Theorem utop2nei 22054

Description: For any symmetrical entourage V and any relation M, build a neighborhood of M. First part of proposition 2 of [BourbakiTop1] p. II.4. (Contributed by Thierry Arnoux, 14-Jan-2018.)

Hypothesis

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

Assertion

Ref	Expression
utop2nei	|- ( ( U e. (UnifOn ` X ) /\ ( V e. U /\ `' V = V ) /\ M C_ ( X X. X ) ) -> ( V o. ( M o. V ) ) e. ( ( nei ` ( J tX J ) ) ` M ) )

Proof of Theorem utop2nei

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

Step	Hyp	Ref	Expression
1		utoptop.1	. . . . . . . 8 |- J = (unifTop ` U )
2		utoptop 22038	. . . . . . . 8 |- ( U e. (UnifOn ` X ) -> (unifTop ` U ) e. Top )
3	1, 2	syl5eqel 2705	. . . . . . 7 |- ( U e. (UnifOn ` X ) -> J e. Top )
4		txtop 21372	. . . . . . 7 |- ( ( J e. Top /\ J e. Top ) -> ( J tX J ) e. Top )
5	3, 3, 4	syl2anc 693	. . . . . 6 |- ( U e. (UnifOn ` X ) -> ( J tX J ) e. Top )
6	5	3ad2ant1 1082	. . . . 5 |- ( ( U e. (UnifOn ` X ) /\ ( V e. U /\ `' V = V ) /\ M C_ ( X X. X ) ) -> ( J tX J ) e. Top )
7	6	adantr 481	. . . 4 |- ( ( ( U e. (UnifOn ` X ) /\ ( V e. U /\ `' V = V ) /\ M C_ ( X X. X ) ) /\ M = (/) ) -> ( J tX J ) e. Top )
8		0nei 20932	. . . 4 |- ( ( J tX J ) e. Top -> (/) e. ( ( nei ` ( J tX J ) ) ` (/) ) )
9	7, 8	syl 17	. . 3 |- ( ( ( U e. (UnifOn ` X ) /\ ( V e. U /\ `' V = V ) /\ M C_ ( X X. X ) ) /\ M = (/) ) -> (/) e. ( ( nei ` ( J tX J ) ) ` (/) ) )
10		coeq1 5279	. . . . . . 7 |- ( M = (/) -> ( M o. V ) = ( (/) o. V ) )
11		co01 5650	. . . . . . 7 |- ( (/) o. V ) = (/)
12	10, 11	syl6eq 2672	. . . . . 6 |- ( M = (/) -> ( M o. V ) = (/) )
13	12	coeq2d 5284	. . . . 5 |- ( M = (/) -> ( V o. ( M o. V ) ) = ( V o. (/) ) )
14		co02 5649	. . . . 5 |- ( V o. (/) ) = (/)
15	13, 14	syl6eq 2672	. . . 4 |- ( M = (/) -> ( V o. ( M o. V ) ) = (/) )
16	15	adantl 482	. . 3 |- ( ( ( U e. (UnifOn ` X ) /\ ( V e. U /\ `' V = V ) /\ M C_ ( X X. X ) ) /\ M = (/) ) -> ( V o. ( M o. V ) ) = (/) )
17		simpr 477	. . . 4 |- ( ( ( U e. (UnifOn ` X ) /\ ( V e. U /\ `' V = V ) /\ M C_ ( X X. X ) ) /\ M = (/) ) -> M = (/) )
18	17	fveq2d 6195	. . 3 |- ( ( ( U e. (UnifOn ` X ) /\ ( V e. U /\ `' V = V ) /\ M C_ ( X X. X ) ) /\ M = (/) ) -> ( ( nei ` ( J tX J ) ) ` M ) = ( ( nei ` ( J tX J ) ) ` (/) ) )
19	9, 16, 18	3eltr4d 2716	. 2 |- ( ( ( U e. (UnifOn ` X ) /\ ( V e. U /\ `' V = V ) /\ M C_ ( X X. X ) ) /\ M = (/) ) -> ( V o. ( M o. V ) ) e. ( ( nei ` ( J tX J ) ) ` M ) )
20	6	adantr 481	. . . . . 6 |- ( ( ( U e. (UnifOn ` X ) /\ ( V e. U /\ `' V = V ) /\ M C_ ( X X. X ) ) /\ r e. M ) -> ( J tX J ) e. Top )
