Theorem utop3cls 22860

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 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ (𝑉 ∈ 𝑈 ∧ ◡𝑉 = 𝑉)) → ((cls‘(𝐽 ×t 𝐽))‘𝑀) ⊆ (𝑉 ∘ (𝑀 ∘ 𝑉)))

Proof of Theorem utop3cls

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

Step	Hyp	Ref	Expression
1		relxp 5573	. . . . 5 ⊢ Rel (𝑋 × 𝑋)
2		utoptop.1	. . . . . . . . . . 11 ⊢ 𝐽 = (unifTop‘𝑈)
3		utoptop 22843	. . . . . . . . . . 11 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (unifTop‘𝑈) ∈ Top)
4	2, 3	eqeltrid 2917	. . . . . . . . . 10 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝐽 ∈ Top)
5		txtop 22177	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ 𝐽 ∈ Top) → (𝐽 ×t 𝐽) ∈ Top)
6	4, 4, 5	syl2anc 586	. . . . . . . . 9 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝐽 ×t 𝐽) ∈ Top)
7	6	ad3antrrr 728	. . . . . . . 8 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ (𝑉 ∈ 𝑈 ∧ ◡𝑉 = 𝑉)) ∧ 𝑧 ∈ ((cls‘(𝐽 ×t 𝐽))‘𝑀)) → (𝐽 ×t 𝐽) ∈ Top)
8		simpllr 774	. . . . . . . . 9 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ (𝑉 ∈ 𝑈 ∧ ◡𝑉 = 𝑉)) ∧ 𝑧 ∈ ((cls‘(𝐽 ×t 𝐽))‘𝑀)) → 𝑀 ⊆ (𝑋 × 𝑋))
9		utoptopon 22845	. . . . . . . . . . . . . 14 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (unifTop‘𝑈) ∈ (TopOn‘𝑋))
10	2, 9	eqeltrid 2917	. . . . . . . . . . . . 13 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝐽 ∈ (TopOn‘𝑋))
11		toponuni 21522	. . . . . . . . . . . . 13 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
12	10, 11	syl 17	. . . . . . . . . . . 12 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = ∪ 𝐽)
13	12	sqxpeqd 5587	. . . . . . . . . . 11 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝑋 × 𝑋) = (∪ 𝐽 × ∪ 𝐽))
14		eqid 2821	. . . . . . . . . . . . 13 ⊢ ∪ 𝐽 = ∪ 𝐽
15	14, 14	txuni 22200	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ Top ∧ 𝐽 ∈ Top) → (∪ 𝐽 × ∪ 𝐽) = ∪ (𝐽 ×t 𝐽))
16	4, 4, 15	syl2anc 586	. . . . . . . . . . 11 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (∪ 𝐽 × ∪ 𝐽) = ∪ (𝐽 ×t 𝐽))
17	13, 16	eqtrd 2856	. . . . . . . . . 10 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝑋 × 𝑋) = ∪ (𝐽 ×t 𝐽))
