Theorem fcluscomplem 15620

Description: Lemma for fcluscomp 15621.

Hypothesis

Ref	Expression
fcluscomp.1	|- X = U.J

Assertion

Ref	Expression
fcluscomplem	|- ((J e. Top /\ f e. Fil /\ X = U.f) -> A.x e. ( fi ` {y | E.s e. f y = ((cls` J)` s)})E.z e. f z C_ x)

Distinct variable groups:   f,s,x,y,z,J   f,X,s,x,y,z

Proof of Theorem fcluscomplem

Step	Hyp	Ref	Expression
1		abrexexg 4837	. . . . 5 |- (f e. Fil -> {y | E.s e. f y = ((cls` J)` s)} e. _V)
2		visset 2295	. . . . . 6 |- x e. _V
3		sppfi 10218	. . . . . 6 |- ((x e. _V /\ {y | E.s e. f y = ((cls` J)` s)} e. _V) -> (x e. ( fi ` {y | E.s e. f y = ((cls` J)` s)}) <-> E.w(w C_ {y | E.s e. f y = ((cls` J)` s)} /\ w e. Fin /\ x = |^|w)))
4	2, 3	mpan 759	. . . . 5 |- ({y | E.s e. f y = ((cls` J)` s)} e. _V -> (x e. ( fi ` {y | E.s e. f y = ((cls` J)` s)}) <-> E.w(w C_ {y | E.s e. f y = ((cls` J)` s)} /\ w e. Fin /\ x = |^|w)))
5	1, 4	syl 12	. . . 4 |- (f e. Fil -> (x e. ( fi ` {y | E.s e. f y = ((cls` J)` s)}) <-> E.w(w C_ {y | E.s e. f y = ((cls` J)` s)} /\ w e. Fin /\ x = |^|w)))
6	5	3ad2ant2 898	. . 3 |- ((J e. Top /\ f e. Fil /\ X = U.f) -> (x e. ( fi ` {y | E.s e. f y = ((cls` J)` s)}) <-> E.w(w C_ {y | E.s e. f y = ((cls` J)` s)} /\ w e. Fin /\ x = |^|w)))
7		sseq2 2639	. . . . . . . 8 |- (x = |^|w -> (z C_ x <-> z C_ |^|w))
8	7	rexbidv 2124	. . . . . . 7 |- (x = |^|w -> (E.z e. f z C_ x <-> E.z e. f z C_ |^|w))
9		breq2 3342	. . . . . . . . . . . . . . . 16 |- (k = (/) -> (w ~~ k <-> w ~~ (/)))
10	9	imbi1d 675	. . . . . . . . . . . . . . 15 |- (k = (/) -> ((w ~~ k -> E.z e. f z C_ |^|w) <-> (w ~~ (/) -> E.z e. f z C_ |^|w)))
11	10	imbi2d 674	. . . . . . . . . . . . . 14 |- (k = (/) -> ((((J e. Top /\ f e. Fil /\ X = U.f) /\ w C_ {y | E.s e. f y = ((cls` J)` s)}) -> (w ~~ k -> E.z e. f z C_ |^|w)) <-> (((J e. Top /\ f e. Fil /\ X = U.f) /\ w C_ {y | E.s e. f y = ((cls` J)` s)}) -> (w ~~ (/) -> E.z e. f z C_ |^|w))))
12	11	albidv 1656	. . . . . . . . . . . . 13 |- (k = (/) -> (A.w(((J e. Top /\ f e. Fil /\ X = U.f) /\ w C_ {y | E.s e. f y = ((cls` J)` s)}) -> (w ~~ k -> E.z e. f z C_ |^|w)) <-> A.w(((J e. Top /\ f e. Fil /\ X = U.f) /\ w C_ {y | E.s e. f y = ((cls` J)` s)}) -> (w ~~ (/) -> E.z e. f z C_ |^|w))))
13		breq2 3342	. . . . . . . . . . . . . . . 16 |- (k = j -> (w ~~ k <-> w ~~ j))
14	13	imbi1d 675	. . . . . . . . . . . . . . 15 |- (k = j -> ((w ~~ k -> E.z e. f z C_ |^|w) <-> (w ~~ j -> E.z e. f z C_ |^|w)))
15	14	imbi2d 674	. . . . . . . . . . . . . 14 |- (k = j -> ((((J e. Top /\ f e. Fil /\ X = U.f) /\ w C_ {y | E.s e. f y = ((cls` J)` s)}) -> (w ~~ k -> E.z e. f z C_ |^|w)) <-> (((J e. Top /\ f e. Fil /\ X = U.f) /\ w C_ {y | E.s e. f y = ((cls` J)` s)}) -> (w ~~ j -> E.z e. f z C_ |^|w))))
16	15	albidv 1656	. . . . . . . . . . . . 13 |- (k = j -> (A.w(((J e. Top /\ f e. Fil /\ X = U.f) /\ w C_ {y | E.s e. f y = ((cls` J)` s)}) -> (w ~~ k -> E.z e. f z C_ |^|w)) <-> A.w(((J e. Top /\ f e. Fil /\ X = U.f) /\ w C_ {y | E.s e. f y = ((cls` J)` s)}) -> (w ~~ j -> E.z e. f z C_ |^|w))))
17		breq2 3342	. . . . . . . . . . . . . . . 16 |- (k = suc j -> (w ~~ k <-> w ~~ suc j))
18	17	imbi1d 675	. . . . . . . . . . . . . . 15 |- (k = suc j -> ((w ~~ k -> E.z e. f z C_ |^|w) <-> (w ~~ suc j -> E.z e. f z C_ |^|w)))
19	18	imbi2d 674	. . . . . . . . . . . . . 14 |- (k = suc j -> ((((J e. Top /\ f e. Fil /\ X = U.f) /\ w C_ {y | E.s e. f y = ((cls` J)` s)}) -> (w ~~ k -> E.z e. f z C_ |^|w)) <-> (((J e. Top /\ f e. Fil /\ X = U.f) /\ w C_ {y | E.s e. f y = ((cls` J)` s)}) -> (w ~~ suc j -> E.z e. f z C_ |^|w))))
20	19	albidv 1656	. . . . . . . . . . . . 13 |- (k = suc j -> (A.w(((J e. Top /\ f e. Fil /\ X = U.f) /\ w C_ {y | E.s e. f y = ((cls` J)` s)}) -> (w ~~ k -> E.z e. f z C_ |^|w)) <-> A.w(((J e. Top /\ f e. Fil /\ X = U.f) /\ w C_ {y | E.s e. f y = ((cls` J)` s)}) -> (w ~~ suc j -> E.z e. f z C_ |^|w))))
21		breq2 3342	. . . . . . . . . . . . . . . 16 |- (k = n -> (w ~~ k <-> w ~~ n))
22	21	imbi1d 675	. . . . . . . . . . . . . . 15 |- (k = n -> ((w ~~ k -> E.z e. f z C_ |^|w) <-> (w ~~ n -> E.z e. f z C_ |^|w)))
