 5597  df-rpss 6565

This theorem is referenced by:  isf34lem7  8760