18	17	ad3antrrr 728	. . . . . . . . 9 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ (𝑉 ∈ 𝑈 ∧ ◡𝑉 = 𝑉)) ∧ 𝑧 ∈ ((cls‘(𝐽 ×t 𝐽))‘𝑀)) → (𝑋 × 𝑋) = ∪ (𝐽 ×t 𝐽))
19	8, 18	sseqtrd 4007	. . . . . . . 8 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ (𝑉 ∈ 𝑈 ∧ ◡𝑉 = 𝑉)) ∧ 𝑧 ∈ ((cls‘(𝐽 ×t 𝐽))‘𝑀)) → 𝑀 ⊆ ∪ (𝐽 ×t 𝐽))
20		eqid 2821	. . . . . . . . 9 ⊢ ∪ (𝐽 ×t 𝐽) = ∪ (𝐽 ×t 𝐽)
21	20	clsss3 21667	. . . . . . . 8 ⊢ (((𝐽 ×t 𝐽) ∈ Top ∧ 𝑀 ⊆ ∪ (𝐽 ×t 𝐽)) → ((cls‘(𝐽 ×t 𝐽))‘𝑀) ⊆ ∪ (𝐽 ×t 𝐽))
22	7, 19, 21	syl2anc 586	. . . . . . 7 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ (𝑉 ∈ 𝑈 ∧ ◡𝑉 = 𝑉)) ∧ 𝑧 ∈ ((cls‘(𝐽 ×t 𝐽))‘𝑀)) → ((cls‘(𝐽 ×t 𝐽))‘𝑀) ⊆ ∪ (𝐽 ×t 𝐽))
23	22, 18	sseqtrrd 4008	. . . . . 6 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ (𝑉 ∈ 𝑈 ∧ ◡𝑉 = 𝑉)) ∧ 𝑧 ∈ ((cls‘(𝐽 ×t 𝐽))‘𝑀)) → ((cls‘(𝐽 ×t 𝐽))‘𝑀) ⊆ (𝑋 × 𝑋))
24		simpr 487	. . . . . 6 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ (𝑉 ∈ 𝑈 ∧ ◡𝑉 = 𝑉)) ∧ 𝑧 ∈ ((cls‘(𝐽 ×t 𝐽))‘𝑀)) → 𝑧 ∈ ((cls‘(𝐽 ×t 𝐽))‘𝑀))
25	23, 24	sseldd 3968	. . . . 5 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ (𝑉 ∈ 𝑈 ∧ ◡𝑉 = 𝑉)) ∧ 𝑧 ∈ ((cls‘(𝐽 ×t 𝐽))‘𝑀)) → 𝑧 ∈ (𝑋 × 𝑋))
26		1st2nd 7738	. . . . 5 ⊢ ((Rel (𝑋 × 𝑋) ∧ 𝑧 ∈ (𝑋 × 𝑋)) → 𝑧 = ⟨(1st ‘𝑧), (2nd ‘𝑧)⟩)
27	1, 25, 26	sylancr 589	. . . 4 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ (𝑉 ∈ 𝑈 ∧ ◡𝑉 = 𝑉)) ∧ 𝑧 ∈ ((cls‘(𝐽 ×t 𝐽))‘𝑀)) → 𝑧 = ⟨(1st ‘𝑧), (2nd ‘𝑧)⟩)
28		simp-4l 781	. . . . . . . . . 10 ⊢ (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ (𝑉 ∈ 𝑈 ∧ ◡𝑉 = 𝑉)) ∧ 𝑧 ∈ ((cls‘(𝐽 ×t 𝐽))‘𝑀)) ∧ 𝑟 ∈ (((𝑉 “ {(1st ‘𝑧)}) × (𝑉 “ {(2nd ‘𝑧)})) ∩ 𝑀)) → 𝑈 ∈ (UnifOn‘𝑋))
29		simpr1l 1226	. . . . . . . . . . 11 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ ((𝑉 ∈ 𝑈 ∧ ◡𝑉 = 𝑉) ∧ 𝑧 ∈ ((cls‘(𝐽 ×t 𝐽))‘𝑀) ∧ 𝑟 ∈ (((𝑉 “ {(1st ‘𝑧)}) × (𝑉 “ {(2nd ‘𝑧)})) ∩ 𝑀))) → 𝑉 ∈ 𝑈)
30	29	3anassrs 1356	. . . . . . . . . 10 ⊢ (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ (𝑉 ∈ 𝑈 ∧ ◡𝑉 = 𝑉)) ∧ 𝑧 ∈ ((cls‘(𝐽 ×t 𝐽))‘𝑀)) ∧ 𝑟 ∈ (((𝑉 “ {(1st ‘𝑧)}) × (𝑉 “ {(2nd ‘𝑧)})) ∩ 𝑀)) → 𝑉 ∈ 𝑈)
31		ustrel 22820	. . . . . . . . . 10 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) → Rel 𝑉)