23	22	imbi2d 674	. . . . . . . . . . . . . 14 |- (k = n -> ((((J e. Top /\ f e. Fil /\ X = U.f) /\ w C_ {y | E.s e. f y = ((cls` J)` s)}) -> (w ~~ k -> E.z e. f z C_ |^|w)) <-> (((J e. Top /\ f e. Fil /\ X = U.f) /\ w C_ {y | E.s e. f y = ((cls` J)` s)}) -> (w ~~ n -> E.z e. f z C_ |^|w))))
24	23	albidv 1656	. . . . . . . . . . . . 13 |- (k = n -> (A.w(((J e. Top /\ f e. Fil /\ X = U.f) /\ w C_ {y | E.s e. f y = ((cls` J)` s)}) -> (w ~~ k -> E.z e. f z C_ |^|w)) <-> A.w(((J e. Top /\ f e. Fil /\ X = U.f) /\ w C_ {y | E.s e. f y = ((cls` J)` s)}) -> (w ~~ n -> E.z e. f z C_ |^|w))))
25		inteq 3217	. . . . . . . . . . . . . . . . . . . 20 |- (w = (/) -> |^|w = |^|(/))
26	25	sseq2d 2645	. . . . . . . . . . . . . . . . . . 19 |- (w = (/) -> (z C_ |^|w <-> z C_ |^|(/)))
27	26	rexbidv 2124	. . . . . . . . . . . . . . . . . 18 |- (w = (/) -> (E.z e. f z C_ |^|w <-> E.z e. f z C_ |^|(/)))
28		eqid 1884	. . . . . . . . . . . . . . . . . . . 20 |- U.f = U.f
29	28	filusb 10267	. . . . . . . . . . . . . . . . . . 19 |- (f e. Fil -> U.f e. f)
30		ssv 2636	. . . . . . . . . . . . . . . . . . . . 21 |- U.f C_ _V
31		int0 3230	. . . . . . . . . . . . . . . . . . . . 21 |- |^|(/) = _V
32	30, 31	sseqtr4i 2650	. . . . . . . . . . . . . . . . . . . 20 |- U.f C_ |^|(/)
33		sseq1 2637	. . . . . . . . . . . . . . . . . . . . 21 |- (z = U.f -> (z C_ |^|(/) <-> U.f C_ |^|(/)))
34	33	rcla4ev 2381	. . . . . . . . . . . . . . . . . . . 20 |- ((U.f e. f /\ U.f C_ |^|(/)) -> E.z e. f z C_ |^|(/))
35	32, 34	mpan2 760	. . . . . . . . . . . . . . . . . . 19 |- (U.f e. f -> E.z e. f z C_ |^|(/))
36	29, 35	syl 12	. . . . . . . . . . . . . . . . . 18 |- (f e. Fil -> E.z e. f z C_ |^|(/))
37	27, 36	syl5cbir 228	. . . . . . . . . . . . . . . . 17 |- (f e. Fil -> (w = (/) -> E.z e. f z C_ |^|w))
38	37	3ad2ant2 898	. . . . . . . . . . . . . . . 16 |- ((J e. Top /\ f e. Fil /\ X = U.f) -> (w = (/) -> E.z e. f z C_ |^|w))
39	38	adantr 425	. . . . . . . . . . . . . . 15 |- (((J e. Top /\ f e. Fil /\ X = U.f) /\ w C_ {y | E.s e. f y = ((cls` J)` s)}) -> (w = (/) -> E.z e. f z C_ |^|w))
40		en0 5482	. . . . . . . . . . . . . . 15 |- (w ~~ (/) <-> w = (/))
41	39, 40	syl5ib 223	. . . . . . . . . . . . . 14 |- (((J e. Top /\ f e. Fil /\ X = U.f) /\ w C_ {y | E.s e. f y = ((cls` J)` s)}) -> (w ~~ (/) -> E.z e. f z C_ |^|w))
42	41	ax-gen 1305	. . . . . . . . . . . . 13 |- A.w(((J e. Top /\ f e. Fil /\ X = U.f) /\ w C_ {y | E.s e. f y = ((cls` J)` s)}) -> (w ~~ (/) -> E.z e. f z C_ |^|w))
43		visset 2295	. . . . . . . . . . . . . . . . . . . 20 |- j e. _V
44	43	sucex 3892	. . . . . . . . . . . . . . . . . . 19 |- suc j e. _V
45	44	bren 5436	. . . . . . . . . . . . . . . . . 18 |- (w ~~ suc j <-> E.g g:w-1-1-onto->suc j)
46		f1ocnv 4651	. . . . . . . . . . . . . . . . . . . 20 |- (g:w-1-1-onto->suc j -> `'g:suc j-1-1-onto->w)
47		f1of1 4634	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- (`'g:suc j-1-1-onto->w -> `'g:suc j-1-1->w)
48		sssucid 3742	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- j C_ suc j
49		f1ores 4654	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ((`'g:suc j-1-1->w /\ j C_ suc j) -> (`'g |` j):j-1-1-onto->(`'g"j))
50	48, 49	mpan2 760	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- (`'g:suc j-1-1->w -> (`'g |` j):j-1-1-onto->(`'g"j))
51	47, 50	syl 12	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- (`'g:suc j-1-1-onto->w -> (`'g |` j):j-1-1-onto->(`'g"j))
52	51	adantr 425	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ((`'g:suc j-1-1-onto->w /\ ((J e. Top /\ f e. Fil /\ X = U.f) /\ w C_ {y | E.s e. f y = ((cls` J)` s)})) -> (`'g |` j):j-1-1-onto->(`'g"j))
53	43	f1oen 5457	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ((`'g |` j):j-1-1-onto->(`'g"j) -> j ~~ (`'g"j))
54		visset 2295	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- g e. _V
55	54	cnvex 4425	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- `'g e. _V
56		imaexg 4279	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- (`'g e. _V -> (`'g"j) e. _V)
