Theorem utop3cls 22055

Description: Relation between a topological closure and a symmetric entourage in an uniform space. Second part of proposition 2 of [BourbakiTop1] p. II.4. (Contributed by Thierry Arnoux, 17-Jan-2018.)

Hypothesis

Ref	Expression
utoptop.1	|- J = (unifTop ` U )

Assertion

Ref	Expression
utop3cls	|- ( ( ( U e. (UnifOn ` X ) /\ M C_ ( X X. X ) ) /\ ( V e. U /\ `' V = V ) ) -> ( ( cls ` ( J tX J ) ) ` M ) C_ ( V o. ( M o. V ) ) )

Proof of Theorem utop3cls

Dummy variables r z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		relxp 5227	. . . . 5 |- Rel ( X X. X )
2		utoptop.1	. . . . . . . . . . 11 |- J = (unifTop ` U )
3		utoptop 22038	. . . . . . . . . . 11 |- ( U e. (UnifOn ` X ) -> (unifTop ` U ) e. Top )
4	2, 3	syl5eqel 2705	. . . . . . . . . 10 |- ( U e. (UnifOn ` X ) -> J e. Top )
5		txtop 21372	. . . . . . . . . 10 |- ( ( J e. Top /\ J e. Top ) -> ( J tX J ) e. Top )
6	4, 4, 5	syl2anc 693	. . . . . . . . 9 |- ( U e. (UnifOn ` X ) -> ( J tX J ) e. Top )
7	6	ad3antrrr 766	. . . . . . . 8 |- ( ( ( ( U e. (UnifOn ` X ) /\ M C_ ( X X. X ) ) /\ ( V e. U /\ `' V = V ) ) /\ z e. ( ( cls ` ( J tX J ) ) ` M ) ) -> ( J tX J ) e. Top )
8		simpllr 799	. . . . . . . . 9 |- ( ( ( ( U e. (UnifOn ` X ) /\ M C_ ( X X. X ) ) /\ ( V e. U /\ `' V = V ) ) /\ z e. ( ( cls ` ( J tX J ) ) ` M ) ) -> M C_ ( X X. X ) )
9		utoptopon 22040	. . . . . . . . . . . . . 14 |- ( U e. (UnifOn ` X ) -> (unifTop ` U ) e. (TopOn ` X ) )
10	2, 9	syl5eqel 2705	. . . . . . . . . . . . 13 |- ( U e. (UnifOn ` X ) -> J e. (TopOn ` X ) )
11		toponuni 20719	. . . . . . . . . . . . 13 |- ( J e. (TopOn ` X ) -> X = U. J )
12	10, 11	syl 17	. . . . . . . . . . . 12 |- ( U e. (UnifOn ` X ) -> X = U. J )
13	12	sqxpeqd 5141	. . . . . . . . . . 11 |- ( U e. (UnifOn ` X ) -> ( X X. X ) = ( U. J X. U. J ) )
14		eqid 2622	. . . . . . . . . . . . 13 |- U. J = U. J
15	14, 14	txuni 21395	. . . . . . . . . . . 12 |- ( ( J e. Top /\ J e. Top ) -> ( U. J X. U. J ) = U. ( J tX J ) )
16	4, 4, 15	syl2anc 693	. . . . . . . . . . 11 |- ( U e. (UnifOn ` X ) -> ( U. J X. U. J ) = U. ( J tX J ) )
17	13, 16	eqtrd 2656	. . . . . . . . . 10 |- ( U e. (UnifOn ` X ) -> ( X X. X ) = U. ( J tX J ) )
18	17	ad3antrrr 766	. . . . . . . . 9 |- ( ( ( ( U e. (UnifOn ` X ) /\ M C_ ( X X. X ) ) /\ ( V e. U /\ `' V = V ) ) /\ z e. ( ( cls ` ( J tX J ) ) ` M ) ) -> ( X X. X ) = U. ( J tX J ) )