21		simpl1 1064	. . . . . . . . . 10 |- ( ( ( U e. (UnifOn ` X ) /\ ( V e. U /\ `' V = V ) /\ M C_ ( X X. X ) ) /\ r e. M ) -> U e. (UnifOn ` X ) )
22	21, 3	syl 17	. . . . . . . . 9 |- ( ( ( U e. (UnifOn ` X ) /\ ( V e. U /\ `' V = V ) /\ M C_ ( X X. X ) ) /\ r e. M ) -> J e. Top )
23		simpl2l 1114	. . . . . . . . . 10 |- ( ( ( U e. (UnifOn ` X ) /\ ( V e. U /\ `' V = V ) /\ M C_ ( X X. X ) ) /\ r e. M ) -> V e. U )
24		simp3 1063	. . . . . . . . . . . 12 |- ( ( U e. (UnifOn ` X ) /\ ( V e. U /\ `' V = V ) /\ M C_ ( X X. X ) ) -> M C_ ( X X. X ) )
25	24	sselda 3603	. . . . . . . . . . 11 |- ( ( ( U e. (UnifOn ` X ) /\ ( V e. U /\ `' V = V ) /\ M C_ ( X X. X ) ) /\ r e. M ) -> r e. ( X X. X ) )
26		xp1st 7198	. . . . . . . . . . 11 |- ( r e. ( X X. X ) -> ( 1st ` r ) e. X )
27	25, 26	syl 17	. . . . . . . . . 10 |- ( ( ( U e. (UnifOn ` X ) /\ ( V e. U /\ `' V = V ) /\ M C_ ( X X. X ) ) /\ r e. M ) -> ( 1st ` r ) e. X )
28	1	utopsnnei 22053	. . . . . . . . . 10 |- ( ( U e. (UnifOn ` X ) /\ V e. U /\ ( 1st ` r ) e. X ) -> ( V " { ( 1st ` r ) } ) e. ( ( nei ` J ) ` { ( 1st ` r ) } ) )
29	21, 23, 27, 28	syl3anc 1326	. . . . . . . . 9 |- ( ( ( U e. (UnifOn ` X ) /\ ( V e. U /\ `' V = V ) /\ M C_ ( X X. X ) ) /\ r e. M ) -> ( V " { ( 1st ` r ) } ) e. ( ( nei ` J ) ` { ( 1st ` r ) } ) )
30		xp2nd 7199	. . . . . . . . . . 11 |- ( r e. ( X X. X ) -> ( 2nd ` r ) e. X )
31	25, 30	syl 17	. . . . . . . . . 10 |- ( ( ( U e. (UnifOn ` X ) /\ ( V e. U /\ `' V = V ) /\ M C_ ( X X. X ) ) /\ r e. M ) -> ( 2nd ` r ) e. X )
32	1	utopsnnei 22053	. . . . . . . . . 10 |- ( ( U e. (UnifOn ` X ) /\ V e. U /\ ( 2nd ` r ) e. X ) -> ( V " { ( 2nd ` r ) } ) e. ( ( nei ` J ) ` { ( 2nd ` r ) } ) )
33	21, 23, 31, 32	syl3anc 1326	. . . . . . . . 9 |- ( ( ( U e. (UnifOn ` X ) /\ ( V e. U /\ `' V = V ) /\ M C_ ( X X. X ) ) /\ r e. M ) -> ( V " { ( 2nd ` r ) } ) e. ( ( nei ` J ) ` { ( 2nd ` r ) } ) )
34		eqid 2622	. . . . . . . . . 10 |- U. J = U. J
35	34, 34	neitx 21410	. . . . . . . . 9 |- ( ( ( J e. Top /\ J e. Top ) /\ ( ( V " { ( 1st ` r ) } ) e. ( ( nei ` J ) ` { ( 1st ` r ) } ) /\ ( V " { ( 2nd ` r ) } ) e. ( ( nei ` J ) ` { ( 2nd ` r ) } ) ) ) -> ( ( V " { ( 1st ` r ) } ) X. ( V " { ( 2nd ` r ) } ) ) e. ( ( nei ` ( J tX J ) ) ` ( { ( 1st ` r ) } X. { ( 2nd ` r ) } ) ) )
36	22, 22, 29, 33, 35	syl22anc 1327	. . . . . . . 8 |- ( ( ( U e. (UnifOn ` X ) /\ ( V e. U /\ `' V = V ) /\ M C_ ( X X. X ) ) /\ r e. M ) -> ( ( V " { ( 1st ` r ) } ) X. ( V " { ( 2nd ` r ) } ) ) e. ( ( nei ` ( J tX J ) ) ` ( { ( 1st ` r ) } X. { ( 2nd ` r ) } ) ) )
37		fvex 6201	. . . . . . . . . 10 |- ( 1st ` r ) e. _V
38		fvex 6201	. . . . . . . . . 10 |- ( 2nd ` r ) e. _V