57	55, 56	ax-mp 7	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- (`'g"j) e. _V
58	57	ensym 5471	. . . . . . . . . . . . . . . . . . . . . . . 24 |- (j ~~ (`'g"j) -> (`'g"j) ~~ j)
59	52, 53, 58	3syl 24	. . . . . . . . . . . . . . . . . . . . . . 23 |- ((`'g:suc j-1-1-onto->w /\ ((J e. Top /\ f e. Fil /\ X = U.f) /\ w C_ {y | E.s e. f y = ((cls` J)` s)})) -> (`'g"j) ~~ j)
60		sseq1 2637	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- (x = (`'g"j) -> (x C_ {y | E.s e. f y = ((cls` J)` s)} <-> (`'g"j) C_ {y | E.s e. f y = ((cls` J)` s)}))
61	60	anbi2d 678	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- (x = (`'g"j) -> (((J e. Top /\ f e. Fil /\ X = U.f) /\ x C_ {y | E.s e. f y = ((cls` J)` s)}) <-> ((J e. Top /\ f e. Fil /\ X = U.f) /\ (`'g"j) C_ {y | E.s e. f y = ((cls` J)` s)})))
62		breq1 3341	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- (x = (`'g"j) -> (x ~~ j <-> (`'g"j) ~~ j))
63		inteq 3217	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 |- (x = (`'g"j) -> |^|x = |^|(`'g"j))
64	63	sseq2d 2645	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- (x = (`'g"j) -> (z C_ |^|x <-> z C_ |^|(`'g"j)))
65	64	rexbidv 2124	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- (x = (`'g"j) -> (E.z e. f z C_ |^|x <-> E.z e. f z C_ |^|(`'g"j)))
66	62, 65	imbi12d 688	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- (x = (`'g"j) -> ((x ~~ j -> E.z e. f z C_ |^|x) <-> ((`'g"j) ~~ j -> E.z e. f z C_ |^|(`'g"j))))
67	61, 66	imbi12d 688	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- (x = (`'g"j) -> ((((J e. Top /\ f e. Fil /\ X = U.f) /\ x C_ {y | E.s e. f y = ((cls` J)` s)}) -> (x ~~ j -> E.z e. f z C_ |^|x)) <-> (((J e. Top /\ f e. Fil /\ X = U.f) /\ (`'g"j) C_ {y | E.s e. f y = ((cls` J)` s)}) -> ((`'g"j) ~~ j -> E.z e. f z C_ |^|(`'g"j)))))
68	57, 67	cla4v 2370	. . . . . . . . . . . . . . . . . . . . . . . 24 |- (A.x(((J e. Top /\ f e. Fil /\ X = U.f) /\ x C_ {y | E.s e. f y = ((cls` J)` s)}) -> (x ~~ j -> E.z e. f z C_ |^|x)) -> (((J e. Top /\ f e. Fil /\ X = U.f) /\ (`'g"j) C_ {y | E.s e. f y = ((cls` J)` s)}) -> ((`'g"j) ~~ j -> E.z e. f z C_ |^|(`'g"j))))
69		simprl 450	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ((`'g:suc j-1-1-onto->w /\ ((J e. Top /\ f e. Fil /\ X = U.f) /\ w C_ {y | E.s e. f y = ((cls` J)` s)})) -> (J e. Top /\ f e. Fil /\ X = U.f))
70		f1ofo 4643	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- (`'g:suc j-1-1-onto->w -> `'g:suc j-onto->w)
71		forn 4620	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- (`'g:suc j-onto->w -> ran `' g = w)
72		imassrn 4278	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 |- (`'g"j) C_ ran `' g
73		sseq2 2639	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 |- (ran `' g = w -> ((`'g"j) C_ ran `' g <-> (`'g"j) C_ w))
74	72, 73	mpbii 210	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- (ran `' g = w -> (`'g"j) C_ w)
75	70, 71, 74	3syl 24	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- (`'g:suc j-1-1-onto->w -> (`'g"j) C_ w)
76	75	adantr 425	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ((`'g:suc j-1-1-onto->w /\ ((J e. Top /\ f e. Fil /\ X = U.f) /\ w C_ {y | E.s e. f y = ((cls` J)` s)})) -> (`'g"j) C_ w)
77		simprr 451	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ((`'g:suc j-1-1-onto->w /\ ((J e. Top /\ f e. Fil /\ X = U.f) /\ w C_ {y | E.s e. f y = ((cls` J)` s)})) -> w C_ {y | E.s e. f y = ((cls` J)` s)})
78	76, 77	sstrd 2627	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ((`'g:suc j-1-1-onto->w /\ ((J e. Top /\ f e. Fil /\ X = U.f) /\ w C_ {y | E.s e. f y = ((cls` J)` s)})) -> (`'g"j) C_ {y | E.s e. f y = ((cls` J)` s)})
79	69, 78	jca 310	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ((`'g:suc j-1-1-onto->w /\ ((J e. Top /\ f e. Fil /\ X = U.f) /\ w C_ {y | E.s e. f y = ((cls` J)` s)})) -> ((J e. Top /\ f e. Fil /\ X = U.f) /\ (`'g"j) C_ {y | E.s e. f y = ((cls` J)` s)}))