32	28, 30, 31	syl2anc 586	. . . . . . . . 9 ⊢ (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ (𝑉 ∈ 𝑈 ∧ ◡𝑉 = 𝑉)) ∧ 𝑧 ∈ ((cls‘(𝐽 ×t 𝐽))‘𝑀)) ∧ 𝑟 ∈ (((𝑉 “ {(1st ‘𝑧)}) × (𝑉 “ {(2nd ‘𝑧)})) ∩ 𝑀)) → Rel 𝑉)
33		simpr 487	. . . . . . . . . . . 12 ⊢ (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ (𝑉 ∈ 𝑈 ∧ ◡𝑉 = 𝑉)) ∧ 𝑧 ∈ ((cls‘(𝐽 ×t 𝐽))‘𝑀)) ∧ 𝑟 ∈ (((𝑉 “ {(1st ‘𝑧)}) × (𝑉 “ {(2nd ‘𝑧)})) ∩ 𝑀)) → 𝑟 ∈ (((𝑉 “ {(1st ‘𝑧)}) × (𝑉 “ {(2nd ‘𝑧)})) ∩ 𝑀))
34		elin 4169	. . . . . . . . . . . 12 ⊢ (𝑟 ∈ (((𝑉 “ {(1st ‘𝑧)}) × (𝑉 “ {(2nd ‘𝑧)})) ∩ 𝑀) ↔ (𝑟 ∈ ((𝑉 “ {(1st ‘𝑧)}) × (𝑉 “ {(2nd ‘𝑧)})) ∧ 𝑟 ∈ 𝑀))
35	33, 34	sylib 220	. . . . . . . . . . 11 ⊢ (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ (𝑉 ∈ 𝑈 ∧ ◡𝑉 = 𝑉)) ∧ 𝑧 ∈ ((cls‘(𝐽 ×t 𝐽))‘𝑀)) ∧ 𝑟 ∈ (((𝑉 “ {(1st ‘𝑧)}) × (𝑉 “ {(2nd ‘𝑧)})) ∩ 𝑀)) → (𝑟 ∈ ((𝑉 “ {(1st ‘𝑧)}) × (𝑉 “ {(2nd ‘𝑧)})) ∧ 𝑟 ∈ 𝑀))
36	35	simpld 497	. . . . . . . . . 10 ⊢ (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ (𝑉 ∈ 𝑈 ∧ ◡𝑉 = 𝑉)) ∧ 𝑧 ∈ ((cls‘(𝐽 ×t 𝐽))‘𝑀)) ∧ 𝑟 ∈ (((𝑉 “ {(1st ‘𝑧)}) × (𝑉 “ {(2nd ‘𝑧)})) ∩ 𝑀)) → 𝑟 ∈ ((𝑉 “ {(1st ‘𝑧)}) × (𝑉 “ {(2nd ‘𝑧)})))
37		xp1st 7721	. . . . . . . . . 10 ⊢ (𝑟 ∈ ((𝑉 “ {(1st ‘𝑧)}) × (𝑉 “ {(2nd ‘𝑧)})) → (1st ‘𝑟) ∈ (𝑉 “ {(1st ‘𝑧)}))
38	36, 37	syl 17	. . . . . . . . 9 ⊢ (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ (𝑉 ∈ 𝑈 ∧ ◡𝑉 = 𝑉)) ∧ 𝑧 ∈ ((cls‘(𝐽 ×t 𝐽))‘𝑀)) ∧ 𝑟 ∈ (((𝑉 “ {(1st ‘𝑧)}) × (𝑉 “ {(2nd ‘𝑧)})) ∩ 𝑀)) → (1st ‘𝑟) ∈ (𝑉 “ {(1st ‘𝑧)}))
39		elrelimasn 5953	. . . . . . . . . 10 ⊢ (Rel 𝑉 → ((1st ‘𝑟) ∈ (𝑉 “ {(1st ‘𝑧)}) ↔ (1st ‘𝑧)𝑉(1st ‘𝑟)))
40	39	biimpa 479	. . . . . . . . 9 ⊢ ((Rel 𝑉 ∧ (1st ‘𝑟) ∈ (𝑉 “ {(1st ‘𝑧)})) → (1st ‘𝑧)𝑉(1st ‘𝑟))
41	32, 38, 40	syl2anc 586	. . . . . . . 8 ⊢ (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ (𝑉 ∈ 𝑈 ∧ ◡𝑉 = 𝑉)) ∧ 𝑧 ∈ ((cls‘(𝐽 ×t 𝐽))‘𝑀)) ∧ 𝑟 ∈ (((𝑉 “ {(1st ‘𝑧)}) × (𝑉 “ {(2nd ‘𝑧)})) ∩ 𝑀)) → (1st ‘𝑧)𝑉(1st ‘𝑟))
42		simp-4r 782	. . . . . . . . . . 11 ⊢ (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ (𝑉 ∈ 𝑈 ∧ ◡𝑉 = 𝑉)) ∧ 𝑧 ∈ ((cls‘(𝐽 ×t 𝐽))‘𝑀)) ∧ 𝑟 ∈ (((𝑉 “ {(1st ‘𝑧)}) × (𝑉 “ {(2nd ‘𝑧)})) ∩ 𝑀)) → 𝑀 ⊆ (𝑋 × 𝑋))
43		xpss 5571	. . . . . . . . . . 11 ⊢ (𝑋 × 𝑋) ⊆ (V × V)