39	37, 38	xpsn 6407	. . . . . . . . 9 |- ( { ( 1st ` r ) } X. { ( 2nd ` r ) } ) = { <. ( 1st ` r ) , ( 2nd ` r ) >. }
40	39	fveq2i 6194	. . . . . . . 8 |- ( ( nei ` ( J tX J ) ) ` ( { ( 1st ` r ) } X. { ( 2nd ` r ) } ) ) = ( ( nei ` ( J tX J ) ) ` { <. ( 1st ` r ) , ( 2nd ` r ) >. } )
41	36, 40	syl6eleq 2711	. . . . . . 7 |- ( ( ( U e. (UnifOn ` X ) /\ ( V e. U /\ `' V = V ) /\ M C_ ( X X. X ) ) /\ r e. M ) -> ( ( V " { ( 1st ` r ) } ) X. ( V " { ( 2nd ` r ) } ) ) e. ( ( nei ` ( J tX J ) ) ` { <. ( 1st ` r ) , ( 2nd ` r ) >. } ) )
42	24	adantr 481	. . . . . . . . . . 11 |- ( ( ( U e. (UnifOn ` X ) /\ ( V e. U /\ `' V = V ) /\ M C_ ( X X. X ) ) /\ r e. M ) -> M C_ ( X X. X ) )
43		xpss 5226	. . . . . . . . . . . . 13 |- ( X X. X ) C_ ( _V X. _V )
44		sstr 3611	. . . . . . . . . . . . 13 |- ( ( M C_ ( X X. X ) /\ ( X X. X ) C_ ( _V X. _V ) ) -> M C_ ( _V X. _V ) )
45	43, 44	mpan2 707	. . . . . . . . . . . 12 |- ( M C_ ( X X. X ) -> M C_ ( _V X. _V ) )
46		df-rel 5121	. . . . . . . . . . . 12 |- ( Rel M <-> M C_ ( _V X. _V ) )
47	45, 46	sylibr 224	. . . . . . . . . . 11 |- ( M C_ ( X X. X ) -> Rel M )
48	42, 47	syl 17	. . . . . . . . . 10 |- ( ( ( U e. (UnifOn ` X ) /\ ( V e. U /\ `' V = V ) /\ M C_ ( X X. X ) ) /\ r e. M ) -> Rel M )
49		1st2nd 7214	. . . . . . . . . 10 |- ( ( Rel M /\ r e. M ) -> r = <. ( 1st ` r ) , ( 2nd ` r ) >. )
50	48, 49	sylancom 701	. . . . . . . . 9 |- ( ( ( U e. (UnifOn ` X ) /\ ( V e. U /\ `' V = V ) /\ M C_ ( X X. X ) ) /\ r e. M ) -> r = <. ( 1st ` r ) , ( 2nd ` r ) >. )
51	50	sneqd 4189	. . . . . . . 8 |- ( ( ( U e. (UnifOn ` X ) /\ ( V e. U /\ `' V = V ) /\ M C_ ( X X. X ) ) /\ r e. M ) -> { r } = { <. ( 1st ` r ) , ( 2nd ` r ) >. } )
52	51	fveq2d 6195	. . . . . . 7 |- ( ( ( U e. (UnifOn ` X ) /\ ( V e. U /\ `' V = V ) /\ M C_ ( X X. X ) ) /\ r e. M ) -> ( ( nei ` ( J tX J ) ) ` { r } ) = ( ( nei ` ( J tX J ) ) ` { <. ( 1st ` r ) , ( 2nd ` r ) >. } ) )
53	41, 52	eleqtrrd 2704	. . . . . 6 |- ( ( ( U e. (UnifOn ` X ) /\ ( V e. U /\ `' V = V ) /\ M C_ ( X X. X ) ) /\ r e. M ) -> ( ( V " { ( 1st ` r ) } ) X. ( V " { ( 2nd ` r ) } ) ) e. ( ( nei ` ( J tX J ) ) ` { r } ) )
54		relxp 5227	. . . . . . . . . . 11 |- Rel ( ( V " { ( 1st ` r ) } ) X. ( V " { ( 2nd ` r ) } ) )
55	54	a1i 11	. . . . . . . . . 10 |- ( ( ( ( U e. (UnifOn ` X ) /\ ( V e. U /\ `' V = V ) /\ M C_ ( X X. X ) ) /\ r e. M ) /\ z e. ( ( V " { ( 1st ` r ) } ) X. ( V " { ( 2nd ` r ) } ) ) ) -> Rel ( ( V " { ( 1st ` r ) } ) X. ( V " { ( 2nd ` r ) } ) ) )
56		1st2nd 7214	. . . . . . . . . 10 |- ( ( Rel ( ( V " { ( 1st ` r ) } ) X. ( V " { ( 2nd ` r ) } ) ) /\ z e. ( ( V " { ( 1st ` r ) } ) X. ( V " { ( 2nd ` r ) } ) ) ) -> z = <. ( 1st ` z ) , ( 2nd ` z ) >. )