80	68, 79	syl5com 63	. . . . . . . . . . . . . . . . . . . . . . 23 |- ((`'g:suc j-1-1-onto->w /\ ((J e. Top /\ f e. Fil /\ X = U.f) /\ w C_ {y | E.s e. f y = ((cls` J)` s)})) -> (A.x(((J e. Top /\ f e. Fil /\ X = U.f) /\ x C_ {y | E.s e. f y = ((cls` J)` s)}) -> (x ~~ j -> E.z e. f z C_ |^|x)) -> ((`'g"j) ~~ j -> E.z e. f z C_ |^|(`'g"j))))
81	59, 80	mpid 58	. . . . . . . . . . . . . . . . . . . . . 22 |- ((`'g:suc j-1-1-onto->w /\ ((J e. Top /\ f e. Fil /\ X = U.f) /\ w C_ {y | E.s e. f y = ((cls` J)` s)})) -> (A.x(((J e. Top /\ f e. Fil /\ X = U.f) /\ x C_ {y | E.s e. f y = ((cls` J)` s)}) -> (x ~~ j -> E.z e. f z C_ |^|x)) -> E.z e. f z C_ |^|(`'g"j)))
82		ssel 2615	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- (w C_ {y | E.s e. f y = ((cls` J)` s)} -> ((`'g` j) e. w -> (`'g` j) e. {y | E.s e. f y = ((cls` J)` s)}))
83		f1of 4635	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 |- (`'g:suc j-1-1-onto->w -> `'g:suc j-->w)
84	43	sucid 3744	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 |- j e. suc j
85		ffvelrn 4787	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 |- ((`'g:suc j-->w /\ j e. suc j) -> (`'g` j) e. w)
86	84, 85	mpan2 760	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 |- (`'g:suc j-->w -> (`'g` j) e. w)
87	83, 86	syl 12	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- (`'g:suc j-1-1-onto->w -> (`'g` j) e. w)
88	82, 87	syl5com 63	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- (`'g:suc j-1-1-onto->w -> (w C_ {y | E.s e. f y = ((cls` J)` s)} -> (`'g` j) e. {y | E.s e. f y = ((cls` J)` s)}))
89	88	adantr 425	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ((`'g:suc j-1-1-onto->w /\ (J e. Top /\ f e. Fil /\ X = U.f)) -> (w C_ {y | E.s e. f y = ((cls` J)` s)} -> (`'g` j) e. {y | E.s e. f y = ((cls` J)` s)}))
90		filint 10269	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 |- ((f e. Fil /\ x e. f /\ s e. f) -> (x i^i s) e. f)
91	90	3com23 1074	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 |- ((f e. Fil /\ s e. f /\ x e. f) -> (x i^i s) e. f)
92	91	3expb 1068	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 |- ((f e. Fil /\ (s e. f /\ x e. f)) -> (x i^i s) e. f)
93	92	3ad2antl2 1039	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 |- (((J e. Top /\ f e. Fil /\ X = U.f) /\ (s e. f /\ x e. f)) -> (x i^i s) e. f)
94	93	anassrs 489	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 |- ((((J e. Top /\ f e. Fil /\ X = U.f) /\ s e. f) /\ x e. f) -> (x i^i s) e. f)
95	94	adantrl 430	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 |- ((((J e. Top /\ f e. Fil /\ X = U.f) /\ s e. f) /\ ((`'g` j) = ((cls` J)` s) /\ x e. f)) -> (x i^i s) e. f)
96	95	adantlll 432	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 |- ((((`'g:suc j-1-1-onto->w /\ (J e. Top /\ f e. Fil /\ X = U.f)) /\ s e. f) /\ ((`'g` j) = ((cls` J)` s) /\ x e. f)) -> (x i^i s) e. f)
97	96	adantrr 431	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 |- ((((`'g:suc j-1-1-onto->w /\ (J e. Top /\ f e. Fil /\ X = U.f)) /\ s e. f) /\ (((`'g` j) = ((cls` J)` s) /\ x e. f) /\ x C_ |^|(`'g"j))) -> (x i^i s) e. f)
98		simprr 451	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 |- ((((`'g:suc j-1-1-onto->w /\ (J e. Top /\ f e. Fil /\ X = U.f)) /\ s e. f) /\ (((`'g` j) = ((cls` J)` s) /\ x e. f) /\ x C_ |^|(`'g"j))) -> x C_ |^|(`'g"j))
99		simpr1 882	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 |- ((`'g:suc j-1-1-onto->w /\ (J e. Top /\ f e. Fil /\ X = U.f)) -> J e. Top)
100	99	ad2antrr 440	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 |- ((((`'g:suc j-1-1-onto->w /\ (J e. Top /\ f e. Fil /\ X = U.f)) /\ s e. f) /\ (((`'g` j) = ((cls` J)` s) /\ x e. f) /\ x C_ |^|(`'g"j))) -> J e. Top)
101		elssuni 3206	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 |- (s e. f -> s C_ U.f)
102	101	adantl 424	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 |- ((X = U.f /\ s e. f) -> s C_ U.f)
103		simpl 346	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 |- ((X = U.f /\ s e. f) -> X = U.f)
104	102, 103	sseqtr4d 2654	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 |- ((X = U.f /\ s e. f) -> s C_ X)
105	104	3ad2antl3 1040	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 |- (((J e. Top /\ f e. Fil /\ X = U.f) /\ s e. f) -> s C_ X)