19	8, 18	sseqtrd 3641	. . . . . . . 8 |- ( ( ( ( U e. (UnifOn ` X ) /\ M C_ ( X X. X ) ) /\ ( V e. U /\ `' V = V ) ) /\ z e. ( ( cls ` ( J tX J ) ) ` M ) ) -> M C_ U. ( J tX J ) )
20		eqid 2622	. . . . . . . . 9 |- U. ( J tX J ) = U. ( J tX J )
21	20	clsss3 20863	. . . . . . . 8 |- ( ( ( J tX J ) e. Top /\ M C_ U. ( J tX J ) ) -> ( ( cls ` ( J tX J ) ) ` M ) C_ U. ( J tX J ) )
22	7, 19, 21	syl2anc 693	. . . . . . 7 |- ( ( ( ( U e. (UnifOn ` X ) /\ M C_ ( X X. X ) ) /\ ( V e. U /\ `' V = V ) ) /\ z e. ( ( cls ` ( J tX J ) ) ` M ) ) -> ( ( cls ` ( J tX J ) ) ` M ) C_ U. ( J tX J ) )
23	22, 18	sseqtr4d 3642	. . . . . 6 |- ( ( ( ( U e. (UnifOn ` X ) /\ M C_ ( X X. X ) ) /\ ( V e. U /\ `' V = V ) ) /\ z e. ( ( cls ` ( J tX J ) ) ` M ) ) -> ( ( cls ` ( J tX J ) ) ` M ) C_ ( X X. X ) )
24		simpr 477	. . . . . 6 |- ( ( ( ( U e. (UnifOn ` X ) /\ M C_ ( X X. X ) ) /\ ( V e. U /\ `' V = V ) ) /\ z e. ( ( cls ` ( J tX J ) ) ` M ) ) -> z e. ( ( cls ` ( J tX J ) ) ` M ) )
25	23, 24	sseldd 3604	. . . . 5 |- ( ( ( ( U e. (UnifOn ` X ) /\ M C_ ( X X. X ) ) /\ ( V e. U /\ `' V = V ) ) /\ z e. ( ( cls ` ( J tX J ) ) ` M ) ) -> z e. ( X X. X ) )
26		1st2nd 7214	. . . . 5 |- ( ( Rel ( X X. X ) /\ z e. ( X X. X ) ) -> z = <. ( 1st ` z ) , ( 2nd ` z ) >. )
27	1, 25, 26	sylancr 695	. . . 4 |- ( ( ( ( U e. (UnifOn ` X ) /\ M C_ ( X X. X ) ) /\ ( V e. U /\ `' V = V ) ) /\ z e. ( ( cls ` ( J tX J ) ) ` M ) ) -> z = <. ( 1st ` z ) , ( 2nd ` z ) >. )
28		simp-4l 806	. . . . . . . . . 10 |- ( ( ( ( ( U e. (UnifOn ` X ) /\ M C_ ( X X. X ) ) /\ ( V e. U /\ `' V = V ) ) /\ z e. ( ( cls ` ( J tX J ) ) ` M ) ) /\ r e. ( ( ( V " { ( 1st ` z ) } ) X. ( V " { ( 2nd ` z ) } ) ) i^i M ) ) -> U e. (UnifOn ` X ) )
29		simpr1l 1118	. . . . . . . . . . 11 |- ( ( ( U e. (UnifOn ` X ) /\ M C_ ( X X. X ) ) /\ ( ( V e. U /\ `' V = V ) /\ z e. ( ( cls ` ( J tX J ) ) ` M ) /\ r e. ( ( ( V " { ( 1st ` z ) } ) X. ( V " { ( 2nd ` z ) } ) ) i^i M ) ) ) -> V e. U )
30	29	3anassrs 1290	. . . . . . . . . 10 |- ( ( ( ( ( U e. (UnifOn ` X ) /\ M C_ ( X X. X ) ) /\ ( V e. U /\ `' V = V ) ) /\ z e. ( ( cls ` ( J tX J ) ) ` M ) ) /\ r e. ( ( ( V " { ( 1st ` z ) } ) X. ( V " { ( 2nd ` z ) } ) ) i^i M ) ) -> V e. U )
31		ustrel 22015	. . . . . . . . . 10 |- ( ( U e. (UnifOn ` X ) /\ V e. U ) -> Rel V )
32	28, 30, 31	syl2anc 693	. . . . . . . . 9 |- ( ( ( ( ( U e. (UnifOn ` X ) /\ M C_ ( X X. X ) ) /\ ( V e. U /\ `' V = V ) ) /\ z e. ( ( cls ` ( J tX J ) ) ` M ) ) /\ r e. ( ( ( V " { ( 1st ` z ) } ) X. ( V " { ( 2nd ` z ) } ) ) i^i M ) ) -> Rel V )
