Theorem isf34lem5 8805

Description: Lemma for isfin3-4 8809. (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 5178	. . . . . . 7 |- ( F " X ) C_ ran F
2		compss.a	. . . . . . . . . 10 |- F = ( x e. ~P A |-> ( A \ x ) )
3	2	isf34lem2 8800	. . . . . . . . 9 |- ( A e. V -> F : ~P A --> ~P A )
4	3	adantr 467	. . . . . . . 8 |- ( ( A e. V /\ ( X C_ ~P A /\ X =/= (/) ) ) -> F : ~P A --> ~P A )
5		frn 5733	. . . . . . . 8 |- ( F : ~P A --> ~P A -> ran F C_ ~P A )
6	4, 5	syl 17	. . . . . . 7 |- ( ( A e. V /\ ( X C_ ~P A /\ X =/= (/) ) ) -> ran F C_ ~P A )
7	1, 6	syl5ss 3442	. . . . . 6 |- ( ( A e. V /\ ( X C_ ~P A /\ X =/= (/) ) ) -> ( F " X ) C_ ~P A )
8		simprl 763	. . . . . . . . . 10 |- ( ( A e. V /\ ( X C_ ~P A /\ X =/= (/) ) ) -> X C_ ~P A )
9		fdm 5731	. . . . . . . . . . 11 |- ( F : ~P A --> ~P A -> dom F = ~P A )
10	4, 9	syl 17	. . . . . . . . . 10 |- ( ( A e. V /\ ( X C_ ~P A /\ X =/= (/) ) ) -> dom F = ~P A )
11	8, 10	sseqtr4d 3468	. . . . . . . . 9 |- ( ( A e. V /\ ( X C_ ~P A /\ X =/= (/) ) ) -> X C_ dom F )
12		dfss1 3636	. . . . . . . . 9 |- ( X C_ dom F <-> ( dom F i^i X ) = X )
13	11, 12	sylib 200	. . . . . . . 8 |- ( ( A e. V /\ ( X C_ ~P A /\ X =/= (/) ) ) -> ( dom F i^i X ) = X )
14		simprr 765	. . . . . . . 8 |- ( ( A e. V /\ ( X C_ ~P A /\ X =/= (/) ) ) -> X =/= (/) )
15	13, 14	eqnetrd 2690	. . . . . . 7 |- ( ( A e. V /\ ( X C_ ~P A /\ X =/= (/) ) ) -> ( dom F i^i X ) =/= (/) )
16		imadisj 5186	. . . . . . . 8 |- ( ( F " X ) = (/) <-> ( dom F i^i X ) = (/) )
17	16	necon3bii 2675	. . . . . . 7 |- ( ( F " X ) =/= (/) <-> ( dom F i^i X ) =/= (/) )
18	15, 17	sylibr 216	. . . . . 6 |- ( ( A e. V /\ ( X C_ ~P A /\ X =/= (/) ) ) -> ( F " X ) =/= (/) )
19	7, 18	jca 535	. . . . 5 |- ( ( A e. V /\ ( X C_ ~P A /\ X =/= (/) ) ) -> ( ( F " X ) C_ ~P A /\ ( F " X ) =/= (/) ) )
20	2	isf34lem4 8804	. . . . 5 |- ( ( A e. V /\ ( ( F " X ) C_ ~P A /\ ( F " X ) =/= (/) ) ) -> ( F ` U. ( F " X ) ) = |^| ( F " ( F " X ) ) )
21	19, 20	syldan 473	. . . 4 |- ( ( A e. V /\ ( X C_ ~P A /\ X =/= (/) ) ) -> ( F ` U. ( F " X ) ) = |^| ( F " ( F " X ) ) )
22	2	isf34lem3 8802	. . . . . 6 |- ( ( A e. V /\ X C_ ~P A ) -> ( F " ( F " X ) ) = X )
23	22	adantrr 722	. . . . 5 |- ( ( A e. V /\ ( X C_ ~P A /\ X =/= (/) ) ) -> ( F " ( F " X ) ) = X )
24	23	inteqd 4238	. . . 4 |- ( ( A e. V /\ ( X C_ ~P A /\ X =/= (/) ) ) -> |^| ( F " ( F " X ) ) = |^| X )