106	105	adantll 428	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 |- (((`'g:suc j-1-1-onto->w /\ (J e. Top /\ f e. Fil /\ X = U.f)) /\ s e. f) -> s C_ X)
107	106	adantr 425	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 |- ((((`'g:suc j-1-1-onto->w /\ (J e. Top /\ f e. Fil /\ X = U.f)) /\ s e. f) /\ (((`'g` j) = ((cls` J)` s) /\ x e. f) /\ x C_ |^|(`'g"j))) -> s C_ X)
108		fcluscomp.1	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 |- X = U.J
109	108	sscls 8965	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 |- ((J e. Top /\ s C_ X) -> s C_ ((cls` J)` s))
110	100, 107, 109	syl11anc 524	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 |- ((((`'g:suc j-1-1-onto->w /\ (J e. Top /\ f e. Fil /\ X = U.f)) /\ s e. f) /\ (((`'g` j) = ((cls` J)` s) /\ x e. f) /\ x C_ |^|(`'g"j))) -> s C_ ((cls` J)` s))
111		simprll 456	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 |- ((((`'g:suc j-1-1-onto->w /\ (J e. Top /\ f e. Fil /\ X = U.f)) /\ s e. f) /\ (((`'g` j) = ((cls` J)` s) /\ x e. f) /\ x C_ |^|(`'g"j))) -> (`'g` j) = ((cls` J)` s))
112	110, 111	sseqtr4d 2654	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 |- ((((`'g:suc j-1-1-onto->w /\ (J e. Top /\ f e. Fil /\ X = U.f)) /\ s e. f) /\ (((`'g` j) = ((cls` J)` s) /\ x e. f) /\ x C_ |^|(`'g"j))) -> s C_ (`'g` j))
113		ss2in 2820	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 |- ((x C_ |^|(`'g"j) /\ s C_ (`'g` j)) -> (x i^i s) C_ (|^|(`'g"j) i^i (`'g` j)))
114	98, 112, 113	syl11anc 524	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 |- ((((`'g:suc j-1-1-onto->w /\ (J e. Top /\ f e. Fil /\ X = U.f)) /\ s e. f) /\ (((`'g` j) = ((cls` J)` s) /\ x e. f) /\ x C_ |^|(`'g"j))) -> (x i^i s) C_ (|^|(`'g"j) i^i (`'g` j)))
115		f1ofn 4636	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 |- (`'g:suc j-1-1-onto->w -> `'g Fn suc j)
116	115	adantr 425	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 |- ((`'g:suc j-1-1-onto->w /\ (J e. Top /\ f e. Fil /\ X = U.f)) -> `'g Fn suc j)
117	116	ad2antrr 440	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 |- ((((`'g:suc j-1-1-onto->w /\ (J e. Top /\ f e. Fil /\ X = U.f)) /\ s e. f) /\ (((`'g` j) = ((cls` J)` s) /\ x e. f) /\ x C_ |^|(`'g"j))) -> `'g Fn suc j)
118	84	a1i 8	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 |- ((((`'g:suc j-1-1-onto->w /\ (J e. Top /\ f e. Fil /\ X = U.f)) /\ s e. f) /\ (((`'g` j) = ((cls` J)` s) /\ x e. f) /\ x C_ |^|(`'g"j))) -> j e. suc j)
119		fnsnfv 4728	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 |- ((`'g Fn suc j /\ j e. suc j) -> {(`'g` j)} = (`'g"{j}))
120	117, 118, 119	syl11anc 524	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 |- ((((`'g:suc j-1-1-onto->w /\ (J e. Top /\ f e. Fil /\ X = U.f)) /\ s e. f) /\ (((`'g` j) = ((cls` J)` s) /\ x e. f) /\ x C_ |^|(`'g"j))) -> {(`'g` j)} = (`'g"{j}))
121	120	inteqd 3219	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 |- ((((`'g:suc j-1-1-onto->w /\ (J e. Top /\ f e. Fil /\ X = U.f)) /\ s e. f) /\ (((`'g` j) = ((cls` J)` s) /\ x e. f) /\ x C_ |^|(`'g"j))) -> |^|{(`'g` j)} = |^|(`'g"{j}))
122		fvex 4689	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 |- (`'g` j) e. _V
123	122	intsn 3252	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 |- |^|{(`'g` j)} = (`'g` j)
124	121, 123	syl5eqr 1942	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 |- ((((`'g:suc j-1-1-onto->w /\ (J e. Top /\ f e. Fil /\ X = U.f)) /\ s e. f) /\ (((`'g` j) = ((cls` J)` s) /\ x e. f) /\ x C_ |^|(`'g"j))) -> (`'g` j) = |^|(`'g"{j}))
125	124	ineq2d 2796	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 |- ((((`'g:suc j-1-1-onto->w /\ (J e. Top /\ f e. Fil /\ X = U.f)) /\ s e. f) /\ (((`'g` j) = ((cls` J)` s) /\ x e. f) /\ x C_ |^|(`'g"j))) -> (|^|(`'g"j) i^i (`'g` j)) = (|^|(`'g"j) i^i |^|(`'g"{j})))
126		intun 3249	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 |- |^|((`'g"j) u. (`'g"{j})) = (|^|(`'g"j) i^i |^|(`'g"{j}))