33		simpr 477	. . . . . . . . . . . 12 |- ( ( ( ( ( U e. (UnifOn ` X ) /\ M C_ ( X X. X ) ) /\ ( V e. U /\ `' V = V ) ) /\ z e. ( ( cls ` ( J tX J ) ) ` M ) ) /\ r e. ( ( ( V " { ( 1st ` z ) } ) X. ( V " { ( 2nd ` z ) } ) ) i^i M ) ) -> r e. ( ( ( V " { ( 1st ` z ) } ) X. ( V " { ( 2nd ` z ) } ) ) i^i M ) )
34		elin 3796	. . . . . . . . . . . 12 |- ( r e. ( ( ( V " { ( 1st ` z ) } ) X. ( V " { ( 2nd ` z ) } ) ) i^i M ) <-> ( r e. ( ( V " { ( 1st ` z ) } ) X. ( V " { ( 2nd ` z ) } ) ) /\ r e. M ) )
35	33, 34	sylib 208	. . . . . . . . . . 11 |- ( ( ( ( ( U e. (UnifOn ` X ) /\ M C_ ( X X. X ) ) /\ ( V e. U /\ `' V = V ) ) /\ z e. ( ( cls ` ( J tX J ) ) ` M ) ) /\ r e. ( ( ( V " { ( 1st ` z ) } ) X. ( V " { ( 2nd ` z ) } ) ) i^i M ) ) -> ( r e. ( ( V " { ( 1st ` z ) } ) X. ( V " { ( 2nd ` z ) } ) ) /\ r e. M ) )
36	35	simpld 475	. . . . . . . . . 10 |- ( ( ( ( ( U e. (UnifOn ` X ) /\ M C_ ( X X. X ) ) /\ ( V e. U /\ `' V = V ) ) /\ z e. ( ( cls ` ( J tX J ) ) ` M ) ) /\ r e. ( ( ( V " { ( 1st ` z ) } ) X. ( V " { ( 2nd ` z ) } ) ) i^i M ) ) -> r e. ( ( V " { ( 1st ` z ) } ) X. ( V " { ( 2nd ` z ) } ) ) )
37		xp1st 7198	. . . . . . . . . 10 |- ( r e. ( ( V " { ( 1st ` z ) } ) X. ( V " { ( 2nd ` z ) } ) ) -> ( 1st ` r ) e. ( V " { ( 1st ` z ) } ) )
38	36, 37	syl 17	. . . . . . . . 9 |- ( ( ( ( ( U e. (UnifOn ` X ) /\ M C_ ( X X. X ) ) /\ ( V e. U /\ `' V = V ) ) /\ z e. ( ( cls ` ( J tX J ) ) ` M ) ) /\ r e. ( ( ( V " { ( 1st ` z ) } ) X. ( V " { ( 2nd ` z ) } ) ) i^i M ) ) -> ( 1st ` r ) e. ( V " { ( 1st ` z ) } ) )
39		elrelimasn 5489	. . . . . . . . . 10 |- ( Rel V -> ( ( 1st ` r ) e. ( V " { ( 1st ` z ) } ) <-> ( 1st ` z ) V ( 1st ` r ) ) )
40	39	biimpa 501	. . . . . . . . 9 |- ( ( Rel V /\ ( 1st ` r ) e. ( V " { ( 1st ` z ) } ) ) -> ( 1st ` z ) V ( 1st ` r ) )
41	32, 38, 40	syl2anc 693	. . . . . . . 8 |- ( ( ( ( ( U e. (UnifOn ` X ) /\ M C_ ( X X. X ) ) /\ ( V e. U /\ `' V = V ) ) /\ z e. ( ( cls ` ( J tX J ) ) ` M ) ) /\ r e. ( ( ( V " { ( 1st ` z ) } ) X. ( V " { ( 2nd ` z ) } ) ) i^i M ) ) -> ( 1st ` z ) V ( 1st ` r ) )
42		simp-4r 807	. . . . . . . . . . 11 |- ( ( ( ( ( U e. (UnifOn ` X ) /\ M C_ ( X X. X ) ) /\ ( V e. U /\ `' V = V ) ) /\ z e. ( ( cls ` ( J tX J ) ) ` M ) ) /\ r e. ( ( ( V " { ( 1st ` z ) } ) X. ( V " { ( 2nd ` z ) } ) ) i^i M ) ) -> M C_ ( X X. X ) )
43		xpss 5226	. . . . . . . . . . 11 |- ( X X. X ) C_ ( _V X. _V )
