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Theorem utop3cls 23756

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 5695	. . . . 5 ⊢ Rel (𝑋 × 𝑋)
2		utoptop.1	. . . . . . . . . . 11 ⊢ 𝐽 = (unifTop‘𝑈)
3		utoptop 23739	. . . . . . . . . . 11 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (unifTop‘𝑈) ∈ Top)
4	2, 3	eqeltrid 2838	. . . . . . . . . 10 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝐽 ∈ Top)
5		txtop 23073	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ 𝐽 ∈ Top) → (𝐽 ×_t 𝐽) ∈ Top)
6	4, 4, 5	syl2anc 585	. . . . . . . . 9 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝐽 ×_t 𝐽) ∈ Top)
7	6	ad3antrrr 729	. . . . . . . 8 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ (𝑉 ∈ 𝑈 ∧ ^◡𝑉 = 𝑉)) ∧ 𝑧 ∈ ((cls‘(𝐽 ×_t 𝐽))‘𝑀)) → (𝐽 ×_t 𝐽) ∈ Top)
8		simpllr 775	. . . . . . . . 9 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ (𝑉 ∈ 𝑈 ∧ ^◡𝑉 = 𝑉)) ∧ 𝑧 ∈ ((cls‘(𝐽 ×_t 𝐽))‘𝑀)) → 𝑀 ⊆ (𝑋 × 𝑋))
9		utoptopon 23741	. . . . . . . . . . . . . 14 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (unifTop‘𝑈) ∈ (TopOn‘𝑋))
10	2, 9	eqeltrid 2838	. . . . . . . . . . . . 13 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝐽 ∈ (TopOn‘𝑋))
11		toponuni 22416	. . . . . . . . . . . . 13 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
12	10, 11	syl 17	. . . . . . . . . . . 12 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = ∪ 𝐽)
13	12	sqxpeqd 5709	. . . . . . . . . . 11 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝑋 × 𝑋) = (∪ 𝐽 × ∪ 𝐽))
14		eqid 2733	. . . . . . . . . . . . 13 ⊢ ∪ 𝐽 = ∪ 𝐽
15	14, 14	txuni 23096	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ Top ∧ 𝐽 ∈ Top) → (∪ 𝐽 × ∪ 𝐽) = ∪ (𝐽 ×_t 𝐽))
16	4, 4, 15	syl2anc 585	. . . . . . . . . . 11 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (∪ 𝐽 × ∪ 𝐽) = ∪ (𝐽 ×_t 𝐽))
17	13, 16	eqtrd 2773	. . . . . . . . . 10 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝑋 × 𝑋) = ∪ (𝐽 ×_t 𝐽))