127	125, 126	syl6eqr 1946	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 |- ((((`'g:suc j-1-1-onto->w /\ (J e. Top /\ f e. Fil /\ X = U.f)) /\ s e. f) /\ (((`'g` j) = ((cls` J)` s) /\ x e. f) /\ x C_ |^|(`'g"j))) -> (|^|(`'g"j) i^i (`'g` j)) = |^|((`'g"j) u. (`'g"{j})))
128		foima 4622	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 |- (`'g:suc j-onto->w -> (`'g"suc j) = w)
129	70, 128	syl 12	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 |- (`'g:suc j-1-1-onto->w -> (`'g"suc j) = w)
130	129	adantr 425	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 |- ((`'g:suc j-1-1-onto->w /\ (J e. Top /\ f e. Fil /\ X = U.f)) -> (`'g"suc j) = w)
131	130	ad2antrr 440	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 |- ((((`'g:suc j-1-1-onto->w /\ (J e. Top /\ f e. Fil /\ X = U.f)) /\ s e. f) /\ (((`'g` j) = ((cls` J)` s) /\ x e. f) /\ x C_ |^|(`'g"j))) -> (`'g"suc j) = w)
132		df-suc 3663	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 |- suc j = (j u. {j})
133	132	imaeq2i 4262	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 |- (`'g"suc j) = (`'g"(j u. {j}))
134		imaundi 4328	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 |- (`'g"(j u. {j})) = ((`'g"j) u. (`'g"{j}))
135	133, 134	eqtr2i 1909	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 |- ((`'g"j) u. (`'g"{j})) = (`'g"suc j)
136	131, 135	syl5eq 1940	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 |- ((((`'g:suc j-1-1-onto->w /\ (J e. Top /\ f e. Fil /\ X = U.f)) /\ s e. f) /\ (((`'g` j) = ((cls` J)` s) /\ x e. f) /\ x C_ |^|(`'g"j))) -> ((`'g"j) u. (`'g"{j})) = w)
137	136	inteqd 3219	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 |- ((((`'g:suc j-1-1-onto->w /\ (J e. Top /\ f e. Fil /\ X = U.f)) /\ s e. f) /\ (((`'g` j) = ((cls` J)` s) /\ x e. f) /\ x C_ |^|(`'g"j))) -> |^|((`'g"j) u. (`'g"{j})) = |^|w)
138	127, 137	eqtrd 1925	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 |- ((((`'g:suc j-1-1-onto->w /\ (J e. Top /\ f e. Fil /\ X = U.f)) /\ s e. f) /\ (((`'g` j) = ((cls` J)` s) /\ x e. f) /\ x C_ |^|(`'g"j))) -> (|^|(`'g"j) i^i (`'g` j)) = |^|w)
139	114, 138	sseqtrd 2653	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 |- ((((`'g:suc j-1-1-onto->w /\ (J e. Top /\ f e. Fil /\ X = U.f)) /\ s e. f) /\ (((`'g` j) = ((cls` J)` s) /\ x e. f) /\ x C_ |^|(`'g"j))) -> (x i^i s) C_ |^|w)
140		sseq1 2637	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 |- (z = (x i^i s) -> (z C_ |^|w <-> (x i^i s) C_ |^|w))
141	140	rcla4ev 2381	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 |- (((x i^i s) e. f /\ (x i^i s) C_ |^|w) -> E.z e. f z C_ |^|w)
142	97, 139, 141	syl11anc 524	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 |- ((((`'g:suc j-1-1-onto->w /\ (J e. Top /\ f e. Fil /\ X = U.f)) /\ s e. f) /\ (((`'g` j) = ((cls` J)` s) /\ x e. f) /\ x C_ |^|(`'g"j))) -> E.z e. f z C_ |^|w)
143	142	exp58 15342	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- ((`'g:suc j-1-1-onto->w /\ (J e. Top /\ f e. Fil /\ X = U.f)) -> (s e. f -> ((`'g` j) = ((cls` J)` s) -> (x e. f -> (x C_ |^|(`'g"j) -> E.z e. f z C_ |^|w)))))
144	143	r19.23adv 2215	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ((`'g:suc j-1-1-onto->w /\ (J e. Top /\ f e. Fil /\ X = U.f)) -> (E.s e. f (`'g` j) = ((cls` J)` s) -> (x e. f -> (x C_ |^|(`'g"j) -> E.z e. f z C_ |^|w))))
145		eqeq1 1890	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 |- (y = (`'g` j) -> (y = ((cls` J)` s) <-> (`'g` j) = ((cls` J)` s)))
146	145	rexbidv 2124	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 |- (y = (`'g` j) -> (E.s e. f y = ((cls` J)` s) <-> E.s e. f (`'g` j) = ((cls` J)` s)))
147	122, 146	elab 2403	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 |- ((`'g` j) e. {y | E.s e. f y = ((cls` J)` s)} <-> E.s e. f (`'g` j) = ((cls` J)` s))
148	144, 147	syl5ib 223	. . . . . . . . . . . . . . . . . . . . . . . . . 26 |- ((`'g:suc j-1-1-onto->w /\ (J e. Top /\ f e. Fil /\ X = U.f)) -> ((`'g` j) e. {y | E.s e. f y = ((cls` J)` s)} -> (x e. f -> (x C_ |^|(`'g"j) -> E.z e. f z C_ |^|w))))