44	42, 43	syl6ss 3615	. . . . . . . . . 10 |- ( ( ( ( ( U e. (UnifOn ` X ) /\ M C_ ( X X. X ) ) /\ ( V e. U /\ `' V = V ) ) /\ z e. ( ( cls ` ( J tX J ) ) ` M ) ) /\ r e. ( ( ( V " { ( 1st ` z ) } ) X. ( V " { ( 2nd ` z ) } ) ) i^i M ) ) -> M C_ ( _V X. _V ) )
45		df-rel 5121	. . . . . . . . . 10 |- ( Rel M <-> M C_ ( _V X. _V ) )
46	44, 45	sylibr 224	. . . . . . . . 9 |- ( ( ( ( ( U e. (UnifOn ` X ) /\ M C_ ( X X. X ) ) /\ ( V e. U /\ `' V = V ) ) /\ z e. ( ( cls ` ( J tX J ) ) ` M ) ) /\ r e. ( ( ( V " { ( 1st ` z ) } ) X. ( V " { ( 2nd ` z ) } ) ) i^i M ) ) -> Rel M )
47	35	simprd 479	. . . . . . . . 9 |- ( ( ( ( ( U e. (UnifOn ` X ) /\ M C_ ( X X. X ) ) /\ ( V e. U /\ `' V = V ) ) /\ z e. ( ( cls ` ( J tX J ) ) ` M ) ) /\ r e. ( ( ( V " { ( 1st ` z ) } ) X. ( V " { ( 2nd ` z ) } ) ) i^i M ) ) -> r e. M )
48		1st2ndbr 7217	. . . . . . . . 9 |- ( ( Rel M /\ r e. M ) -> ( 1st ` r ) M ( 2nd ` r ) )
49	46, 47, 48	syl2anc 693	. . . . . . . 8 |- ( ( ( ( ( U e. (UnifOn ` X ) /\ M C_ ( X X. X ) ) /\ ( V e. U /\ `' V = V ) ) /\ z e. ( ( cls ` ( J tX J ) ) ` M ) ) /\ r e. ( ( ( V " { ( 1st ` z ) } ) X. ( V " { ( 2nd ` z ) } ) ) i^i M ) ) -> ( 1st ` r ) M ( 2nd ` r ) )
50		xp2nd 7199	. . . . . . . . . . 11 |- ( r e. ( ( V " { ( 1st ` z ) } ) X. ( V " { ( 2nd ` z ) } ) ) -> ( 2nd ` r ) e. ( V " { ( 2nd ` z ) } ) )
51	36, 50	syl 17	. . . . . . . . . 10 |- ( ( ( ( ( U e. (UnifOn ` X ) /\ M C_ ( X X. X ) ) /\ ( V e. U /\ `' V = V ) ) /\ z e. ( ( cls ` ( J tX J ) ) ` M ) ) /\ r e. ( ( ( V " { ( 1st ` z ) } ) X. ( V " { ( 2nd ` z ) } ) ) i^i M ) ) -> ( 2nd ` r ) e. ( V " { ( 2nd ` z ) } ) )
52		elrelimasn 5489	. . . . . . . . . . 11 |- ( Rel V -> ( ( 2nd ` r ) e. ( V " { ( 2nd ` z ) } ) <-> ( 2nd ` z ) V ( 2nd ` r ) ) )
53	52	biimpa 501	. . . . . . . . . 10 |- ( ( Rel V /\ ( 2nd ` r ) e. ( V " { ( 2nd ` z ) } ) ) -> ( 2nd ` z ) V ( 2nd ` r ) )
54	32, 51, 53	syl2anc 693	. . . . . . . . 9 |- ( ( ( ( ( U e. (UnifOn ` X ) /\ M C_ ( X X. X ) ) /\ ( V e. U /\ `' V = V ) ) /\ z e. ( ( cls ` ( J tX J ) ) ` M ) ) /\ r e. ( ( ( V " { ( 1st ` z ) } ) X. ( V " { ( 2nd ` z ) } ) ) i^i M ) ) -> ( 2nd ` z ) V ( 2nd ` r ) )
55		simpr1r 1119	. . . . . . . . . . 11 |- ( ( ( U e. (UnifOn ` X ) /\ M C_ ( X X. X ) ) /\ ( ( V e. U /\ `' V = V ) /\ z e. ( ( cls ` ( J tX J ) ) ` M ) /\ r e. ( ( ( V " { ( 1st ` z ) } ) X. ( V " { ( 2nd ` z ) } ) ) i^i M ) ) ) -> `' V = V )