44	42, 43	sstrdi 3979	. . . . . . . . . 10 ⊢ (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ (𝑉 ∈ 𝑈 ∧ ◡𝑉 = 𝑉)) ∧ 𝑧 ∈ ((cls‘(𝐽 ×t 𝐽))‘𝑀)) ∧ 𝑟 ∈ (((𝑉 “ {(1st ‘𝑧)}) × (𝑉 “ {(2nd ‘𝑧)})) ∩ 𝑀)) → 𝑀 ⊆ (V × V))
45		df-rel 5562	. . . . . . . . . 10 ⊢ (Rel 𝑀 ↔ 𝑀 ⊆ (V × V))
46	44, 45	sylibr 236	. . . . . . . . 9 ⊢ (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ (𝑉 ∈ 𝑈 ∧ ◡𝑉 = 𝑉)) ∧ 𝑧 ∈ ((cls‘(𝐽 ×t 𝐽))‘𝑀)) ∧ 𝑟 ∈ (((𝑉 “ {(1st ‘𝑧)}) × (𝑉 “ {(2nd ‘𝑧)})) ∩ 𝑀)) → Rel 𝑀)
47	35	simprd 498	. . . . . . . . 9 ⊢ (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ (𝑉 ∈ 𝑈 ∧ ◡𝑉 = 𝑉)) ∧ 𝑧 ∈ ((cls‘(𝐽 ×t 𝐽))‘𝑀)) ∧ 𝑟 ∈ (((𝑉 “ {(1st ‘𝑧)}) × (𝑉 “ {(2nd ‘𝑧)})) ∩ 𝑀)) → 𝑟 ∈ 𝑀)
48		1st2ndbr 7741	. . . . . . . . 9 ⊢ ((Rel 𝑀 ∧ 𝑟 ∈ 𝑀) → (1st ‘𝑟)𝑀(2nd ‘𝑟))
49	46, 47, 48	syl2anc 586	. . . . . . . 8 ⊢ (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ (𝑉 ∈ 𝑈 ∧ ◡𝑉 = 𝑉)) ∧ 𝑧 ∈ ((cls‘(𝐽 ×t 𝐽))‘𝑀)) ∧ 𝑟 ∈ (((𝑉 “ {(1st ‘𝑧)}) × (𝑉 “ {(2nd ‘𝑧)})) ∩ 𝑀)) → (1st ‘𝑟)𝑀(2nd ‘𝑟))
50		xp2nd 7722	. . . . . . . . . . 11 ⊢ (𝑟 ∈ ((𝑉 “ {(1st ‘𝑧)}) × (𝑉 “ {(2nd ‘𝑧)})) → (2nd ‘𝑟) ∈ (𝑉 “ {(2nd ‘𝑧)}))
51	36, 50	syl 17	. . . . . . . . . 10 ⊢ (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ (𝑉 ∈ 𝑈 ∧ ◡𝑉 = 𝑉)) ∧ 𝑧 ∈ ((cls‘(𝐽 ×t 𝐽))‘𝑀)) ∧ 𝑟 ∈ (((𝑉 “ {(1st ‘𝑧)}) × (𝑉 “ {(2nd ‘𝑧)})) ∩ 𝑀)) → (2nd ‘𝑟) ∈ (𝑉 “ {(2nd ‘𝑧)}))
52		elrelimasn 5953	. . . . . . . . . . 11 ⊢ (Rel 𝑉 → ((2nd ‘𝑟) ∈ (𝑉 “ {(2nd ‘𝑧)}) ↔ (2nd ‘𝑧)𝑉(2nd ‘𝑟)))
53	52	biimpa 479	. . . . . . . . . 10 ⊢ ((Rel 𝑉 ∧ (2nd ‘𝑟) ∈ (𝑉 “ {(2nd ‘𝑧)})) → (2nd ‘𝑧)𝑉(2nd ‘𝑟))
54	32, 51, 53	syl2anc 586	. . . . . . . . 9 ⊢ (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ (𝑉 ∈ 𝑈 ∧ ◡𝑉 = 𝑉)) ∧ 𝑧 ∈ ((cls‘(𝐽 ×t 𝐽))‘𝑀)) ∧ 𝑟 ∈ (((𝑉 “ {(1st ‘𝑧)}) × (𝑉 “ {(2nd ‘𝑧)})) ∩ 𝑀)) → (2nd ‘𝑧)𝑉(2nd ‘𝑟))
55		simpr1r 1227	. . . . . . . . . . 11 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ ((𝑉 ∈ 𝑈 ∧ ◡𝑉 = 𝑉) ∧ 𝑧 ∈ ((cls‘(𝐽 ×t 𝐽))‘𝑀) ∧ 𝑟 ∈ (((𝑉 “ {(1st ‘𝑧)}) × (𝑉 “ {(2nd ‘𝑧)})) ∩ 𝑀))) → ◡𝑉 = 𝑉)
56	55	3anassrs 1356	. . . . . . . . . 10 ⊢ (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ (𝑉 ∈ 𝑈 ∧ ◡𝑉 = 𝑉)) ∧ 𝑧 ∈ ((cls‘(𝐽 ×t 𝐽))‘𝑀)) ∧ 𝑟 ∈ (((𝑉 “ {(1st ‘𝑧)}) × (𝑉 “ {(2nd ‘𝑧)})) ∩ 𝑀)) → ◡𝑉 = 𝑉)