18	17	ad3antrrr 729	. . . . . . . . 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 4023	. . . . . . . 8 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ (𝑉 ∈ 𝑈 ∧ ^◡𝑉 = 𝑉)) ∧ 𝑧 ∈ ((cls‘(𝐽 ×_t 𝐽))‘𝑀)) → 𝑀 ⊆ ∪ (𝐽 ×_t 𝐽))
20		eqid 2733	. . . . . . . . 9 ⊢ ∪ (𝐽 ×_t 𝐽) = ∪ (𝐽 ×_t 𝐽)
21	20	clsss3 22563	. . . . . . . 8 ⊢ (((𝐽 ×_t 𝐽) ∈ Top ∧ 𝑀 ⊆ ∪ (𝐽 ×_t 𝐽)) → ((cls‘(𝐽 ×_t 𝐽))‘𝑀) ⊆ ∪ (𝐽 ×_t 𝐽))
22	7, 19, 21	syl2anc 585	. . . . . . 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 4024	. . . . . 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 486	. . . . . 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 3984	. . . . 5 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ (𝑉 ∈ 𝑈 ∧ ^◡𝑉 = 𝑉)) ∧ 𝑧 ∈ ((cls‘(𝐽 ×_t 𝐽))‘𝑀)) → 𝑧 ∈ (𝑋 × 𝑋))
26		1st2nd 8025	. . . . 5 ⊢ ((Rel (𝑋 × 𝑋) ∧ 𝑧 ∈ (𝑋 × 𝑋)) → 𝑧 = ⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩)
27	1, 25, 26	sylancr 588	. . . 4 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ (𝑉 ∈ 𝑈 ∧ ^◡𝑉 = 𝑉)) ∧ 𝑧 ∈ ((cls‘(𝐽 ×_t 𝐽))‘𝑀)) → 𝑧 = ⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩)
28		simp-4l 782	. . . . . . . . . 10 ⊢ (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ (𝑉 ∈ 𝑈 ∧ ^◡𝑉 = 𝑉)) ∧ 𝑧 ∈ ((cls‘(𝐽 ×_t 𝐽))‘𝑀)) ∧ 𝑟 ∈ (((𝑉 “ {(1^st ‘𝑧)}) × (𝑉 “ {(2^nd ‘𝑧)})) ∩ 𝑀)) → 𝑈 ∈ (UnifOn‘𝑋))
29		simpr1l 1231	. . . . . . . . . . 11 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ ((𝑉 ∈ 𝑈 ∧ ^◡𝑉 = 𝑉) ∧ 𝑧 ∈ ((cls‘(𝐽 ×_t 𝐽))‘𝑀) ∧ 𝑟 ∈ (((𝑉 “ {(1^st ‘𝑧)}) × (𝑉 “ {(2^nd ‘𝑧)})) ∩ 𝑀))) → 𝑉 ∈ 𝑈)
30	29	3anassrs 1361	. . . . . . . . . 10 ⊢ (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ (𝑉 ∈ 𝑈 ∧ ^◡𝑉 = 𝑉)) ∧ 𝑧 ∈ ((cls‘(𝐽 ×_t 𝐽))‘𝑀)) ∧ 𝑟 ∈ (((𝑉 “ {(1^st ‘𝑧)}) × (𝑉 “ {(2^nd ‘𝑧)})) ∩ 𝑀)) → 𝑉 ∈ 𝑈)
31		ustrel 23716	. . . . . . . . . 10 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) → Rel 𝑉)
32	28, 30, 31	syl2anc 585	. . . . . . . . 9 ⊢ (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ (𝑉 ∈ 𝑈 ∧ ^◡𝑉 = 𝑉)) ∧ 𝑧 ∈ ((cls‘(𝐽 ×_t 𝐽))‘𝑀)) ∧ 𝑟 ∈ (((𝑉 “ {(1^st ‘𝑧)}) × (𝑉 “ {(2^nd ‘𝑧)})) ∩ 𝑀)) → Rel 𝑉)
