Theorem isf34lem5 9200

Description: Lemma for isfin3-4 9204. (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 5477	. . . . . . 7 |- ( F " X ) C_ ran F
2		compss.a	. . . . . . . . . 10 |- F = ( x e. ~P A |-> ( A \ x ) )
3	2	isf34lem2 9195	. . . . . . . . 9 |- ( A e. V -> F : ~P A --> ~P A )
4	3	adantr 481	. . . . . . . 8 |- ( ( A e. V /\ ( X C_ ~P A /\ X =/= (/) ) ) -> F : ~P A --> ~P A )
5		frn 6053	. . . . . . . 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 3614	. . . . . 6 |- ( ( A e. V /\ ( X C_ ~P A /\ X =/= (/) ) ) -> ( F " X ) C_ ~P A )
8		simprl 794	. . . . . . . . . 10 |- ( ( A e. V /\ ( X C_ ~P A /\ X =/= (/) ) ) -> X C_ ~P A )
9		fdm 6051	. . . . . . . . . . 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 3642	. . . . . . . . 9 |- ( ( A e. V /\ ( X C_ ~P A /\ X =/= (/) ) ) -> X C_ dom F )
12		sseqin2 3817	. . . . . . . . 9 |- ( X C_ dom F <-> ( dom F i^i X ) = X )
13	11, 12	sylib 208	. . . . . . . 8 |- ( ( A e. V /\ ( X C_ ~P A /\ X =/= (/) ) ) -> ( dom F i^i X ) = X )
14		simprr 796	. . . . . . . 8 |- ( ( A e. V /\ ( X C_ ~P A /\ X =/= (/) ) ) -> X =/= (/) )
15	13, 14	eqnetrd 2861	. . . . . . 7 |- ( ( A e. V /\ ( X C_ ~P A /\ X =/= (/) ) ) -> ( dom F i^i X ) =/= (/) )
16		imadisj 5484	. . . . . . . 8 |- ( ( F " X ) = (/) <-> ( dom F i^i X ) = (/) )
17	16	necon3bii 2846	. . . . . . 7 |- ( ( F " X ) =/= (/) <-> ( dom F i^i X ) =/= (/) )
18	15, 17	sylibr 224	. . . . . 6 |- ( ( A e. V /\ ( X C_ ~P A /\ X =/= (/) ) ) -> ( F " X ) =/= (/) )
19	7, 18	jca 554	. . . . 5 |- ( ( A e. V /\ ( X C_ ~P A /\ X =/= (/) ) ) -> ( ( F " X ) C_ ~P A /\ ( F " X ) =/= (/) ) )
20	2	isf34lem4 9199	. . . . 5 |- ( ( A e. V /\ ( ( F " X ) C_ ~P A /\ ( F " X ) =/= (/) ) ) -> ( F ` U. ( F " X ) ) = |^| ( F " ( F " X ) ) )
21	19, 20	syldan 487	. . . 4 |- ( ( A e. V /\ ( X C_ ~P A /\ X =/= (/) ) ) -> ( F ` U. ( F " X ) ) = |^| ( F " ( F " X ) ) )
22	2	isf34lem3 9197	. . . . . 6 |- ( ( A e. V /\ X C_ ~P A ) -> ( F " ( F " X ) ) = X )
23	22	adantrr 753	. . . . 5 |- ( ( A e. V /\ ( X C_ ~P A /\ X =/= (/) ) ) -> ( F " ( F " X ) ) = X )
24	23	inteqd 4480	. . . 4 |- ( ( A e. V /\ ( X C_ ~P A /\ X =/= (/) ) ) -> |^| ( F " ( F " X ) ) = |^| X )
25	21, 24	eqtrd 2656	. . 3 |- ( ( A e. V /\ ( X C_ ~P A /\ X =/= (/) ) ) -> ( F ` U. ( F " X ) ) = |^| X )
26	25	fveq2d 6195	. 2 |- ( ( A e. V /\ ( X C_ ~P A /\ X =/= (/) ) ) -> ( F ` ( F ` U. ( F " X ) ) ) = ( F ` |^| X ) )
27	2	compsscnv 9193	. . . 4 |- `' F = F
28	27	fveq1i 6192	. . 3 |- ( `' F ` ( F ` U. ( F " X ) ) ) = ( F ` ( F ` U. ( F " X ) ) )
29	2	compssiso 9196	. . . . . 6 |- ( A e. V -> F Isom [ C.] , `' [ C.] ( ~P A , ~P A ) )
30		isof1o 6573	. . . . . 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 481	. . . 4 |- ( ( A e. V /\ ( X C_ ~P A /\ X =/= (/) ) ) -> F : ~P A -1-1-onto-> ~P A )
33		sspwuni 4611	. . . . . 6 |- ( ( F " X ) C_ ~P A <-> U. ( F " X ) C_ A )
34	7, 33	sylib 208	. . . . 5 |- ( ( A e. V /\ ( X C_ ~P A /\ X =/= (/) ) ) -> U. ( F " X ) C_ A )
35		elpw2g 4827	. . . . . 6 |- ( A e. V -> ( U. ( F " X ) e. ~P A <-> U. ( F " X ) C_ A ) )
36	35	adantr 481	. . . . 5 |- ( ( A e. V /\ ( X C_ ~P A /\ X =/= (/) ) ) -> ( U. ( F " X ) e. ~P A <-> U. ( F " X ) C_ A ) )
37	34, 36	mpbird 247	. . . 4 |- ( ( A e. V /\ ( X C_ ~P A /\ X =/= (/) ) ) -> U. ( F " X ) e. ~P A )
38		f1ocnvfv1 6532	. . . 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 693	. . 3 |- ( ( A e. V /\ ( X C_ ~P A /\ X =/= (/) ) ) -> ( `' F ` ( F ` U. ( F " X ) ) ) = U. ( F " X ) )
40	28, 39	syl5eqr 2670	. 2 |- ( ( A e. V /\ ( X C_ ~P A /\ X =/= (/) ) ) -> ( F ` ( F ` U. ( F " X ) ) ) = U. ( F " X ) )
41	26, 40	eqtr3d 2658	1 |- ( ( A e. V /\ ( X C_ ~P A /\ X =/= (/) ) ) -> ( F ` |^| X ) = U. ( F " X ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   \ cdif 3571   i^i cin 3573   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   U.cuni 4436   |^|cint 4475   |-> cmpt 4729   `'ccnv 5113   dom cdm 5114   ran crn 5115   "cima 5117   -->wf 5884   -1-1-onto->wf1o 5887   ` cfv 5888   Isom wiso 5889   [ C.] crpss 6936

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-int 4476  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-rpss 6937

This theorem is referenced by:  isf34lem7  9201