149	89, 148	syld 30	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ((`'g:suc j-1-1-onto->w /\ (J e. Top /\ f e. Fil /\ X = U.f)) -> (w C_ {y | E.s e. f y = ((cls` J)` s)} -> (x e. f -> (x C_ |^|(`'g"j) -> E.z e. f z C_ |^|w))))
150	149	impr 422	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ((`'g:suc j-1-1-onto->w /\ ((J e. Top /\ f e. Fil /\ X = U.f) /\ w C_ {y | E.s e. f y = ((cls` J)` s)})) -> (x e. f -> (x C_ |^|(`'g"j) -> E.z e. f z C_ |^|w)))
151	150	r19.23adv 2215	. . . . . . . . . . . . . . . . . . . . . . 23 |- ((`'g:suc j-1-1-onto->w /\ ((J e. Top /\ f e. Fil /\ X = U.f) /\ w C_ {y | E.s e. f y = ((cls` J)` s)})) -> (E.x e. f x C_ |^|(`'g"j) -> E.z e. f z C_ |^|w))
152		sseq1 2637	. . . . . . . . . . . . . . . . . . . . . . . 24 |- (z = x -> (z C_ |^|(`'g"j) <-> x C_ |^|(`'g"j)))
153	152	cbvrexv 2281	. . . . . . . . . . . . . . . . . . . . . . 23 |- (E.z e. f z C_ |^|(`'g"j) <-> E.x e. f x C_ |^|(`'g"j))
154	151, 153	syl5ib 223	. . . . . . . . . . . . . . . . . . . . . 22 |- ((`'g:suc j-1-1-onto->w /\ ((J e. Top /\ f e. Fil /\ X = U.f) /\ w C_ {y | E.s e. f y = ((cls` J)` s)})) -> (E.z e. f z C_ |^|(`'g"j) -> E.z e. f z C_ |^|w))
155	81, 154	syld 30	. . . . . . . . . . . . . . . . . . . . 21 |- ((`'g:suc j-1-1-onto->w /\ ((J e. Top /\ f e. Fil /\ X = U.f) /\ w C_ {y | E.s e. f y = ((cls` J)` s)})) -> (A.x(((J e. Top /\ f e. Fil /\ X = U.f) /\ x C_ {y | E.s e. f y = ((cls` J)` s)}) -> (x ~~ j -> E.z e. f z C_ |^|x)) -> E.z e. f z C_ |^|w))
156	155	ex 402	. . . . . . . . . . . . . . . . . . . 20 |- (`'g:suc j-1-1-onto->w -> (((J e. Top /\ f e. Fil /\ X = U.f) /\ w C_ {y | E.s e. f y = ((cls` J)` s)}) -> (A.x(((J e. Top /\ f e. Fil /\ X = U.f) /\ x C_ {y | E.s e. f y = ((cls` J)` s)}) -> (x ~~ j -> E.z e. f z C_ |^|x)) -> E.z e. f z C_ |^|w)))
157	46, 156	syl 12	. . . . . . . . . . . . . . . . . . 19 |- (g:w-1-1-onto->suc j -> (((J e. Top /\ f e. Fil /\ X = U.f) /\ w C_ {y | E.s e. f y = ((cls` J)` s)}) -> (A.x(((J e. Top /\ f e. Fil /\ X = U.f) /\ x C_ {y | E.s e. f y = ((cls` J)` s)}) -> (x ~~ j -> E.z e. f z C_ |^|x)) -> E.z e. f z C_ |^|w)))
158	157	19.23aiv 1674	. . . . . . . . . . . . . . . . . 18 |- (E.g g:w-1-1-onto->suc j -> (((J e. Top /\ f e. Fil /\ X = U.f) /\ w C_ {y | E.s e. f y = ((cls` J)` s)}) -> (A.x(((J e. Top /\ f e. Fil /\ X = U.f) /\ x C_ {y | E.s e. f y = ((cls` J)` s)}) -> (x ~~ j -> E.z e. f z C_ |^|x)) -> E.z e. f z C_ |^|w)))
159	45, 158	sylbi 216	. . . . . . . . . . . . . . . . 17 |- (w ~~ suc j -> (((J e. Top /\ f e. Fil /\ X = U.f) /\ w C_ {y | E.s e. f y = ((cls` J)` s)}) -> (A.x(((J e. Top /\ f e. Fil /\ X = U.f) /\ x C_ {y | E.s e. f y = ((cls` J)` s)}) -> (x ~~ j -> E.z e. f z C_ |^|x)) -> E.z e. f z C_ |^|w)))
160	159	com13 37	. . . . . . . . . . . . . . . 16 |- (A.x(((J e. Top /\ f e. Fil /\ X = U.f) /\ x C_ {y | E.s e. f y = ((cls` J)` s)}) -> (x ~~ j -> E.z e. f z C_ |^|x)) -> (((J e. Top /\ f e. Fil /\ X = U.f) /\ w C_ {y | E.s e. f y = ((cls` J)` s)}) -> (w ~~ suc j -> E.z e. f z C_ |^|w)))
161	160	a1i 8	. . . . . . . . . . . . . . 15 |- (j e. om -> (A.x(((J e. Top /\ f e. Fil /\ X = U.f) /\ x C_ {y | E.s e. f y = ((cls` J)` s)}) -> (x ~~ j -> E.z e. f z C_ |^|x)) -> (((J e. Top /\ f e. Fil /\ X = U.f) /\ w C_ {y | E.s e. f y = ((cls` J)` s)}) -> (w ~~ suc j -> E.z e. f z C_ |^|w))))
162	161	19.21adv 1666	. . . . . . . . . . . . . 14 |- (j e. om -> (A.x(((J e. Top /\ f e. Fil /\ X = U.f) /\ x C_ {y | E.s e. f y = ((cls` J)` s)}) -> (x ~~ j -> E.z e. f z C_ |^|x)) -> A.w(((J e. Top /\ f e. Fil /\ X = U.f) /\ w C_ {y | E.s e. f y = ((cls` J)` s)}) -> (w ~~ suc j -> E.z e. f z C_ |^|w))))
163		sseq1 2637	. . . . . . . . . . . . . . . . 17 |- (w = x -> (w C_ {y | E.s e. f y = ((cls` J)` s)} <-> x C_ {y | E.s e. f y = ((cls` J)` s)}))
164	163	anbi2d 678	. . . . . . . . . . . . . . . 16 |- (w = x -> (((J e. Top /\ f e. Fil /\ X = U.f) /\ w C_ {y | E.s e. f y = ((cls` J)` s)}) <-> ((J e. Top /\ f e. Fil /\ X = U.f) /\ x C_ {y | E.s e. f y = ((cls` J)` s)})))
165		breq1 3341	. . . . . . . . . . . . . . . . 17 |- (w = x -> (w ~~ j <-> x ~~ j))
166		inteq 3217	. . . . . . . . . . . . . . . . . . 19 |- (w = x -> |^|w = |^|x)