33		simpr 486	. . . . . . . . . . . 12 ⊢ (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ (𝑉 ∈ 𝑈 ∧ ^◡𝑉 = 𝑉)) ∧ 𝑧 ∈ ((cls‘(𝐽 ×_t 𝐽))‘𝑀)) ∧ 𝑟 ∈ (((𝑉 “ {(1^st ‘𝑧)}) × (𝑉 “ {(2^nd ‘𝑧)})) ∩ 𝑀)) → 𝑟 ∈ (((𝑉 “ {(1^st ‘𝑧)}) × (𝑉 “ {(2^nd ‘𝑧)})) ∩ 𝑀))
34		elin 3965	. . . . . . . . . . . 12 ⊢ (𝑟 ∈ (((𝑉 “ {(1^st ‘𝑧)}) × (𝑉 “ {(2^nd ‘𝑧)})) ∩ 𝑀) ↔ (𝑟 ∈ ((𝑉 “ {(1^st ‘𝑧)}) × (𝑉 “ {(2^nd ‘𝑧)})) ∧ 𝑟 ∈ 𝑀))
35	33, 34	sylib 217	. . . . . . . . . . 11 ⊢ (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ (𝑉 ∈ 𝑈 ∧ ^◡𝑉 = 𝑉)) ∧ 𝑧 ∈ ((cls‘(𝐽 ×_t 𝐽))‘𝑀)) ∧ 𝑟 ∈ (((𝑉 “ {(1^st ‘𝑧)}) × (𝑉 “ {(2^nd ‘𝑧)})) ∩ 𝑀)) → (𝑟 ∈ ((𝑉 “ {(1^st ‘𝑧)}) × (𝑉 “ {(2^nd ‘𝑧)})) ∧ 𝑟 ∈ 𝑀))
36	35	simpld 496	. . . . . . . . . 10 ⊢ (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ (𝑉 ∈ 𝑈 ∧ ^◡𝑉 = 𝑉)) ∧ 𝑧 ∈ ((cls‘(𝐽 ×_t 𝐽))‘𝑀)) ∧ 𝑟 ∈ (((𝑉 “ {(1^st ‘𝑧)}) × (𝑉 “ {(2^nd ‘𝑧)})) ∩ 𝑀)) → 𝑟 ∈ ((𝑉 “ {(1^st ‘𝑧)}) × (𝑉 “ {(2^nd ‘𝑧)})))
37		xp1st 8007	. . . . . . . . . 10 ⊢ (𝑟 ∈ ((𝑉 “ {(1^st ‘𝑧)}) × (𝑉 “ {(2^nd ‘𝑧)})) → (1^st ‘𝑟) ∈ (𝑉 “ {(1^st ‘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 “ {(1^st ‘𝑧)}) × (𝑉 “ {(2^nd ‘𝑧)})) ∩ 𝑀)) → (1^st ‘𝑟) ∈ (𝑉 “ {(1^st ‘𝑧)}))
39		elrelimasn 6085	. . . . . . . . . 10 ⊢ (Rel 𝑉 → ((1^st ‘𝑟) ∈ (𝑉 “ {(1^st ‘𝑧)}) ↔ (1^st ‘𝑧)𝑉(1^st ‘𝑟)))
40	39	biimpa 478	. . . . . . . . 9 ⊢ ((Rel 𝑉 ∧ (1^st ‘𝑟) ∈ (𝑉 “ {(1^st ‘𝑧)})) → (1^st ‘𝑧)𝑉(1^st ‘𝑟))
41	32, 38, 40	syl2anc 585	. . . . . . . 8 ⊢ (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ (𝑉 ∈ 𝑈 ∧ ^◡𝑉 = 𝑉)) ∧ 𝑧 ∈ ((cls‘(𝐽 ×_t 𝐽))‘𝑀)) ∧ 𝑟 ∈ (((𝑉 “ {(1^st ‘𝑧)}) × (𝑉 “ {(2^nd ‘𝑧)})) ∩ 𝑀)) → (1^st ‘𝑧)𝑉(1^st ‘𝑟))
42		simp-4r 783	. . . . . . . . . . 11 ⊢ (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ (𝑉 ∈ 𝑈 ∧ ^◡𝑉 = 𝑉)) ∧ 𝑧 ∈ ((cls‘(𝐽 ×_t 𝐽))‘𝑀)) ∧ 𝑟 ∈ (((𝑉 “ {(1^st ‘𝑧)}) × (𝑉 “ {(2^nd ‘𝑧)})) ∩ 𝑀)) → 𝑀 ⊆ (𝑋 × 𝑋))
43		xpss 5693	. . . . . . . . . . 11 ⊢ (𝑋 × 𝑋) ⊆ (V × V)
44	42, 43	sstrdi 3995	. . . . . . . . . 10 ⊢ (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ (𝑉 ∈ 𝑈 ∧ ^◡𝑉 = 𝑉)) ∧ 𝑧 ∈ ((cls‘(𝐽 ×_t 𝐽))‘𝑀)) ∧ 𝑟 ∈ (((𝑉 “ {(1^st ‘𝑧)}) × (𝑉 “ {(2^nd ‘𝑧)})) ∩ 𝑀)) → 𝑀 ⊆ (V × V))