57		fvex 6683	. . . . . . . . . . . 12 ⊢ (2nd ‘𝑟) ∈ V
58		fvex 6683	. . . . . . . . . . . 12 ⊢ (2nd ‘𝑧) ∈ V
59	57, 58	brcnv 5753	. . . . . . . . . . 11 ⊢ ((2nd ‘𝑟)◡𝑉(2nd ‘𝑧) ↔ (2nd ‘𝑧)𝑉(2nd ‘𝑟))
60		breq 5068	. . . . . . . . . . 11 ⊢ (◡𝑉 = 𝑉 → ((2nd ‘𝑟)◡𝑉(2nd ‘𝑧) ↔ (2nd ‘𝑟)𝑉(2nd ‘𝑧)))
61	59, 60	syl5rbbr 288	. . . . . . . . . 10 ⊢ (◡𝑉 = 𝑉 → ((2nd ‘𝑟)𝑉(2nd ‘𝑧) ↔ (2nd ‘𝑧)𝑉(2nd ‘𝑟)))
62	56, 61	syl 17	. . . . . . . . 9 ⊢ (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ (𝑉 ∈ 𝑈 ∧ ◡𝑉 = 𝑉)) ∧ 𝑧 ∈ ((cls‘(𝐽 ×t 𝐽))‘𝑀)) ∧ 𝑟 ∈ (((𝑉 “ {(1st ‘𝑧)}) × (𝑉 “ {(2nd ‘𝑧)})) ∩ 𝑀)) → ((2nd ‘𝑟)𝑉(2nd ‘𝑧) ↔ (2nd ‘𝑧)𝑉(2nd ‘𝑟)))
63	54, 62	mpbird 259	. . . . . . . 8 ⊢ (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ (𝑉 ∈ 𝑈 ∧ ◡𝑉 = 𝑉)) ∧ 𝑧 ∈ ((cls‘(𝐽 ×t 𝐽))‘𝑀)) ∧ 𝑟 ∈ (((𝑉 “ {(1st ‘𝑧)}) × (𝑉 “ {(2nd ‘𝑧)})) ∩ 𝑀)) → (2nd ‘𝑟)𝑉(2nd ‘𝑧))
64		fvex 6683	. . . . . . . . . 10 ⊢ (1st ‘𝑧) ∈ V
65		fvex 6683	. . . . . . . . . 10 ⊢ (1st ‘𝑟) ∈ V
66		brcogw 5739	. . . . . . . . . . 11 ⊢ ((((1st ‘𝑧) ∈ V ∧ (2nd ‘𝑟) ∈ V ∧ (1st ‘𝑟) ∈ V) ∧ ((1st ‘𝑧)𝑉(1st ‘𝑟) ∧ (1st ‘𝑟)𝑀(2nd ‘𝑟))) → (1st ‘𝑧)(𝑀 ∘ 𝑉)(2nd ‘𝑟))
67	66	ex 415	. . . . . . . . . 10 ⊢ (((1st ‘𝑧) ∈ V ∧ (2nd ‘𝑟) ∈ V ∧ (1st ‘𝑟) ∈ V) → (((1st ‘𝑧)𝑉(1st ‘𝑟) ∧ (1st ‘𝑟)𝑀(2nd ‘𝑟)) → (1st ‘𝑧)(𝑀 ∘ 𝑉)(2nd ‘𝑟)))
68	64, 57, 65, 67	mp3an 1457	. . . . . . . . 9 ⊢ (((1st ‘𝑧)𝑉(1st ‘𝑟) ∧ (1st ‘𝑟)𝑀(2nd ‘𝑟)) → (1st ‘𝑧)(𝑀 ∘ 𝑉)(2nd ‘𝑟))
69		brcogw 5739	. . . . . . . . . . 11 ⊢ ((((1st ‘𝑧) ∈ V ∧ (2nd ‘𝑧) ∈ V ∧ (2nd ‘𝑟) ∈ V) ∧ ((1st ‘𝑧)(𝑀 ∘ 𝑉)(2nd ‘𝑟) ∧ (2nd ‘𝑟)𝑉(2nd ‘𝑧))) → (1st ‘𝑧)(𝑉 ∘ (𝑀 ∘ 𝑉))(2nd ‘𝑧))
70	69	ex 415	. . . . . . . . . 10 ⊢ (((1st ‘𝑧) ∈ V ∧ (2nd ‘𝑧) ∈ V ∧ (2nd ‘𝑟) ∈ V) → (((1st ‘𝑧)(𝑀 ∘ 𝑉)(2nd ‘𝑟) ∧ (2nd ‘𝑟)𝑉(2nd ‘𝑧)) → (1st ‘𝑧)(𝑉 ∘ (𝑀 ∘ 𝑉))(2nd ‘𝑧)))
71	64, 58, 57, 70	mp3an 1457	. . . . . . . . 9 ⊢ (((1st ‘𝑧)(𝑀 ∘ 𝑉)(2nd ‘𝑟) ∧ (2nd ‘𝑟)𝑉(2nd ‘𝑧)) → (1st ‘𝑧)(𝑉 ∘ (𝑀 ∘ 𝑉))(2nd ‘𝑧))
72	68, 71	sylan 582	. . . . . . . 8 ⊢ ((((1st ‘𝑧)𝑉(1st ‘𝑟) ∧ (1st ‘𝑟)𝑀(2nd ‘𝑟)) ∧ (2nd ‘𝑟)𝑉(2nd ‘𝑧)) → (1st ‘𝑧)(𝑉 ∘ (𝑀 ∘ 𝑉))(2nd ‘𝑧))