57	55, 56	sylancom 701	. . . . . . . . 9 |- ( ( ( ( U e. (UnifOn ` X ) /\ ( V e. U /\ `' V = V ) /\ M C_ ( X X. X ) ) /\ r e. M ) /\ z e. ( ( V " { ( 1st ` r ) } ) X. ( V " { ( 2nd ` r ) } ) ) ) -> z = <. ( 1st ` z ) , ( 2nd ` z ) >. )
58		simpll2 1101	. . . . . . . . . . . . 13 |- ( ( ( ( U e. (UnifOn ` X ) /\ ( V e. U /\ `' V = V ) /\ M C_ ( X X. X ) ) /\ r e. M ) /\ z e. ( ( V " { ( 1st ` r ) } ) X. ( V " { ( 2nd ` r ) } ) ) ) -> ( V e. U /\ `' V = V ) )
59	58	simprd 479	. . . . . . . . . . . 12 |- ( ( ( ( U e. (UnifOn ` X ) /\ ( V e. U /\ `' V = V ) /\ M C_ ( X X. X ) ) /\ r e. M ) /\ z e. ( ( V " { ( 1st ` r ) } ) X. ( V " { ( 2nd ` r ) } ) ) ) -> `' V = V )
60		simpll1 1100	. . . . . . . . . . . . . 14 |- ( ( ( ( U e. (UnifOn ` X ) /\ ( V e. U /\ `' V = V ) /\ M C_ ( X X. X ) ) /\ r e. M ) /\ z e. ( ( V " { ( 1st ` r ) } ) X. ( V " { ( 2nd ` r ) } ) ) ) -> U e. (UnifOn ` X ) )
61	58	simpld 475	. . . . . . . . . . . . . 14 |- ( ( ( ( U e. (UnifOn ` X ) /\ ( V e. U /\ `' V = V ) /\ M C_ ( X X. X ) ) /\ r e. M ) /\ z e. ( ( V " { ( 1st ` r ) } ) X. ( V " { ( 2nd ` r ) } ) ) ) -> V e. U )
62		ustrel 22015	. . . . . . . . . . . . . 14 |- ( ( U e. (UnifOn ` X ) /\ V e. U ) -> Rel V )
63	60, 61, 62	syl2anc 693	. . . . . . . . . . . . 13 |- ( ( ( ( U e. (UnifOn ` X ) /\ ( V e. U /\ `' V = V ) /\ M C_ ( X X. X ) ) /\ r e. M ) /\ z e. ( ( V " { ( 1st ` r ) } ) X. ( V " { ( 2nd ` r ) } ) ) ) -> Rel V )
64		xp1st 7198	. . . . . . . . . . . . . 14 |- ( z e. ( ( V " { ( 1st ` r ) } ) X. ( V " { ( 2nd ` r ) } ) ) -> ( 1st ` z ) e. ( V " { ( 1st ` r ) } ) )
65	64	adantl 482	. . . . . . . . . . . . 13 |- ( ( ( ( U e. (UnifOn ` X ) /\ ( V e. U /\ `' V = V ) /\ M C_ ( X X. X ) ) /\ r e. M ) /\ z e. ( ( V " { ( 1st ` r ) } ) X. ( V " { ( 2nd ` r ) } ) ) ) -> ( 1st ` z ) e. ( V " { ( 1st ` r ) } ) )
66		elrelimasn 5489	. . . . . . . . . . . . . 14 |- ( Rel V -> ( ( 1st ` z ) e. ( V " { ( 1st ` r ) } ) <-> ( 1st ` r ) V ( 1st ` z ) ) )
67	66	biimpa 501	. . . . . . . . . . . . 13 |- ( ( Rel V /\ ( 1st ` z ) e. ( V " { ( 1st ` r ) } ) ) -> ( 1st ` r ) V ( 1st ` z ) )
68	63, 65, 67	syl2anc 693	. . . . . . . . . . . 12 |- ( ( ( ( U e. (UnifOn ` X ) /\ ( V e. U /\ `' V = V ) /\ M C_ ( X X. X ) ) /\ r e. M ) /\ z e. ( ( V " { ( 1st ` r ) } ) X. ( V " { ( 2nd ` r ) } ) ) ) -> ( 1st ` r ) V ( 1st ` z ) )
69		fvex 6201	. . . . . . . . . . . . . . 15 |- ( 1st ` z ) e. _V
70	37, 69	brcnv 5305	. . . . . . . . . . . . . 14 |- ( ( 1st ` r ) `' V ( 1st ` z ) <-> ( 1st ` z ) V ( 1st ` r ) )
71		breq 4655	. . . . . . . . . . . . . 14 |- ( `' V = V -> ( ( 1st ` r ) `' V ( 1st ` z ) <-> ( 1st ` r ) V ( 1st ` z ) ) )