45		df-rel 5684	. . . . . . . . . 10 ⊢ (Rel 𝑀 ↔ 𝑀 ⊆ (V × V))
46	44, 45	sylibr 233	. . . . . . . . 9 ⊢ (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ (𝑉 ∈ 𝑈 ∧ ^◡𝑉 = 𝑉)) ∧ 𝑧 ∈ ((cls‘(𝐽 ×_t 𝐽))‘𝑀)) ∧ 𝑟 ∈ (((𝑉 “ {(1^st ‘𝑧)}) × (𝑉 “ {(2^nd ‘𝑧)})) ∩ 𝑀)) → Rel 𝑀)
47	35	simprd 497	. . . . . . . . 9 ⊢ (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ (𝑉 ∈ 𝑈 ∧ ^◡𝑉 = 𝑉)) ∧ 𝑧 ∈ ((cls‘(𝐽 ×_t 𝐽))‘𝑀)) ∧ 𝑟 ∈ (((𝑉 “ {(1^st ‘𝑧)}) × (𝑉 “ {(2^nd ‘𝑧)})) ∩ 𝑀)) → 𝑟 ∈ 𝑀)
48		1st2ndbr 8028	. . . . . . . . 9 ⊢ ((Rel 𝑀 ∧ 𝑟 ∈ 𝑀) → (1^st ‘𝑟)𝑀(2^nd ‘𝑟))
49	46, 47, 48	syl2anc 585	. . . . . . . 8 ⊢ (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ (𝑉 ∈ 𝑈 ∧ ^◡𝑉 = 𝑉)) ∧ 𝑧 ∈ ((cls‘(𝐽 ×_t 𝐽))‘𝑀)) ∧ 𝑟 ∈ (((𝑉 “ {(1^st ‘𝑧)}) × (𝑉 “ {(2^nd ‘𝑧)})) ∩ 𝑀)) → (1^st ‘𝑟)𝑀(2^nd ‘𝑟))
50		xp2nd 8008	. . . . . . . . . . 11 ⊢ (𝑟 ∈ ((𝑉 “ {(1^st ‘𝑧)}) × (𝑉 “ {(2^nd ‘𝑧)})) → (2^nd ‘𝑟) ∈ (𝑉 “ {(2^nd ‘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 “ {(1^st ‘𝑧)}) × (𝑉 “ {(2^nd ‘𝑧)})) ∩ 𝑀)) → (2^nd ‘𝑟) ∈ (𝑉 “ {(2^nd ‘𝑧)}))
52		elrelimasn 6085	. . . . . . . . . . 11 ⊢ (Rel 𝑉 → ((2^nd ‘𝑟) ∈ (𝑉 “ {(2^nd ‘𝑧)}) ↔ (2^nd ‘𝑧)𝑉(2^nd ‘𝑟)))
53	52	biimpa 478	. . . . . . . . . 10 ⊢ ((Rel 𝑉 ∧ (2^nd ‘𝑟) ∈ (𝑉 “ {(2^nd ‘𝑧)})) → (2^nd ‘𝑧)𝑉(2^nd ‘𝑟))
54	32, 51, 53	syl2anc 585	. . . . . . . . 9 ⊢ (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ (𝑉 ∈ 𝑈 ∧ ^◡𝑉 = 𝑉)) ∧ 𝑧 ∈ ((cls‘(𝐽 ×_t 𝐽))‘𝑀)) ∧ 𝑟 ∈ (((𝑉 “ {(1^st ‘𝑧)}) × (𝑉 “ {(2^nd ‘𝑧)})) ∩ 𝑀)) → (2^nd ‘𝑧)𝑉(2^nd ‘𝑟))
55		simpr1r 1232	. . . . . . . . . . 11 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ ((𝑉 ∈ 𝑈 ∧ ^◡𝑉 = 𝑉) ∧ 𝑧 ∈ ((cls‘(𝐽 ×_t 𝐽))‘𝑀) ∧ 𝑟 ∈ (((𝑉 “ {(1^st ‘𝑧)}) × (𝑉 “ {(2^nd ‘𝑧)})) ∩ 𝑀))) → ^◡𝑉 = 𝑉)
56	55	3anassrs 1361	. . . . . . . . . 10 ⊢ (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ (𝑉 ∈ 𝑈 ∧ ^◡𝑉 = 𝑉)) ∧ 𝑧 ∈ ((cls‘(𝐽 ×_t 𝐽))‘𝑀)) ∧ 𝑟 ∈ (((𝑉 “ {(1^st ‘𝑧)}) × (𝑉 “ {(2^nd ‘𝑧)})) ∩ 𝑀)) → ^◡𝑉 = 𝑉)
57		breq 5151	. . . . . . . . . . 11 ⊢ (^◡𝑉 = 𝑉 → ((2^nd ‘𝑟)^◡𝑉(2^nd ‘𝑧) ↔ (2^nd ‘𝑟)𝑉(2^nd ‘𝑧)))