56	55	3anassrs 1290	. . . . . . . . . 10 |- ( ( ( ( ( U e. (UnifOn ` X ) /\ M C_ ( X X. X ) ) /\ ( V e. U /\ `' V = V ) ) /\ z e. ( ( cls ` ( J tX J ) ) ` M ) ) /\ r e. ( ( ( V " { ( 1st ` z ) } ) X. ( V " { ( 2nd ` z ) } ) ) i^i M ) ) -> `' V = V )
57		fvex 6201	. . . . . . . . . . . 12 |- ( 2nd ` r ) e. _V
58		fvex 6201	. . . . . . . . . . . 12 |- ( 2nd ` z ) e. _V
59	57, 58	brcnv 5305	. . . . . . . . . . 11 |- ( ( 2nd ` r ) `' V ( 2nd ` z ) <-> ( 2nd ` z ) V ( 2nd ` r ) )
60		breq 4655	. . . . . . . . . . 11 |- ( `' V = V -> ( ( 2nd ` r ) `' V ( 2nd ` z ) <-> ( 2nd ` r ) V ( 2nd ` z ) ) )
61	59, 60	syl5rbbr 275	. . . . . . . . . 10 |- ( `' V = V -> ( ( 2nd ` r ) V ( 2nd ` z ) <-> ( 2nd ` z ) V ( 2nd ` r ) ) )
62	56, 61	syl 17	. . . . . . . . 9 |- ( ( ( ( ( U e. (UnifOn ` X ) /\ M C_ ( X X. X ) ) /\ ( V e. U /\ `' V = V ) ) /\ z e. ( ( cls ` ( J tX J ) ) ` M ) ) /\ r e. ( ( ( V " { ( 1st ` z ) } ) X. ( V " { ( 2nd ` z ) } ) ) i^i M ) ) -> ( ( 2nd ` r ) V ( 2nd ` z ) <-> ( 2nd ` z ) V ( 2nd ` r ) ) )
63	54, 62	mpbird 247	. . . . . . . 8 |- ( ( ( ( ( U e. (UnifOn ` X ) /\ M C_ ( X X. X ) ) /\ ( V e. U /\ `' V = V ) ) /\ z e. ( ( cls ` ( J tX J ) ) ` M ) ) /\ r e. ( ( ( V " { ( 1st ` z ) } ) X. ( V " { ( 2nd ` z ) } ) ) i^i M ) ) -> ( 2nd ` r ) V ( 2nd ` z ) )
64		fvex 6201	. . . . . . . . . 10 |- ( 1st ` z ) e. _V
65		fvex 6201	. . . . . . . . . 10 |- ( 1st ` r ) e. _V
66		brcogw 5290	. . . . . . . . . . 11 |- ( ( ( ( 1st ` z ) e. _V /\ ( 2nd ` r ) e. _V /\ ( 1st ` r ) e. _V ) /\ ( ( 1st ` z ) V ( 1st ` r ) /\ ( 1st ` r ) M ( 2nd ` r ) ) ) -> ( 1st ` z ) ( M o. V ) ( 2nd ` r ) )
67	66	ex 450	. . . . . . . . . 10 |- ( ( ( 1st ` z ) e. _V /\ ( 2nd ` r ) e. _V /\ ( 1st ` r ) e. _V ) -> ( ( ( 1st ` z ) V ( 1st ` r ) /\ ( 1st ` r ) M ( 2nd ` r ) ) -> ( 1st ` z ) ( M o. V ) ( 2nd ` r ) ) )
68	64, 57, 65, 67	mp3an 1424	. . . . . . . . 9 |- ( ( ( 1st ` z ) V ( 1st ` r ) /\ ( 1st ` r ) M ( 2nd ` r ) ) -> ( 1st ` z ) ( M o. V ) ( 2nd ` r ) )
69		brcogw 5290	. . . . . . . . . . 11 |- ( ( ( ( 1st ` z ) e. _V /\ ( 2nd ` z ) e. _V /\ ( 2nd ` r ) e. _V ) /\ ( ( 1st ` z ) ( M o. V ) ( 2nd ` r ) /\ ( 2nd ` r ) V ( 2nd ` z ) ) ) -> ( 1st ` z ) ( V o. ( M o. V ) ) ( 2nd ` z ) )
70	69	ex 450	. . . . . . . . . 10 |- ( ( ( 1st ` z ) e. _V /\ ( 2nd ` z ) e. _V /\ ( 2nd ` r ) e. _V ) -> ( ( ( 1st ` z ) ( M o. V ) ( 2nd ` r ) /\ ( 2nd ` r ) V ( 2nd ` z ) ) -> ( 1st ` z ) ( V o. ( M o. V ) ) ( 2nd ` z ) ) )
71	64, 58, 57, 70	mp3an 1424	. . . . . . . . 9 |- ( ( ( 1st ` z ) ( M o. V ) ( 2nd ` r ) /\ ( 2nd ` r ) V ( 2nd ` z ) ) -> ( 1st ` z ) ( V o. ( M o. V ) ) ( 2nd ` z ) )