73	41, 49, 63, 72	syl21anc 835	. . . . . . 7 ⊢ (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ (𝑉 ∈ 𝑈 ∧ ◡𝑉 = 𝑉)) ∧ 𝑧 ∈ ((cls‘(𝐽 ×t 𝐽))‘𝑀)) ∧ 𝑟 ∈ (((𝑉 “ {(1st ‘𝑧)}) × (𝑉 “ {(2nd ‘𝑧)})) ∩ 𝑀)) → (1st ‘𝑧)(𝑉 ∘ (𝑀 ∘ 𝑉))(2nd ‘𝑧))
74	73	ralrimiva 3182	. . . . . 6 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ (𝑉 ∈ 𝑈 ∧ ◡𝑉 = 𝑉)) ∧ 𝑧 ∈ ((cls‘(𝐽 ×t 𝐽))‘𝑀)) → ∀𝑟 ∈ (((𝑉 “ {(1st ‘𝑧)}) × (𝑉 “ {(2nd ‘𝑧)})) ∩ 𝑀)(1st ‘𝑧)(𝑉 ∘ (𝑀 ∘ 𝑉))(2nd ‘𝑧))
75		simplll 773	. . . . . . . . 9 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ (𝑉 ∈ 𝑈 ∧ ◡𝑉 = 𝑉)) ∧ 𝑧 ∈ ((cls‘(𝐽 ×t 𝐽))‘𝑀)) → 𝑈 ∈ (UnifOn‘𝑋))
76		simplrl 775	. . . . . . . . 9 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ (𝑉 ∈ 𝑈 ∧ ◡𝑉 = 𝑉)) ∧ 𝑧 ∈ ((cls‘(𝐽 ×t 𝐽))‘𝑀)) → 𝑉 ∈ 𝑈)
77	4	3ad2ant1 1129	. . . . . . . . . . 11 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈 ∧ 𝑧 ∈ (𝑋 × 𝑋)) → 𝐽 ∈ Top)
78		xp1st 7721	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ (𝑋 × 𝑋) → (1st ‘𝑧) ∈ 𝑋)
79	2	utopsnnei 22858	. . . . . . . . . . . 12 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈 ∧ (1st ‘𝑧) ∈ 𝑋) → (𝑉 “ {(1st ‘𝑧)}) ∈ ((nei‘𝐽)‘{(1st ‘𝑧)}))
80	78, 79	syl3an3 1161	. . . . . . . . . . 11 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈 ∧ 𝑧 ∈ (𝑋 × 𝑋)) → (𝑉 “ {(1st ‘𝑧)}) ∈ ((nei‘𝐽)‘{(1st ‘𝑧)}))
81		xp2nd 7722	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ (𝑋 × 𝑋) → (2nd ‘𝑧) ∈ 𝑋)
82	2	utopsnnei 22858	. . . . . . . . . . . 12 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈 ∧ (2nd ‘𝑧) ∈ 𝑋) → (𝑉 “ {(2nd ‘𝑧)}) ∈ ((nei‘𝐽)‘{(2nd ‘𝑧)}))
83	81, 82	syl3an3 1161	. . . . . . . . . . 11 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈 ∧ 𝑧 ∈ (𝑋 × 𝑋)) → (𝑉 “ {(2nd ‘𝑧)}) ∈ ((nei‘𝐽)‘{(2nd ‘𝑧)}))
84	14, 14	neitx 22215	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ Top ∧ 𝐽 ∈ Top) ∧ ((𝑉 “ {(1st ‘𝑧)}) ∈ ((nei‘𝐽)‘{(1st ‘𝑧)}) ∧ (𝑉 “ {(2nd ‘𝑧)}) ∈ ((nei‘𝐽)‘{(2nd ‘𝑧)}))) → ((𝑉 “ {(1st ‘𝑧)}) × (𝑉 “ {(2nd ‘𝑧)})) ∈ ((nei‘(𝐽 ×t 𝐽))‘({(1st ‘𝑧)} × {(2nd ‘𝑧)})))
85	77, 77, 80, 83, 84	syl22anc 836	. . . . . . . . . 10 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈 ∧ 𝑧 ∈ (𝑋 × 𝑋)) → ((𝑉 “ {(1st ‘𝑧)}) × (𝑉 “ {(2nd ‘𝑧)})) ∈ ((nei‘(𝐽 ×t 𝐽))‘({(1st ‘𝑧)} × {(2nd ‘𝑧)})))