167	166	sseq2d 2645	. . . . . . . . . . . . . . . . . 18 |- (w = x -> (z C_ |^|w <-> z C_ |^|x))
168	167	rexbidv 2124	. . . . . . . . . . . . . . . . 17 |- (w = x -> (E.z e. f z C_ |^|w <-> E.z e. f z C_ |^|x))
169	165, 168	imbi12d 688	. . . . . . . . . . . . . . . 16 |- (w = x -> ((w ~~ j -> E.z e. f z C_ |^|w) <-> (x ~~ j -> E.z e. f z C_ |^|x)))
170	164, 169	imbi12d 688	. . . . . . . . . . . . . . 15 |- (w = x -> ((((J e. Top /\ f e. Fil /\ X = U.f) /\ w C_ {y | E.s e. f y = ((cls` J)` s)}) -> (w ~~ j -> E.z e. f z C_ |^|w)) <-> (((J e. Top /\ f e. Fil /\ X = U.f) /\ x C_ {y | E.s e. f y = ((cls` J)` s)}) -> (x ~~ j -> E.z e. f z C_ |^|x))))
171	170	cbvalv 1696	. . . . . . . . . . . . . 14 |- (A.w(((J e. Top /\ f e. Fil /\ X = U.f) /\ w C_ {y | E.s e. f y = ((cls` J)` s)}) -> (w ~~ j -> E.z e. f z C_ |^|w)) <-> A.x(((J e. Top /\ f e. Fil /\ X = U.f) /\ x C_ {y | E.s e. f y = ((cls` J)` s)}) -> (x ~~ j -> E.z e. f z C_ |^|x)))
172	162, 171	syl5ib 223	. . . . . . . . . . . . 13 |- (j e. om -> (A.w(((J e. Top /\ f e. Fil /\ X = U.f) /\ w C_ {y | E.s e. f y = ((cls` J)` s)}) -> (w ~~ j -> E.z e. f z C_ |^|w)) -> A.w(((J e. Top /\ f e. Fil /\ X = U.f) /\ w C_ {y | E.s e. f y = ((cls` J)` s)}) -> (w ~~ suc j -> E.z e. f z C_ |^|w))))
173	12, 16, 20, 24, 42, 172	finds 3979	. . . . . . . . . . . 12 |- (n e. om -> A.w(((J e. Top /\ f e. Fil /\ X = U.f) /\ w C_ {y | E.s e. f y = ((cls` J)` s)}) -> (w ~~ n -> E.z e. f z C_ |^|w)))
174	173	19.21bi 1408	. . . . . . . . . . 11 |- (n e. om -> (((J e. Top /\ f e. Fil /\ X = U.f) /\ w C_ {y | E.s e. f y = ((cls` J)` s)}) -> (w ~~ n -> E.z e. f z C_ |^|w)))
175	174	com12 14	. . . . . . . . . 10 |- (((J e. Top /\ f e. Fil /\ X = U.f) /\ w C_ {y | E.s e. f y = ((cls` J)` s)}) -> (n e. om -> (w ~~ n -> E.z e. f z C_ |^|w)))
176	175	r19.23adv 2215	. . . . . . . . 9 |- (((J e. Top /\ f e. Fil /\ X = U.f) /\ w C_ {y | E.s e. f y = ((cls` J)` s)}) -> (E.n e. om w ~~ n -> E.z e. f z C_ |^|w))
177		isfi 5441	. . . . . . . . 9 |- (w e. Fin <-> E.n e. om w ~~ n)
178	176, 177	syl5ib 223	. . . . . . . 8 |- (((J e. Top /\ f e. Fil /\ X = U.f) /\ w C_ {y | E.s e. f y = ((cls` J)` s)}) -> (w e. Fin -> E.z e. f z C_ |^|w))
179	178	impr 422	. . . . . . 7 |- (((J e. Top /\ f e. Fil /\ X = U.f) /\ (w C_ {y | E.s e. f y = ((cls` J)` s)} /\ w e. Fin)) -> E.z e. f z C_ |^|w)
180	8, 179	syl5cbir 228	. . . . . 6 |- (((J e. Top /\ f e. Fil /\ X = U.f) /\ (w C_ {y | E.s e. f y = ((cls` J)` s)} /\ w e. Fin)) -> (x = |^|w -> E.z e. f z C_ x))
181	180	exp32 408	. . . . 5 |- ((J e. Top /\ f e. Fil /\ X = U.f) -> (w C_ {y | E.s e. f y = ((cls` J)` s)} -> (w e. Fin -> (x = |^|w -> E.z e. f z C_ x))))
182	181	3impd 1082	. . . 4 |- ((J e. Top /\ f e. Fil /\ X = U.f) -> ((w C_ {y | E.s e. f y = ((cls` J)` s)} /\ w e. Fin /\ x = |^|w) -> E.z e. f z C_ x))
183	182	19.23adv 1584	. . 3 |- ((J e. Top /\ f e. Fil /\ X = U.f) -> (E.w(w C_ {y | E.s e. f y = ((cls` J)` s)} /\ w e. Fin /\ x = |^|w) -> E.z e. f z C_ x))
184	6, 183	sylbid 220	. 2 |- ((J e. Top /\ f e. Fil /\ X = U.f) -> (x e. ( fi ` {y | E.s e. f y = ((cls` J)` s)}) -> E.z e. f z C_ x))
185	184	r19.21aiv 2175	1 |- ((J e. Top /\ f e. Fil /\ X = U.f) -> A.x e. ( fi ` {y | E.s e. f y = ((cls` J)` s)})E.z e. f z C_ x)

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858  A.wal 1296   = wceq 1298   e. wcel 1300  E.wex 1326  {cab 1871  A.wral 2105  E.wrex 2106  _Vcvv 2292   u. cun 2591   i^i cin 2592   C_ wss 2593  (/)c0 2875  {csn 3044  U.cuni 3177  |^|cint 3214   class class class wbr 3338  suc csuc 3659  omcom 3949  `'ccnv 3985  ran crn 3987   |` cres 3988  "cima 3989   Fn wfn 3993  -->wf 3994  -1-1->wf1 3995  -onto->wfo 3996  -1-1-onto->wf1o 3997  ` cfv 3998   ~~ cen 5423  Fincfn 5426  Topctop 8857  clsccl 8938   fi cfi 10210  Filcfil 10264

This theorem is referenced by:  fcluscomp 15621

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-er 5318  df-en 5427  df-fin 5430  df-top 8861  df-cld 8939  df-cls 8941  df-fi 10211  df-fil 10265

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