58		fvex 6905	. . . . . . . . . . . 12 ⊢ (2^nd ‘𝑟) ∈ V
59		fvex 6905	. . . . . . . . . . . 12 ⊢ (2^nd ‘𝑧) ∈ V
60	58, 59	brcnv 5883	. . . . . . . . . . 11 ⊢ ((2^nd ‘𝑟)^◡𝑉(2^nd ‘𝑧) ↔ (2^nd ‘𝑧)𝑉(2^nd ‘𝑟))
61	57, 60	bitr3di 286	. . . . . . . . . 10 ⊢ (^◡𝑉 = 𝑉 → ((2^nd ‘𝑟)𝑉(2^nd ‘𝑧) ↔ (2^nd ‘𝑧)𝑉(2^nd ‘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 “ {(1^st ‘𝑧)}) × (𝑉 “ {(2^nd ‘𝑧)})) ∩ 𝑀)) → ((2^nd ‘𝑟)𝑉(2^nd ‘𝑧) ↔ (2^nd ‘𝑧)𝑉(2^nd ‘𝑟)))
63	54, 62	mpbird 257	. . . . . . . 8 ⊢ (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ (𝑉 ∈ 𝑈 ∧ ^◡𝑉 = 𝑉)) ∧ 𝑧 ∈ ((cls‘(𝐽 ×_t 𝐽))‘𝑀)) ∧ 𝑟 ∈ (((𝑉 “ {(1^st ‘𝑧)}) × (𝑉 “ {(2^nd ‘𝑧)})) ∩ 𝑀)) → (2^nd ‘𝑟)𝑉(2^nd ‘𝑧))
64		fvex 6905	. . . . . . . . . 10 ⊢ (1^st ‘𝑧) ∈ V
65		fvex 6905	. . . . . . . . . 10 ⊢ (1^st ‘𝑟) ∈ V
66		brcogw 5869	. . . . . . . . . . 11 ⊢ ((((1^st ‘𝑧) ∈ V ∧ (2^nd ‘𝑟) ∈ V ∧ (1^st ‘𝑟) ∈ V) ∧ ((1^st ‘𝑧)𝑉(1^st ‘𝑟) ∧ (1^st ‘𝑟)𝑀(2^nd ‘𝑟))) → (1^st ‘𝑧)(𝑀 ∘ 𝑉)(2^nd ‘𝑟))
67	66	ex 414	. . . . . . . . . 10 ⊢ (((1^st ‘𝑧) ∈ V ∧ (2^nd ‘𝑟) ∈ V ∧ (1^st ‘𝑟) ∈ V) → (((1^st ‘𝑧)𝑉(1^st ‘𝑟) ∧ (1^st ‘𝑟)𝑀(2^nd ‘𝑟)) → (1^st ‘𝑧)(𝑀 ∘ 𝑉)(2^nd ‘𝑟)))
68	64, 58, 65, 67	mp3an 1462	. . . . . . . . 9 ⊢ (((1^st ‘𝑧)𝑉(1^st ‘𝑟) ∧ (1^st ‘𝑟)𝑀(2^nd ‘𝑟)) → (1^st ‘𝑧)(𝑀 ∘ 𝑉)(2^nd ‘𝑟))
69		brcogw 5869	. . . . . . . . . . 11 ⊢ ((((1^st ‘𝑧) ∈ V ∧ (2^nd ‘𝑧) ∈ V ∧ (2^nd ‘𝑟) ∈ V) ∧ ((1^st ‘𝑧)(𝑀 ∘ 𝑉)(2^nd ‘𝑟) ∧ (2^nd ‘𝑟)𝑉(2^nd ‘𝑧))) → (1^st ‘𝑧)(𝑉 ∘ (𝑀 ∘ 𝑉))(2^nd ‘𝑧))
70	69	ex 414	. . . . . . . . . 10 ⊢ (((1^st ‘𝑧) ∈ V ∧ (2^nd ‘𝑧) ∈ V ∧ (2^nd ‘𝑟) ∈ V) → (((1^st ‘𝑧)(𝑀 ∘ 𝑉)(2^nd ‘𝑟) ∧ (2^nd ‘𝑟)𝑉(2^nd ‘𝑧)) → (1^st ‘𝑧)(𝑉 ∘ (𝑀 ∘ 𝑉))(2^nd ‘𝑧)))
71	64, 59, 58, 70	mp3an 1462	. . . . . . . . 9 ⊢ (((1^st ‘𝑧)(𝑀 ∘ 𝑉)(2^nd ‘𝑟) ∧ (2^nd ‘𝑟)𝑉(2^nd ‘𝑧)) → (1^st ‘𝑧)(𝑉 ∘ (𝑀 ∘ 𝑉))(2^nd ‘𝑧))
72	68, 71	sylan 581	. . . . . . . 8 ⊢ ((((1^st ‘𝑧)𝑉(1^st ‘𝑟) ∧ (1^st ‘𝑟)𝑀(2^nd ‘𝑟)) ∧ (2^nd ‘𝑟)𝑉(2^nd ‘𝑧)) → (1^st ‘𝑧)(𝑉 ∘ (𝑀 ∘ 𝑉))(2^nd ‘𝑧))