72	68, 71	sylan 488	. . . . . . . 8 |- ( ( ( ( 1st ` z ) V ( 1st ` r ) /\ ( 1st ` r ) M ( 2nd ` r ) ) /\ ( 2nd ` r ) V ( 2nd ` z ) ) -> ( 1st ` z ) ( V o. ( M o. V ) ) ( 2nd ` z ) )
73	41, 49, 63, 72	syl21anc 1325	. . . . . . 7 |- ( ( ( ( ( U e. (UnifOn ` X ) /\ M C_ ( X X. X ) ) /\ ( V e. U /\ `' V = V ) ) /\ z e. ( ( cls ` ( J tX J ) ) ` M ) ) /\ r e. ( ( ( V " { ( 1st ` z ) } ) X. ( V " { ( 2nd ` z ) } ) ) i^i M ) ) -> ( 1st ` z ) ( V o. ( M o. V ) ) ( 2nd ` z ) )
74	73	ralrimiva 2966	. . . . . 6 |- ( ( ( ( U e. (UnifOn ` X ) /\ M C_ ( X X. X ) ) /\ ( V e. U /\ `' V = V ) ) /\ z e. ( ( cls ` ( J tX J ) ) ` M ) ) -> A. r e. ( ( ( V " { ( 1st ` z ) } ) X. ( V " { ( 2nd ` z ) } ) ) i^i M ) ( 1st ` z ) ( V o. ( M o. V ) ) ( 2nd ` z ) )
75		simplll 798	. . . . . . . . 9 |- ( ( ( ( U e. (UnifOn ` X ) /\ M C_ ( X X. X ) ) /\ ( V e. U /\ `' V = V ) ) /\ z e. ( ( cls ` ( J tX J ) ) ` M ) ) -> U e. (UnifOn ` X ) )
76		simplrl 800	. . . . . . . . 9 |- ( ( ( ( U e. (UnifOn ` X ) /\ M C_ ( X X. X ) ) /\ ( V e. U /\ `' V = V ) ) /\ z e. ( ( cls ` ( J tX J ) ) ` M ) ) -> V e. U )
77	4	3ad2ant1 1082	. . . . . . . . . . 11 |- ( ( U e. (UnifOn ` X ) /\ V e. U /\ z e. ( X X. X ) ) -> J e. Top )
78		xp1st 7198	. . . . . . . . . . . 12 |- ( z e. ( X X. X ) -> ( 1st ` z ) e. X )
79	2	utopsnnei 22053	. . . . . . . . . . . 12 |- ( ( U e. (UnifOn ` X ) /\ V e. U /\ ( 1st ` z ) e. X ) -> ( V " { ( 1st ` z ) } ) e. ( ( nei ` J ) ` { ( 1st ` z ) } ) )
80	78, 79	syl3an3 1361	. . . . . . . . . . 11 |- ( ( U e. (UnifOn ` X ) /\ V e. U /\ z e. ( X X. X ) ) -> ( V " { ( 1st ` z ) } ) e. ( ( nei ` J ) ` { ( 1st ` z ) } ) )
81		xp2nd 7199	. . . . . . . . . . . 12 |- ( z e. ( X X. X ) -> ( 2nd ` z ) e. X )
82	2	utopsnnei 22053	. . . . . . . . . . . 12 |- ( ( U e. (UnifOn ` X ) /\ V e. U /\ ( 2nd ` z ) e. X ) -> ( V " { ( 2nd ` z ) } ) e. ( ( nei ` J ) ` { ( 2nd ` z ) } ) )
83	81, 82	syl3an3 1361	. . . . . . . . . . 11 |- ( ( U e. (UnifOn ` X ) /\ V e. U /\ z e. ( X X. X ) ) -> ( V " { ( 2nd ` z ) } ) e. ( ( nei ` J ) ` { ( 2nd ` z ) } ) )
84	14, 14	neitx 21410	. . . . . . . . . . 11 |- ( ( ( J e. Top /\ J e. Top ) /\ ( ( V " { ( 1st ` z ) } ) e. ( ( nei ` J ) ` { ( 1st ` z ) } ) /\ ( V " { ( 2nd ` z ) } ) e. ( ( nei ` J ) ` { ( 2nd ` z ) } ) ) ) -> ( ( V " { ( 1st ` z ) } ) X. ( V " { ( 2nd ` z ) } ) ) e. ( ( nei ` ( J tX J ) ) ` ( { ( 1st ` z ) } X. { ( 2nd ` z ) } ) ) )