86		1st2nd2 7728	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ (𝑋 × 𝑋) → 𝑧 = ⟨(1st ‘𝑧), (2nd ‘𝑧)⟩)
87	86	sneqd 4579	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ (𝑋 × 𝑋) → {𝑧} = {⟨(1st ‘𝑧), (2nd ‘𝑧)⟩})
88	64, 58	xpsn 6903	. . . . . . . . . . . . 13 ⊢ ({(1st ‘𝑧)} × {(2nd ‘𝑧)}) = {⟨(1st ‘𝑧), (2nd ‘𝑧)⟩}
89	87, 88	syl6eqr 2874	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ (𝑋 × 𝑋) → {𝑧} = ({(1st ‘𝑧)} × {(2nd ‘𝑧)}))
90	89	fveq2d 6674	. . . . . . . . . . 11 ⊢ (𝑧 ∈ (𝑋 × 𝑋) → ((nei‘(𝐽 ×t 𝐽))‘{𝑧}) = ((nei‘(𝐽 ×t 𝐽))‘({(1st ‘𝑧)} × {(2nd ‘𝑧)})))
91	90	3ad2ant3 1131	. . . . . . . . . 10 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈 ∧ 𝑧 ∈ (𝑋 × 𝑋)) → ((nei‘(𝐽 ×t 𝐽))‘{𝑧}) = ((nei‘(𝐽 ×t 𝐽))‘({(1st ‘𝑧)} × {(2nd ‘𝑧)})))
92	85, 91	eleqtrrd 2916	. . . . . . . . 9 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈 ∧ 𝑧 ∈ (𝑋 × 𝑋)) → ((𝑉 “ {(1st ‘𝑧)}) × (𝑉 “ {(2nd ‘𝑧)})) ∈ ((nei‘(𝐽 ×t 𝐽))‘{𝑧}))
93	75, 76, 25, 92	syl3anc 1367	. . . . . . . 8 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ (𝑉 ∈ 𝑈 ∧ ◡𝑉 = 𝑉)) ∧ 𝑧 ∈ ((cls‘(𝐽 ×t 𝐽))‘𝑀)) → ((𝑉 “ {(1st ‘𝑧)}) × (𝑉 “ {(2nd ‘𝑧)})) ∈ ((nei‘(𝐽 ×t 𝐽))‘{𝑧}))
94	20	neindisj 21725	. . . . . . . 8 ⊢ ((((𝐽 ×t 𝐽) ∈ Top ∧ 𝑀 ⊆ ∪ (𝐽 ×t 𝐽)) ∧ (𝑧 ∈ ((cls‘(𝐽 ×t 𝐽))‘𝑀) ∧ ((𝑉 “ {(1st ‘𝑧)}) × (𝑉 “ {(2nd ‘𝑧)})) ∈ ((nei‘(𝐽 ×t 𝐽))‘{𝑧}))) → (((𝑉 “ {(1st ‘𝑧)}) × (𝑉 “ {(2nd ‘𝑧)})) ∩ 𝑀) ≠ ∅)
95	7, 19, 24, 93, 94	syl22anc 836	. . . . . . 7 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ (𝑉 ∈ 𝑈 ∧ ◡𝑉 = 𝑉)) ∧ 𝑧 ∈ ((cls‘(𝐽 ×t 𝐽))‘𝑀)) → (((𝑉 “ {(1st ‘𝑧)}) × (𝑉 “ {(2nd ‘𝑧)})) ∩ 𝑀) ≠ ∅)
96		r19.3rzv 4444	. . . . . . 7 ⊢ ((((𝑉 “ {(1st ‘𝑧)}) × (𝑉 “ {(2nd ‘𝑧)})) ∩ 𝑀) ≠ ∅ → ((1st ‘𝑧)(𝑉 ∘ (𝑀 ∘ 𝑉))(2nd ‘𝑧) ↔ ∀𝑟 ∈ (((𝑉 “ {(1st ‘𝑧)}) × (𝑉 “ {(2nd ‘𝑧)})) ∩ 𝑀)(1st ‘𝑧)(𝑉 ∘ (𝑀 ∘ 𝑉))(2nd ‘𝑧)))
97	95, 96	syl 17	. . . . . 6 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ (𝑉 ∈ 𝑈 ∧ ◡𝑉 = 𝑉)) ∧ 𝑧 ∈ ((cls‘(𝐽 ×t 𝐽))‘𝑀)) → ((1st ‘𝑧)(𝑉 ∘ (𝑀 ∘ 𝑉))(2nd ‘𝑧) ↔ ∀𝑟 ∈ (((𝑉 “ {(1st ‘𝑧)}) × (𝑉 “ {(2nd ‘𝑧)})) ∩ 𝑀)(1st ‘𝑧)(𝑉 ∘ (𝑀 ∘ 𝑉))(2nd ‘𝑧)))