73	41, 49, 63, 72	syl21anc 837	. . . . . . 7 ⊢ (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ (𝑉 ∈ 𝑈 ∧ ^◡𝑉 = 𝑉)) ∧ 𝑧 ∈ ((cls‘(𝐽 ×_t 𝐽))‘𝑀)) ∧ 𝑟 ∈ (((𝑉 “ {(1^st ‘𝑧)}) × (𝑉 “ {(2^nd ‘𝑧)})) ∩ 𝑀)) → (1^st ‘𝑧)(𝑉 ∘ (𝑀 ∘ 𝑉))(2^nd ‘𝑧))
74	73	ralrimiva 3147	. . . . . 6 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ (𝑉 ∈ 𝑈 ∧ ^◡𝑉 = 𝑉)) ∧ 𝑧 ∈ ((cls‘(𝐽 ×_t 𝐽))‘𝑀)) → ∀𝑟 ∈ (((𝑉 “ {(1^st ‘𝑧)}) × (𝑉 “ {(2^nd ‘𝑧)})) ∩ 𝑀)(1^st ‘𝑧)(𝑉 ∘ (𝑀 ∘ 𝑉))(2^nd ‘𝑧))
75		simplll 774	. . . . . . . . 9 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ (𝑉 ∈ 𝑈 ∧ ^◡𝑉 = 𝑉)) ∧ 𝑧 ∈ ((cls‘(𝐽 ×_t 𝐽))‘𝑀)) → 𝑈 ∈ (UnifOn‘𝑋))
76		simplrl 776	. . . . . . . . 9 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ (𝑉 ∈ 𝑈 ∧ ^◡𝑉 = 𝑉)) ∧ 𝑧 ∈ ((cls‘(𝐽 ×_t 𝐽))‘𝑀)) → 𝑉 ∈ 𝑈)
77	4	3ad2ant1 1134	. . . . . . . . . . 11 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈 ∧ 𝑧 ∈ (𝑋 × 𝑋)) → 𝐽 ∈ Top)
78		xp1st 8007	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ (𝑋 × 𝑋) → (1^st ‘𝑧) ∈ 𝑋)
79	2	utopsnnei 23754	. . . . . . . . . . . 12 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈 ∧ (1^st ‘𝑧) ∈ 𝑋) → (𝑉 “ {(1^st ‘𝑧)}) ∈ ((nei‘𝐽)‘{(1^st ‘𝑧)}))
80	78, 79	syl3an3 1166	. . . . . . . . . . 11 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈 ∧ 𝑧 ∈ (𝑋 × 𝑋)) → (𝑉 “ {(1^st ‘𝑧)}) ∈ ((nei‘𝐽)‘{(1^st ‘𝑧)}))
81		xp2nd 8008	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ (𝑋 × 𝑋) → (2^nd ‘𝑧) ∈ 𝑋)
82	2	utopsnnei 23754	. . . . . . . . . . . 12 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈 ∧ (2^nd ‘𝑧) ∈ 𝑋) → (𝑉 “ {(2^nd ‘𝑧)}) ∈ ((nei‘𝐽)‘{(2^nd ‘𝑧)}))
83	81, 82	syl3an3 1166	. . . . . . . . . . 11 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈 ∧ 𝑧 ∈ (𝑋 × 𝑋)) → (𝑉 “ {(2^nd ‘𝑧)}) ∈ ((nei‘𝐽)‘{(2^nd ‘𝑧)}))
84	14, 14	neitx 23111	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ Top ∧ 𝐽 ∈ Top) ∧ ((𝑉 “ {(1^st ‘𝑧)}) ∈ ((nei‘𝐽)‘{(1^st ‘𝑧)}) ∧ (𝑉 “ {(2^nd ‘𝑧)}) ∈ ((nei‘𝐽)‘{(2^nd ‘𝑧)}))) → ((𝑉 “ {(1^st ‘𝑧)}) × (𝑉 “ {(2^nd ‘𝑧)})) ∈ ((nei‘(𝐽 ×_t 𝐽))‘({(1^st ‘𝑧)} × {(2^nd ‘𝑧)})))
