Theorem isf34lem5 8797

Description: Lemma for isfin3-4 8801. (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 5190	. . . . . . 7 |- ( F " X ) C_ ran F
2		compss.a	. . . . . . . . . 10 |- F = ( x e. ~P A |-> ( A \ x ) )
3	2	isf34lem2 8792	. . . . . . . . 9 |- ( A e. V -> F : ~P A --> ~P A )
4	3	adantr 466	. . . . . . . 8 |- ( ( A e. V /\ ( X C_ ~P A /\ X =/= (/) ) ) -> F : ~P A --> ~P A )
5		frn 5743	. . . . . . . 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 3472	. . . . . 6 |- ( ( A e. V /\ ( X C_ ~P A /\ X =/= (/) ) ) -> ( F " X ) C_ ~P A )
8		simprl 762	. . . . . . . . . 10 |- ( ( A e. V /\ ( X C_ ~P A /\ X =/= (/) ) ) -> X C_ ~P A )
9		fdm 5741	. . . . . . . . . . 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 3498	. . . . . . . . 9 |- ( ( A e. V /\ ( X C_ ~P A /\ X =/= (/) ) ) -> X C_ dom F )
12		dfss1 3664	. . . . . . . . 9 |- ( X C_ dom F <-> ( dom F i^i X ) = X )
13	11, 12	sylib 199	. . . . . . . 8 |- ( ( A e. V /\ ( X C_ ~P A /\ X =/= (/) ) ) -> ( dom F i^i X ) = X )
14		simprr 764	. . . . . . . 8 |- ( ( A e. V /\ ( X C_ ~P A /\ X =/= (/) ) ) -> X =/= (/) )
15	13, 14	eqnetrd 2715	. . . . . . 7 |- ( ( A e. V /\ ( X C_ ~P A /\ X =/= (/) ) ) -> ( dom F i^i X ) =/= (/) )
16		imadisj 5198	. . . . . . . 8 |- ( ( F " X ) = (/) <-> ( dom F i^i X ) = (/) )
17	16	necon3bii 2690	. . . . . . 7 |- ( ( F " X ) =/= (/) <-> ( dom F i^i X ) =/= (/) )
18	15, 17	sylibr 215	. . . . . 6 |- ( ( A e. V /\ ( X C_ ~P A /\ X =/= (/) ) ) -> ( F " X ) =/= (/) )
19	7, 18	jca 534	. . . . 5 |- ( ( A e. V /\ ( X C_ ~P A /\ X =/= (/) ) ) -> ( ( F " X ) C_ ~P A /\ ( F " X ) =/= (/) ) )
20	2	isf34lem4 8796	. . . . 5 |- ( ( A e. V /\ ( ( F " X ) C_ ~P A /\ ( F " X ) =/= (/) ) ) -> ( F ` U. ( F " X ) ) = |^| ( F " ( F " X ) ) )
21	19, 20	syldan 472	. . . 4 |- ( ( A e. V /\ ( X C_ ~P A /\ X =/= (/) ) ) -> ( F ` U. ( F " X ) ) = |^| ( F " ( F " X ) ) )
22	2	isf34lem3 8794	. . . . . 6 |- ( ( A e. V /\ X C_ ~P A ) -> ( F " ( F " X ) ) = X )
23	22	adantrr 721	. . . . 5 |- ( ( A e. V /\ ( X C_ ~P A /\ X =/= (/) ) ) -> ( F " ( F " X ) ) = X )
24	23	inteqd 4254	. . . 4 |- ( ( A e. V /\ ( X C_ ~P A /\ X =/= (/) ) ) -> |^| ( F " ( F " X ) ) = |^| X )
25	21, 24	eqtrd 2461	. . 3 |- ( ( A e. V /\ ( X C_ ~P A /\ X =/= (/) ) ) -> ( F ` U. ( F " X ) ) = |^| X )
26	25	fveq2d 5876	. 2 |- ( ( A e. V /\ ( X C_ ~P A /\ X =/= (/) ) ) -> ( F ` ( F ` U. ( F " X ) ) ) = ( F ` |^| X ) )
27	2	compsscnv 8790	. . . 4 |- `' F = F
28	27	fveq1i 5873	. . 3 |- ( `' F ` ( F ` U. ( F " X ) ) ) = ( F ` ( F ` U. ( F " X ) ) )
29	2	compssiso 8793	. . . . . 6 |- ( A e. V -> F Isom [ C.] , `' [ C.] ( ~P A , ~P A ) )
30		isof1o 6222	. . . . . 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 466	. . . 4 |- ( ( A e. V /\ ( X C_ ~P A /\ X =/= (/) ) ) -> F : ~P A -1-1-onto-> ~P A )
33		sspwuni 4382	. . . . . 6 |- ( ( F " X ) C_ ~P A <-> U. ( F " X ) C_ A )
34	7, 33	sylib 199	. . . . 5 |- ( ( A e. V /\ ( X C_ ~P A /\ X =/= (/) ) ) -> U. ( F " X ) C_ A )
35		elpw2g 4579	. . . . . 6 |- ( A e. V -> ( U. ( F " X ) e. ~P A <-> U. ( F " X ) C_ A ) )
36	35	adantr 466	. . . . 5 |- ( ( A e. V /\ ( X C_ ~P A /\ X =/= (/) ) ) -> ( U. ( F " X ) e. ~P A <-> U. ( F " X ) C_ A ) )
37	34, 36	mpbird 235	. . . 4 |- ( ( A e. V /\ ( X C_ ~P A /\ X =/= (/) ) ) -> U. ( F " X ) e. ~P A )
38		f1ocnvfv1 6181	. . . 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 665	. . 3 |- ( ( A e. V /\ ( X C_ ~P A /\ X =/= (/) ) ) -> ( `' F ` ( F ` U. ( F " X ) ) ) = U. ( F " X ) )
40	28, 39	syl5eqr 2475	. 2 |- ( ( A e. V /\ ( X C_ ~P A /\ X =/= (/) ) ) -> ( F ` ( F ` U. ( F " X ) ) ) = U. ( F " X ) )
41	26, 40	eqtr3d 2463	1 |- ( ( A e. V /\ ( X C_ ~P A /\ X =/= (/) ) ) -> ( F ` |^| X ) = U. ( F " X ) )

Syntax hints:   -> wi 4   <-> wb 187   /\ wa 370   = wceq 1437   e. wcel 1867   =/= wne 2616   \ cdif 3430   i^i cin 3432   C_ wss 3433   (/)c0 3758   ~Pcpw 3976   U.cuni 4213   |^|cint 4249   |-> cmpt 4475   `'ccnv 4844   dom cdm 4845   ran crn 4846   "cima 4848   -->wf 5588   -1-1-onto->wf1o 5591   ` cfv 5592   Isom wiso 5593   [ C.] crpss 6575

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-rab 2782  df-v 3080  df-sbc 3297  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-pss 3449  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-int 4250  df-br 4418  df-opab 4476  df-mpt 4477  df-id 4760  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-isom 5601  df-rpss 6576

This theorem is referenced by:  isf34lem7  8798