72	70, 71	syl5bbr 274	. . . . . . . . . . . . 13 |- ( `' V = V -> ( ( 1st ` z ) V ( 1st ` r ) <-> ( 1st ` r ) V ( 1st ` z ) ) )
73	72	biimpar 502	. . . . . . . . . . . 12 |- ( ( `' V = V /\ ( 1st ` r ) V ( 1st ` z ) ) -> ( 1st ` z ) V ( 1st ` r ) )
74	59, 68, 73	syl2anc 693	. . . . . . . . . . 11 |- ( ( ( ( U e. (UnifOn ` X ) /\ ( V e. U /\ `' V = V ) /\ M C_ ( X X. X ) ) /\ r e. M ) /\ z e. ( ( V " { ( 1st ` r ) } ) X. ( V " { ( 2nd ` r ) } ) ) ) -> ( 1st ` z ) V ( 1st ` r ) )
75		simpll3 1102	. . . . . . . . . . . 12 |- ( ( ( ( U e. (UnifOn ` X ) /\ ( V e. U /\ `' V = V ) /\ M C_ ( X X. X ) ) /\ r e. M ) /\ z e. ( ( V " { ( 1st ` r ) } ) X. ( V " { ( 2nd ` r ) } ) ) ) -> M C_ ( X X. X ) )
76		simplr 792	. . . . . . . . . . . 12 |- ( ( ( ( U e. (UnifOn ` X ) /\ ( V e. U /\ `' V = V ) /\ M C_ ( X X. X ) ) /\ r e. M ) /\ z e. ( ( V " { ( 1st ` r ) } ) X. ( V " { ( 2nd ` r ) } ) ) ) -> r e. M )
77		1st2ndbr 7217	. . . . . . . . . . . . 13 |- ( ( Rel M /\ r e. M ) -> ( 1st ` r ) M ( 2nd ` r ) )
78	47, 77	sylan 488	. . . . . . . . . . . 12 |- ( ( M C_ ( X X. X ) /\ r e. M ) -> ( 1st ` r ) M ( 2nd ` r ) )
79	75, 76, 78	syl2anc 693	. . . . . . . . . . 11 |- ( ( ( ( U e. (UnifOn ` X ) /\ ( V e. U /\ `' V = V ) /\ M C_ ( X X. X ) ) /\ r e. M ) /\ z e. ( ( V " { ( 1st ` r ) } ) X. ( V " { ( 2nd ` r ) } ) ) ) -> ( 1st ` r ) M ( 2nd ` r ) )
80		xp2nd 7199	. . . . . . . . . . . . 13 |- ( z e. ( ( V " { ( 1st ` r ) } ) X. ( V " { ( 2nd ` r ) } ) ) -> ( 2nd ` z ) e. ( V " { ( 2nd ` r ) } ) )
81	80	adantl 482	. . . . . . . . . . . 12 |- ( ( ( ( U e. (UnifOn ` X ) /\ ( V e. U /\ `' V = V ) /\ M C_ ( X X. X ) ) /\ r e. M ) /\ z e. ( ( V " { ( 1st ` r ) } ) X. ( V " { ( 2nd ` r ) } ) ) ) -> ( 2nd ` z ) e. ( V " { ( 2nd ` r ) } ) )
82		elrelimasn 5489	. . . . . . . . . . . . 13 |- ( Rel V -> ( ( 2nd ` z ) e. ( V " { ( 2nd ` r ) } ) <-> ( 2nd ` r ) V ( 2nd ` z ) ) )
83	82	biimpa 501	. . . . . . . . . . . 12 |- ( ( Rel V /\ ( 2nd ` z ) e. ( V " { ( 2nd ` r ) } ) ) -> ( 2nd ` r ) V ( 2nd ` z ) )
84	63, 81, 83	syl2anc 693	. . . . . . . . . . 11 |- ( ( ( ( U e. (UnifOn ` X ) /\ ( V e. U /\ `' V = V ) /\ M C_ ( X X. X ) ) /\ r e. M ) /\ z e. ( ( V " { ( 1st ` r ) } ) X. ( V " { ( 2nd ` r ) } ) ) ) -> ( 2nd ` r ) V ( 2nd ` z ) )
85	69, 38, 37	3pm3.2i 1239	. . . . . . . . . . . . 13 |- ( ( 1st ` z ) e. _V /\ ( 2nd ` r ) e. _V /\ ( 1st ` r ) e. _V )
86		brcogw 5290	. . . . . . . . . . . . 13 |- ( ( ( ( 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 ) )
87	85, 86	mpan 706	. . . . . . . . . . . 12 |- ( ( ( 1st ` z ) V ( 1st ` r ) /\ ( 1st ` r ) M ( 2nd ` r ) ) -> ( 1st ` z ) ( M o. V ) ( 2nd ` r ) )