98	74, 97	mpbird 259	. . . . 5 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ (𝑉 ∈ 𝑈 ∧ ◡𝑉 = 𝑉)) ∧ 𝑧 ∈ ((cls‘(𝐽 ×t 𝐽))‘𝑀)) → (1st ‘𝑧)(𝑉 ∘ (𝑀 ∘ 𝑉))(2nd ‘𝑧))
99		df-br 5067	. . . . 5 ⊢ ((1st ‘𝑧)(𝑉 ∘ (𝑀 ∘ 𝑉))(2nd ‘𝑧) ↔ ⟨(1st ‘𝑧), (2nd ‘𝑧)⟩ ∈ (𝑉 ∘ (𝑀 ∘ 𝑉)))
100	98, 99	sylib 220	. . . 4 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ (𝑉 ∈ 𝑈 ∧ ◡𝑉 = 𝑉)) ∧ 𝑧 ∈ ((cls‘(𝐽 ×t 𝐽))‘𝑀)) → ⟨(1st ‘𝑧), (2nd ‘𝑧)⟩ ∈ (𝑉 ∘ (𝑀 ∘ 𝑉)))
101	27, 100	eqeltrd 2913	. . 3 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ (𝑉 ∈ 𝑈 ∧ ◡𝑉 = 𝑉)) ∧ 𝑧 ∈ ((cls‘(𝐽 ×t 𝐽))‘𝑀)) → 𝑧 ∈ (𝑉 ∘ (𝑀 ∘ 𝑉)))
102	101	ex 415	. 2 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ (𝑉 ∈ 𝑈 ∧ ◡𝑉 = 𝑉)) → (𝑧 ∈ ((cls‘(𝐽 ×t 𝐽))‘𝑀) → 𝑧 ∈ (𝑉 ∘ (𝑀 ∘ 𝑉))))
103	102	ssrdv 3973	1 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ (𝑉 ∈ 𝑈 ∧ ◡𝑉 = 𝑉)) → ((cls‘(𝐽 ×t 𝐽))‘𝑀) ⊆ (𝑉 ∘ (𝑀 ∘ 𝑉)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114   ≠ wne 3016  ∀wral 3138  Vcvv 3494   ∩ cin 3935   ⊆ wss 3936  ∅c0 4291  {csn 4567  ⟨cop 4573  ∪ cuni 4838   class class class wbr 5066   × cxp 5553  ^◡ccnv 5554   “ cima 5558   ∘ ccom 5559  Rel wrel 5560  ‘cfv 6355  (class class class)co 7156  1^st c1st 7687  2^nd c2nd 7688  Topctop 21501  TopOnctopon 21518  clsccl 21626  neicnei 21705   ×_t ctx 22168  UnifOncust 22808  unifTopcutop 22839

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-rep 5190  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4839  df-int 4877  df-iun 4921  df-iin 4922  df-br 5067  df-opab 5129  df-mpt 5147  df-tr 5173  df-id 5460  df-eprel 5465  df-po 5474  df-so 5475  df-fr 5514  df-we 5516  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-pred 6148  df-ord 6194  df-on 6195  df-lim 6196  df-suc 6197  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-ov 7159  df-oprab 7160  df-mpo 7161  df-om 7581  df-1st 7689  df-2nd 7690  df-wrecs 7947  df-recs 8008  df-rdg 8046  df-1o 8102  df-oadd 8106  df-er 8289  df-en 8510  df-fin 8513  df-fi 8875  df-topgen 16717  df-top 21502  df-topon 21519  df-bases 21554  df-cld 21627  df-ntr 21628  df-cls 21629  df-nei 21706  df-tx 22170  df-ust 22809  df-utop 22840

This theorem is referenced by:  utopreg  22861