85	77, 77, 80, 83, 84	syl22anc 838	. . . . . . . . . 10 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈 ∧ 𝑧 ∈ (𝑋 × 𝑋)) → ((𝑉 “ {(1^st ‘𝑧)}) × (𝑉 “ {(2^nd ‘𝑧)})) ∈ ((nei‘(𝐽 ×_t 𝐽))‘({(1^st ‘𝑧)} × {(2^nd ‘𝑧)})))
86		1st2nd2 8014	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ (𝑋 × 𝑋) → 𝑧 = ⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩)
87	86	sneqd 4641	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ (𝑋 × 𝑋) → {𝑧} = {⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩})
88	64, 59	xpsn 7139	. . . . . . . . . . . . 13 ⊢ ({(1^st ‘𝑧)} × {(2^nd ‘𝑧)}) = {⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩}
89	87, 88	eqtr4di 2791	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ (𝑋 × 𝑋) → {𝑧} = ({(1^st ‘𝑧)} × {(2^nd ‘𝑧)}))
90	89	fveq2d 6896	. . . . . . . . . . 11 ⊢ (𝑧 ∈ (𝑋 × 𝑋) → ((nei‘(𝐽 ×_t 𝐽))‘{𝑧}) = ((nei‘(𝐽 ×_t 𝐽))‘({(1^st ‘𝑧)} × {(2^nd ‘𝑧)})))
91	90	3ad2ant3 1136	. . . . . . . . . 10 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈 ∧ 𝑧 ∈ (𝑋 × 𝑋)) → ((nei‘(𝐽 ×_t 𝐽))‘{𝑧}) = ((nei‘(𝐽 ×_t 𝐽))‘({(1^st ‘𝑧)} × {(2^nd ‘𝑧)})))
92	85, 91	eleqtrrd 2837	. . . . . . . . 9 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈 ∧ 𝑧 ∈ (𝑋 × 𝑋)) → ((𝑉 “ {(1^st ‘𝑧)}) × (𝑉 “ {(2^nd ‘𝑧)})) ∈ ((nei‘(𝐽 ×_t 𝐽))‘{𝑧}))
93	75, 76, 25, 92	syl3anc 1372	. . . . . . . 8 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ (𝑉 ∈ 𝑈 ∧ ^◡𝑉 = 𝑉)) ∧ 𝑧 ∈ ((cls‘(𝐽 ×_t 𝐽))‘𝑀)) → ((𝑉 “ {(1^st ‘𝑧)}) × (𝑉 “ {(2^nd ‘𝑧)})) ∈ ((nei‘(𝐽 ×_t 𝐽))‘{𝑧}))
94	20	neindisj 22621	. . . . . . . 8 ⊢ ((((𝐽 ×_t 𝐽) ∈ Top ∧ 𝑀 ⊆ ∪ (𝐽 ×_t 𝐽)) ∧ (𝑧 ∈ ((cls‘(𝐽 ×_t 𝐽))‘𝑀) ∧ ((𝑉 “ {(1^st ‘𝑧)}) × (𝑉 “ {(2^nd ‘𝑧)})) ∈ ((nei‘(𝐽 ×_t 𝐽))‘{𝑧}))) → (((𝑉 “ {(1^st ‘𝑧)}) × (𝑉 “ {(2^nd ‘𝑧)})) ∩ 𝑀) ≠ ∅)
95	7, 19, 24, 93, 94	syl22anc 838	. . . . . . 7 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ (𝑉 ∈ 𝑈 ∧ ^◡𝑉 = 𝑉)) ∧ 𝑧 ∈ ((cls‘(𝐽 ×_t 𝐽))‘𝑀)) → (((𝑉 “ {(1^st ‘𝑧)}) × (𝑉 “ {(2^nd ‘𝑧)})) ∩ 𝑀) ≠ ∅)
96		r19.3rzv 4499	. . . . . . 7 ⊢ ((((𝑉 “ {(1^st ‘𝑧)}) × (𝑉 “ {(2^nd ‘𝑧)})) ∩ 𝑀) ≠ ∅ → ((1^st ‘𝑧)(𝑉 ∘ (𝑀 ∘ 𝑉))(2^nd ‘𝑧) ↔ ∀𝑟 ∈ (((𝑉 “ {(1^st ‘𝑧)}) × (𝑉 “ {(2^nd ‘𝑧)})) ∩ 𝑀)(1^st ‘𝑧)(𝑉 ∘ (𝑀 ∘ 𝑉))(2^nd ‘𝑧)))