85	77, 77, 80, 83, 84	syl22anc 1327	. . . . . . . . . 10 |- ( ( U e. (UnifOn ` X ) /\ V e. U /\ z e. ( X X. X ) ) -> ( ( V " { ( 1st ` z ) } ) X. ( V " { ( 2nd ` z ) } ) ) e. ( ( nei ` ( J tX J ) ) ` ( { ( 1st ` z ) } X. { ( 2nd ` z ) } ) ) )
86		1st2nd2 7205	. . . . . . . . . . . . . 14 |- ( z e. ( X X. X ) -> z = <. ( 1st ` z ) , ( 2nd ` z ) >. )
87	86	sneqd 4189	. . . . . . . . . . . . 13 |- ( z e. ( X X. X ) -> { z } = { <. ( 1st ` z ) , ( 2nd ` z ) >. } )
88	64, 58	xpsn 6407	. . . . . . . . . . . . 13 |- ( { ( 1st ` z ) } X. { ( 2nd ` z ) } ) = { <. ( 1st ` z ) , ( 2nd ` z ) >. }
89	87, 88	syl6eqr 2674	. . . . . . . . . . . 12 |- ( z e. ( X X. X ) -> { z } = ( { ( 1st ` z ) } X. { ( 2nd ` z ) } ) )
90	89	fveq2d 6195	. . . . . . . . . . 11 |- ( z e. ( X X. X ) -> ( ( nei ` ( J tX J ) ) ` { z } ) = ( ( nei ` ( J tX J ) ) ` ( { ( 1st ` z ) } X. { ( 2nd ` z ) } ) ) )
91	90	3ad2ant3 1084	. . . . . . . . . 10 |- ( ( U e. (UnifOn ` X ) /\ V e. U /\ z e. ( X X. X ) ) -> ( ( nei ` ( J tX J ) ) ` { z } ) = ( ( nei ` ( J tX J ) ) ` ( { ( 1st ` z ) } X. { ( 2nd ` z ) } ) ) )
92	85, 91	eleqtrrd 2704	. . . . . . . . 9 |- ( ( U e. (UnifOn ` X ) /\ V e. U /\ z e. ( X X. X ) ) -> ( ( V " { ( 1st ` z ) } ) X. ( V " { ( 2nd ` z ) } ) ) e. ( ( nei ` ( J tX J ) ) ` { z } ) )
93	75, 76, 25, 92	syl3anc 1326	. . . . . . . 8 |- ( ( ( ( U e. (UnifOn ` X ) /\ M C_ ( X X. X ) ) /\ ( V e. U /\ `' V = V ) ) /\ z e. ( ( cls ` ( J tX J ) ) ` M ) ) -> ( ( V " { ( 1st ` z ) } ) X. ( V " { ( 2nd ` z ) } ) ) e. ( ( nei ` ( J tX J ) ) ` { z } ) )
94	20	neindisj 20921	. . . . . . . 8 |- ( ( ( ( J tX J ) e. Top /\ M C_ U. ( J tX J ) ) /\ ( z e. ( ( cls ` ( J tX J ) ) ` M ) /\ ( ( V " { ( 1st ` z ) } ) X. ( V " { ( 2nd ` z ) } ) ) e. ( ( nei ` ( J tX J ) ) ` { z } ) ) ) -> ( ( ( V " { ( 1st ` z ) } ) X. ( V " { ( 2nd ` z ) } ) ) i^i M ) =/= (/) )
95	7, 19, 24, 93, 94	syl22anc 1327	. . . . . . 7 |- ( ( ( ( U e. (UnifOn ` X ) /\ M C_ ( X X. X ) ) /\ ( V e. U /\ `' V = V ) ) /\ z e. ( ( cls ` ( J tX J ) ) ` M ) ) -> ( ( ( V " { ( 1st ` z ) } ) X. ( V " { ( 2nd ` z ) } ) ) i^i M ) =/= (/) )
96		r19.3rzv 4064	. . . . . . 7 |- ( ( ( ( V " { ( 1st ` z ) } ) X. ( V " { ( 2nd ` z ) } ) ) i^i M ) =/= (/) -> ( ( 1st ` z ) ( V o. ( M o. V ) ) ( 2nd ` z ) <-> A. r e. ( ( ( V " { ( 1st ` z ) } ) X. ( V " { ( 2nd ` z ) } ) ) i^i M ) ( 1st ` z ) ( V o. ( M o. V ) ) ( 2nd ` z ) ) )