88		fvex 6201	. . . . . . . . . . . . . 14 |- ( 2nd ` z ) e. _V
89	69, 88, 38	3pm3.2i 1239	. . . . . . . . . . . . 13 |- ( ( 1st ` z ) e. _V /\ ( 2nd ` z ) e. _V /\ ( 2nd ` r ) e. _V )
90		brcogw 5290	. . . . . . . . . . . . 13 |- ( ( ( ( 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 ) )
91	89, 90	mpan 706	. . . . . . . . . . . 12 |- ( ( ( 1st ` z ) ( M o. V ) ( 2nd ` r ) /\ ( 2nd ` r ) V ( 2nd ` z ) ) -> ( 1st ` z ) ( V o. ( M o. V ) ) ( 2nd ` z ) )
92	87, 91	sylan 488	. . . . . . . . . . 11 |- ( ( ( ( 1st ` z ) V ( 1st ` r ) /\ ( 1st ` r ) M ( 2nd ` r ) ) /\ ( 2nd ` r ) V ( 2nd ` z ) ) -> ( 1st ` z ) ( V o. ( M o. V ) ) ( 2nd ` z ) )
93	74, 79, 84, 92	syl21anc 1325	. . . . . . . . . 10 |- ( ( ( ( U e. (UnifOn ` X ) /\ ( V e. U /\ `' V = V ) /\ M C_ ( X X. X ) ) /\ r e. M ) /\ z e. ( ( V " { ( 1st ` r ) } ) X. ( V " { ( 2nd ` r ) } ) ) ) -> ( 1st ` z ) ( V o. ( M o. V ) ) ( 2nd ` z ) )
94		df-br 4654	. . . . . . . . . 10 |- ( ( 1st ` z ) ( V o. ( M o. V ) ) ( 2nd ` z ) <-> <. ( 1st ` z ) , ( 2nd ` z ) >. e. ( V o. ( M o. V ) ) )
95	93, 94	sylib 208	. . . . . . . . 9 |- ( ( ( ( U e. (UnifOn ` X ) /\ ( V e. U /\ `' V = V ) /\ M C_ ( X X. X ) ) /\ r e. M ) /\ z e. ( ( V " { ( 1st ` r ) } ) X. ( V " { ( 2nd ` r ) } ) ) ) -> <. ( 1st ` z ) , ( 2nd ` z ) >. e. ( V o. ( M o. V ) ) )
96	57, 95	eqeltrd 2701	. . . . . . . 8 |- ( ( ( ( U e. (UnifOn ` X ) /\ ( V e. U /\ `' V = V ) /\ M C_ ( X X. X ) ) /\ r e. M ) /\ z e. ( ( V " { ( 1st ` r ) } ) X. ( V " { ( 2nd ` r ) } ) ) ) -> z e. ( V o. ( M o. V ) ) )
97	96	ex 450	. . . . . . 7 |- ( ( ( U e. (UnifOn ` X ) /\ ( V e. U /\ `' V = V ) /\ M C_ ( X X. X ) ) /\ r e. M ) -> ( z e. ( ( V " { ( 1st ` r ) } ) X. ( V " { ( 2nd ` r ) } ) ) -> z e. ( V o. ( M o. V ) ) ) )
98	97	ssrdv 3609	. . . . . 6 |- ( ( ( U e. (UnifOn ` X ) /\ ( V e. U /\ `' V = V ) /\ M C_ ( X X. X ) ) /\ r e. M ) -> ( ( V " { ( 1st ` r ) } ) X. ( V " { ( 2nd ` r ) } ) ) C_ ( V o. ( M o. V ) ) )
99		simp1 1061	. . . . . . . . . . 11 |- ( ( U e. (UnifOn ` X ) /\ ( V e. U /\ `' V = V ) /\ M C_ ( X X. X ) ) -> U e. (UnifOn ` X ) )
100		simp2l 1087	. . . . . . . . . . 11 |- ( ( U e. (UnifOn ` X ) /\ ( V e. U /\ `' V = V ) /\ M C_ ( X X. X ) ) -> V e. U )
101		ustssxp 22008	. . . . . . . . . . 11 |- ( ( U e. (UnifOn ` X ) /\ V e. U ) -> V C_ ( X X. X ) )
102	99, 100, 101	syl2anc 693	. . . . . . . . . 10 |- ( ( U e. (UnifOn ` X ) /\ ( V e. U /\ `' V = V ) /\ M C_ ( X X. X ) ) -> V C_ ( X X. X ) )
103		coss1 5277	. . . . . . . . . 10 |- ( V C_ ( X X. X ) -> ( V o. ( M o. V ) ) C_ ( ( X X. X ) o. ( M o. V ) ) )
104	102, 103	syl 17	. . . . . . . . 9 |- ( ( U e. (UnifOn ` X ) /\ ( V e. U /\ `' V = V ) /\ M C_ ( X X. X ) ) -> ( V o. ( M o. V ) ) C_ ( ( X X. X ) o. ( M o. V ) ) )