25	21, 24	eqtrd 2484	. . 3 |- ( ( A e. V /\ ( X C_ ~P A /\ X =/= (/) ) ) -> ( F ` U. ( F " X ) ) = |^| X )
26	25	fveq2d 5867	. 2 |- ( ( A e. V /\ ( X C_ ~P A /\ X =/= (/) ) ) -> ( F ` ( F ` U. ( F " X ) ) ) = ( F ` |^| X ) )
27	2	compsscnv 8798	. . . 4 |- `' F = F
28	27	fveq1i 5864	. . 3 |- ( `' F ` ( F ` U. ( F " X ) ) ) = ( F ` ( F ` U. ( F " X ) ) )
29	2	compssiso 8801	. . . . . 6 |- ( A e. V -> F Isom [ C.] , `' [ C.] ( ~P A , ~P A ) )
30		isof1o 6214	. . . . . 6 |- ( F Isom [ C.] , `' [ C.] ( ~P A , ~P A ) -> F : ~P A -1-1-onto-> ~P A )
31	29, 30	syl 17	. . . . 5 |- ( A e. V -> F : ~P A -1-1-onto-> ~P A )
32	31	adantr 467	. . . 4 |- ( ( A e. V /\ ( X C_ ~P A /\ X =/= (/) ) ) -> F : ~P A -1-1-onto-> ~P A )
33		sspwuni 4366	. . . . . 6 |- ( ( F " X ) C_ ~P A <-> U. ( F " X ) C_ A )
34	7, 33	sylib 200	. . . . 5 |- ( ( A e. V /\ ( X C_ ~P A /\ X =/= (/) ) ) -> U. ( F " X ) C_ A )
35		elpw2g 4565	. . . . . 6 |- ( A e. V -> ( U. ( F " X ) e. ~P A <-> U. ( F " X ) C_ A ) )
36	35	adantr 467	. . . . 5 |- ( ( A e. V /\ ( X C_ ~P A /\ X =/= (/) ) ) -> ( U. ( F " X ) e. ~P A <-> U. ( F " X ) C_ A ) )
37	34, 36	mpbird 236	. . . 4 |- ( ( A e. V /\ ( X C_ ~P A /\ X =/= (/) ) ) -> U. ( F " X ) e. ~P A )
38		f1ocnvfv1 6173	. . . 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 666	. . 3 |- ( ( A e. V /\ ( X C_ ~P A /\ X =/= (/) ) ) -> ( `' F ` ( F ` U. ( F " X ) ) ) = U. ( F " X ) )
40	28, 39	syl5eqr 2498	. 2 |- ( ( A e. V /\ ( X C_ ~P A /\ X =/= (/) ) ) -> ( F ` ( F ` U. ( F " X ) ) ) = U. ( F " X ) )
41	26, 40	eqtr3d 2486	1 |- ( ( A e. V /\ ( X C_ ~P A /\ X =/= (/) ) ) -> ( F ` |^| X ) = U. ( F " X ) )

Syntax hints:   -> wi 4   <-> wb 188   /\ wa 371   = wceq 1443   e. wcel 1886   =/= wne 2621   \ cdif 3400   i^i cin 3402   C_ wss 3403   (/)c0 3730   ~Pcpw 3950   U.cuni 4197   |^|cint 4233   |-> cmpt 4460   `'ccnv 4832   dom cdm 4833   ran crn 4834   "cima 4836   -->wf 5577   -1-1-onto->wf1o 5580   ` cfv 5581   Isom wiso 5582   [ C.] crpss 6567

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-ral 2741  df-rex 2742  df-rab 2745  df-v 3046  df-sbc 3267  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-pss 3419  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-op 3974  df-uni 4198  df-int 4234  df-br 4402  df-opab 4461  df-mpt 4462  df-id 4748  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-isom 5590  df-rpss 6568

This theorem is referenced by:  isf34lem7  8806