97	95, 96	syl 17	. . . . . 6 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ (𝑉 ∈ 𝑈 ∧ ^◡𝑉 = 𝑉)) ∧ 𝑧 ∈ ((cls‘(𝐽 ×_t 𝐽))‘𝑀)) → ((1^st ‘𝑧)(𝑉 ∘ (𝑀 ∘ 𝑉))(2^nd ‘𝑧) ↔ ∀𝑟 ∈ (((𝑉 “ {(1^st ‘𝑧)}) × (𝑉 “ {(2^nd ‘𝑧)})) ∩ 𝑀)(1^st ‘𝑧)(𝑉 ∘ (𝑀 ∘ 𝑉))(2^nd ‘𝑧)))
98	74, 97	mpbird 257	. . . . 5 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ (𝑉 ∈ 𝑈 ∧ ^◡𝑉 = 𝑉)) ∧ 𝑧 ∈ ((cls‘(𝐽 ×_t 𝐽))‘𝑀)) → (1^st ‘𝑧)(𝑉 ∘ (𝑀 ∘ 𝑉))(2^nd ‘𝑧))
99		df-br 5150	. . . . 5 ⊢ ((1^st ‘𝑧)(𝑉 ∘ (𝑀 ∘ 𝑉))(2^nd ‘𝑧) ↔ ⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩ ∈ (𝑉 ∘ (𝑀 ∘ 𝑉)))
100	98, 99	sylib 217	. . . 4 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ (𝑉 ∈ 𝑈 ∧ ^◡𝑉 = 𝑉)) ∧ 𝑧 ∈ ((cls‘(𝐽 ×_t 𝐽))‘𝑀)) → ⟨(1^st ‘𝑧), (2^nd ‘𝑧)⟩ ∈ (𝑉 ∘ (𝑀 ∘ 𝑉)))
101	27, 100	eqeltrd 2834	. . 3 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ (𝑉 ∈ 𝑈 ∧ ^◡𝑉 = 𝑉)) ∧ 𝑧 ∈ ((cls‘(𝐽 ×_t 𝐽))‘𝑀)) → 𝑧 ∈ (𝑉 ∘ (𝑀 ∘ 𝑉)))
102	101	ex 414	. 2 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ (𝑉 ∈ 𝑈 ∧ ^◡𝑉 = 𝑉)) → (𝑧 ∈ ((cls‘(𝐽 ×_t 𝐽))‘𝑀) → 𝑧 ∈ (𝑉 ∘ (𝑀 ∘ 𝑉))))
103	102	ssrdv 3989	1 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ (𝑉 ∈ 𝑈 ∧ ^◡𝑉 = 𝑉)) → ((cls‘(𝐽 ×_t 𝐽))‘𝑀) ⊆ (𝑉 ∘ (𝑀 ∘ 𝑉)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 ∀wral 3062 Vcvv 3475 ∩ cin 3948 ⊆ wss 3949 ∅c0 4323 {csn 4629 ⟨cop 4635 ∪ cuni 4909 class class class wbr 5149 × cxp 5675 ^◡ccnv 5676 “ cima 5680 ∘ ccom 5681 Rel wrel 5682 ‘cfv 6544 (class class class)co 7409 1^st c1st 7973 2^nd c2nd 7974 Topctop 22395 TopOnctopon 22412 clsccl 22522 neicnei 22601 ×_t ctx 23064 UnifOncust 23704 unifTopcutop 23735

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-iin 5001 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-1st 7975 df-2nd 7976 df-1o 8466 df-er 8703 df-en 8940 df-fin 8943 df-fi 9406 df-topgen 17389 df-top 22396 df-topon 22413 df-bases 22449 df-cld 22523 df-ntr 22524 df-cls 22525 df-nei 22602 df-tx 23066 df-ust 23705 df-utop 23736

This theorem is referenced by: utopreg 23757

W3C validator