105		coss1 5277	. . . . . . . . . . . 12 |- ( M C_ ( X X. X ) -> ( M o. V ) C_ ( ( X X. X ) o. V ) )
106	24, 105	syl 17	. . . . . . . . . . 11 |- ( ( U e. (UnifOn ` X ) /\ ( V e. U /\ `' V = V ) /\ M C_ ( X X. X ) ) -> ( M o. V ) C_ ( ( X X. X ) o. V ) )
107		coss2 5278	. . . . . . . . . . . . 13 |- ( V C_ ( X X. X ) -> ( ( X X. X ) o. V ) C_ ( ( X X. X ) o. ( X X. X ) ) )
108		xpcoid 5676	. . . . . . . . . . . . 13 |- ( ( X X. X ) o. ( X X. X ) ) = ( X X. X )
109	107, 108	syl6sseq 3651	. . . . . . . . . . . 12 |- ( V C_ ( X X. X ) -> ( ( X X. X ) o. V ) C_ ( X X. X ) )
110	102, 109	syl 17	. . . . . . . . . . 11 |- ( ( U e. (UnifOn ` X ) /\ ( V e. U /\ `' V = V ) /\ M C_ ( X X. X ) ) -> ( ( X X. X ) o. V ) C_ ( X X. X ) )
111	106, 110	sstrd 3613	. . . . . . . . . 10 |- ( ( U e. (UnifOn ` X ) /\ ( V e. U /\ `' V = V ) /\ M C_ ( X X. X ) ) -> ( M o. V ) C_ ( X X. X ) )
112		coss2 5278	. . . . . . . . . . 11 |- ( ( M o. V ) C_ ( X X. X ) -> ( ( X X. X ) o. ( M o. V ) ) C_ ( ( X X. X ) o. ( X X. X ) ) )
113	112, 108	syl6sseq 3651	. . . . . . . . . 10 |- ( ( M o. V ) C_ ( X X. X ) -> ( ( X X. X ) o. ( M o. V ) ) C_ ( X X. X ) )
114	111, 113	syl 17	. . . . . . . . 9 |- ( ( U e. (UnifOn ` X ) /\ ( V e. U /\ `' V = V ) /\ M C_ ( X X. X ) ) -> ( ( X X. X ) o. ( M o. V ) ) C_ ( X X. X ) )
115	104, 114	sstrd 3613	. . . . . . . 8 |- ( ( U e. (UnifOn ` X ) /\ ( V e. U /\ `' V = V ) /\ M C_ ( X X. X ) ) -> ( V o. ( M o. V ) ) C_ ( X X. X ) )
116		utopbas 22039	. . . . . . . . . . . 12 |- ( U e. (UnifOn ` X ) -> X = U. (unifTop ` U ) )
117	1	unieqi 4445	. . . . . . . . . . . 12 |- U. J = U. (unifTop ` U )
118	116, 117	syl6eqr 2674	. . . . . . . . . . 11 |- ( U e. (UnifOn ` X ) -> X = U. J )
119	118	sqxpeqd 5141	. . . . . . . . . 10 |- ( U e. (UnifOn ` X ) -> ( X X. X ) = ( U. J X. U. J ) )
120	34, 34	txuni 21395	. . . . . . . . . . 11 |- ( ( J e. Top /\ J e. Top ) -> ( U. J X. U. J ) = U. ( J tX J ) )
121	3, 3, 120	syl2anc 693	. . . . . . . . . 10 |- ( U e. (UnifOn ` X ) -> ( U. J X. U. J ) = U. ( J tX J ) )
122	119, 121	eqtrd 2656	. . . . . . . . 9 |- ( U e. (UnifOn ` X ) -> ( X X. X ) = U. ( J tX J ) )
123	122	3ad2ant1 1082	. . . . . . . 8 |- ( ( U e. (UnifOn ` X ) /\ ( V e. U /\ `' V = V ) /\ M C_ ( X X. X ) ) -> ( X X. X ) = U. ( J tX J ) )
124	115, 123	sseqtrd 3641	. . . . . . 7 |- ( ( U e. (UnifOn ` X ) /\ ( V e. U /\ `' V = V ) /\ M C_ ( X X. X ) ) -> ( V o. ( M o. V ) ) C_ U. ( J tX J ) )
125	124	adantr 481	. . . . . 6 |- ( ( ( U e. (UnifOn ` X ) /\ ( V e. U /\ `' V = V ) /\ M C_ ( X X. X ) ) /\ r e. M ) -> ( V o. ( M o. V ) ) C_ U. ( J tX J ) )