97	95, 96	syl 17	. . . . . 6 |- ( ( ( ( U e. (UnifOn ` X ) /\ M C_ ( X X. X ) ) /\ ( V e. U /\ `' V = V ) ) /\ z e. ( ( cls ` ( J tX J ) ) ` M ) ) -> ( ( 1st ` z ) ( V o. ( M o. V ) ) ( 2nd ` z ) <-> A. r e. ( ( ( V " { ( 1st ` z ) } ) X. ( V " { ( 2nd ` z ) } ) ) i^i M ) ( 1st ` z ) ( V o. ( M o. V ) ) ( 2nd ` z ) ) )
98	74, 97	mpbird 247	. . . . 5 |- ( ( ( ( U e. (UnifOn ` X ) /\ M C_ ( X X. X ) ) /\ ( V e. U /\ `' V = V ) ) /\ z e. ( ( cls ` ( J tX J ) ) ` M ) ) -> ( 1st ` z ) ( V o. ( M o. V ) ) ( 2nd ` z ) )
99		df-br 4654	. . . . 5 |- ( ( 1st ` z ) ( V o. ( M o. V ) ) ( 2nd ` z ) <-> <. ( 1st ` z ) , ( 2nd ` z ) >. e. ( V o. ( M o. V ) ) )
100	98, 99	sylib 208	. . . 4 |- ( ( ( ( U e. (UnifOn ` X ) /\ M C_ ( X X. X ) ) /\ ( V e. U /\ `' V = V ) ) /\ z e. ( ( cls ` ( J tX J ) ) ` M ) ) -> <. ( 1st ` z ) , ( 2nd ` z ) >. e. ( V o. ( M o. V ) ) )
101	27, 100	eqeltrd 2701	. . 3 |- ( ( ( ( U e. (UnifOn ` X ) /\ M C_ ( X X. X ) ) /\ ( V e. U /\ `' V = V ) ) /\ z e. ( ( cls ` ( J tX J ) ) ` M ) ) -> z e. ( V o. ( M o. V ) ) )
102	101	ex 450	. 2 |- ( ( ( U e. (UnifOn ` X ) /\ M C_ ( X X. X ) ) /\ ( V e. U /\ `' V = V ) ) -> ( z e. ( ( cls ` ( J tX J ) ) ` M ) -> z e. ( V o. ( M o. V ) ) ) )
103	102	ssrdv 3609	1 |- ( ( ( U e. (UnifOn ` X ) /\ M C_ ( X X. X ) ) /\ ( V e. U /\ `' V = V ) ) -> ( ( cls ` ( J tX J ) ) ` M ) C_ ( V o. ( M o. V ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   _Vcvv 3200   i^i cin 3573   C_ wss 3574   (/)c0 3915   {csn 4177   <.cop 4183   U.cuni 4436   class class class wbr 4653   X. cxp 5112   `'ccnv 5113   "cima 5117   o. ccom 5118   Rel wrel 5119   ` cfv 5888  (class class class)co 6650   1stc1st 7166   2ndc2nd 7167   Topctop 20698  TopOnctopon 20715   clsccl 20822   neicnei 20901   tX ctx 21363  UnifOncust 22003  unifTopcutop 22034

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-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  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-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  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-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  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-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  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-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-fin 7959  df-fi 8317  df-topgen 16104  df-top 20699  df-topon 20716  df-bases 20750  df-cld 20823  df-ntr 20824  df-cls 20825  df-nei 20902  df-tx 21365  df-ust 22004  df-utop 22035

This theorem is referenced by:  utopreg  22056