126		eqid 2622	. . . . . . 7 |- U. ( J tX J ) = U. ( J tX J )
127	126	ssnei2 20920	. . . . . 6 |- ( ( ( ( J tX J ) e. Top /\ ( ( V " { ( 1st ` r ) } ) X. ( V " { ( 2nd ` r ) } ) ) e. ( ( nei ` ( J tX J ) ) ` { r } ) ) /\ ( ( ( V " { ( 1st ` r ) } ) X. ( V " { ( 2nd ` r ) } ) ) C_ ( V o. ( M o. V ) ) /\ ( V o. ( M o. V ) ) C_ U. ( J tX J ) ) ) -> ( V o. ( M o. V ) ) e. ( ( nei ` ( J tX J ) ) ` { r } ) )
128	20, 53, 98, 125, 127	syl22anc 1327	. . . . 5 |- ( ( ( U e. (UnifOn ` X ) /\ ( V e. U /\ `' V = V ) /\ M C_ ( X X. X ) ) /\ r e. M ) -> ( V o. ( M o. V ) ) e. ( ( nei ` ( J tX J ) ) ` { r } ) )
129	128	ralrimiva 2966	. . . 4 |- ( ( U e. (UnifOn ` X ) /\ ( V e. U /\ `' V = V ) /\ M C_ ( X X. X ) ) -> A. r e. M ( V o. ( M o. V ) ) e. ( ( nei ` ( J tX J ) ) ` { r } ) )
130	129	adantr 481	. . 3 |- ( ( ( U e. (UnifOn ` X ) /\ ( V e. U /\ `' V = V ) /\ M C_ ( X X. X ) ) /\ M =/= (/) ) -> A. r e. M ( V o. ( M o. V ) ) e. ( ( nei ` ( J tX J ) ) ` { r } ) )
131	6	adantr 481	. . . 4 |- ( ( ( U e. (UnifOn ` X ) /\ ( V e. U /\ `' V = V ) /\ M C_ ( X X. X ) ) /\ M =/= (/) ) -> ( J tX J ) e. Top )
132	24, 123	sseqtrd 3641	. . . . 5 |- ( ( U e. (UnifOn ` X ) /\ ( V e. U /\ `' V = V ) /\ M C_ ( X X. X ) ) -> M C_ U. ( J tX J ) )
133	132	adantr 481	. . . 4 |- ( ( ( U e. (UnifOn ` X ) /\ ( V e. U /\ `' V = V ) /\ M C_ ( X X. X ) ) /\ M =/= (/) ) -> M C_ U. ( J tX J ) )
134		simpr 477	. . . 4 |- ( ( ( U e. (UnifOn ` X ) /\ ( V e. U /\ `' V = V ) /\ M C_ ( X X. X ) ) /\ M =/= (/) ) -> M =/= (/) )
135	126	neips 20917	. . . 4 |- ( ( ( J tX J ) e. Top /\ M C_ U. ( J tX J ) /\ M =/= (/) ) -> ( ( V o. ( M o. V ) ) e. ( ( nei ` ( J tX J ) ) ` M ) <-> A. r e. M ( V o. ( M o. V ) ) e. ( ( nei ` ( J tX J ) ) ` { r } ) ) )
136	131, 133, 134, 135	syl3anc 1326	. . 3 |- ( ( ( U e. (UnifOn ` X ) /\ ( V e. U /\ `' V = V ) /\ M C_ ( X X. X ) ) /\ M =/= (/) ) -> ( ( V o. ( M o. V ) ) e. ( ( nei ` ( J tX J ) ) ` M ) <-> A. r e. M ( V o. ( M o. V ) ) e. ( ( nei ` ( J tX J ) ) ` { r } ) ) )
137	130, 136	mpbird 247	. 2 |- ( ( ( U e. (UnifOn ` X ) /\ ( V e. U /\ `' V = V ) /\ M C_ ( X X. X ) ) /\ M =/= (/) ) -> ( V o. ( M o. V ) ) e. ( ( nei ` ( J tX J ) ) ` M ) )
138	19, 137	pm2.61dane 2881	1 |- ( ( U e. (UnifOn ` X ) /\ ( V e. U /\ `' V = V ) /\ M C_ ( X X. X ) ) -> ( V o. ( M o. V ) ) e. ( ( nei ` ( J tX J ) ) ` M ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   _Vcvv 3200   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   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-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-nei 20902  df-tx 21365  df-ust 22004  df-utop 22035

This theorem is referenced by: (None)