Theorem utop3cls 24115

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 5649	. . . . 5 ⊢ Rel (𝑋 × 𝑋)
2		utoptop.1	. . . . . . . . . . 11 ⊢ 𝐽 = (unifTop‘𝑈)
3		utoptop 24098	. . . . . . . . . . 11 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (unifTop‘𝑈) ∈ Top)
4	2, 3	eqeltrid 2832	. . . . . . . . . 10 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝐽 ∈ Top)
5		txtop 23432	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ 𝐽 ∈ Top) → (𝐽 ×t 𝐽) ∈ Top)
6	4, 4, 5	syl2anc 584	. . . . . . . . 9 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝐽 ×t 𝐽) ∈ Top)
7	6	ad3antrrr 730	. . . . . . . 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 24100	. . . . . . . . . . . . . 14 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (unifTop‘𝑈) ∈ (TopOn‘𝑋))
10	2, 9	eqeltrid 2832	. . . . . . . . . . . . 13 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝐽 ∈ (TopOn‘𝑋))
11		toponuni 22777	. . . . . . . . . . . . 13 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
12	10, 11	syl 17	. . . . . . . . . . . 12 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = ∪ 𝐽)
13	12	sqxpeqd 5663	. . . . . . . . . . 11 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝑋 × 𝑋) = (∪ 𝐽 × ∪ 𝐽))
14		eqid 2729	. . . . . . . . . . . . 13 ⊢ ∪ 𝐽 = ∪ 𝐽
15	14, 14	txuni 23455	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ Top ∧ 𝐽 ∈ Top) → (∪ 𝐽 × ∪ 𝐽) = ∪ (𝐽 ×t 𝐽))
16	4, 4, 15	syl2anc 584	. . . . . . . . . . 11 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (∪ 𝐽 × ∪ 𝐽) = ∪ (𝐽 ×t 𝐽))
17	13, 16	eqtrd 2764	. . . . . . . . . 10 ⊢ (𝑈 ∈ (UnifOn‘𝑋) → (𝑋 × 𝑋) = ∪ (𝐽 ×t 𝐽))
18	17	ad3antrrr 730	. . . . . . . . 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 3980	. . . . . . . 8 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ (𝑉 ∈ 𝑈 ∧ ◡𝑉 = 𝑉)) ∧ 𝑧 ∈ ((cls‘(𝐽 ×t 𝐽))‘𝑀)) → 𝑀 ⊆ ∪ (𝐽 ×t 𝐽))
20		eqid 2729	. . . . . . . . 9 ⊢ ∪ (𝐽 ×t 𝐽) = ∪ (𝐽 ×t 𝐽)
21	20	clsss3 22922	. . . . . . . 8 ⊢ (((𝐽 ×t 𝐽) ∈ Top ∧ 𝑀 ⊆ ∪ (𝐽 ×t 𝐽)) → ((cls‘(𝐽 ×t 𝐽))‘𝑀) ⊆ ∪ (𝐽 ×t 𝐽))
22	7, 19, 21	syl2anc 584	. . . . . . 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 3981	. . . . . 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 484	. . . . . 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 3944	. . . . 5 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ (𝑉 ∈ 𝑈 ∧ ◡𝑉 = 𝑉)) ∧ 𝑧 ∈ ((cls‘(𝐽 ×t 𝐽))‘𝑀)) → 𝑧 ∈ (𝑋 × 𝑋))
26		1st2nd 7997	. . . . 5 ⊢ ((Rel (𝑋 × 𝑋) ∧ 𝑧 ∈ (𝑋 × 𝑋)) → 𝑧 = ⟨(1st ‘𝑧), (2nd ‘𝑧)⟩)
27	1, 25, 26	sylancr 587	. . . 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 782	. . . . . . . . . 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 1231	. . . . . . . . . . 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 1361	. . . . . . . . . 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 24075	. . . . . . . . . 10 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈) → Rel 𝑉)
32	28, 30, 31	syl2anc 584	. . . . . . . . 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 484	. . . . . . . . . . . 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 3927	. . . . . . . . . . . 12 ⊢ (𝑟 ∈ (((𝑉 “ {(1st ‘𝑧)}) × (𝑉 “ {(2nd ‘𝑧)})) ∩ 𝑀) ↔ (𝑟 ∈ ((𝑉 “ {(1st ‘𝑧)}) × (𝑉 “ {(2nd ‘𝑧)})) ∧ 𝑟 ∈ 𝑀))
35	33, 34	sylib 218	. . . . . . . . . . 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 494	. . . . . . . . . 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 7979	. . . . . . . . . 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 6046	. . . . . . . . . 10 ⊢ (Rel 𝑉 → ((1st ‘𝑟) ∈ (𝑉 “ {(1st ‘𝑧)}) ↔ (1st ‘𝑧)𝑉(1st ‘𝑟)))
40	39	biimpa 476	. . . . . . . . 9 ⊢ ((Rel 𝑉 ∧ (1st ‘𝑟) ∈ (𝑉 “ {(1st ‘𝑧)})) → (1st ‘𝑧)𝑉(1st ‘𝑟))
41	32, 38, 40	syl2anc 584	. . . . . . . 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 783	. . . . . . . . . . 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 5647	. . . . . . . . . . 11 ⊢ (𝑋 × 𝑋) ⊆ (V × V)
44	42, 43	sstrdi 3956	. . . . . . . . . 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 5638	. . . . . . . . . 10 ⊢ (Rel 𝑀 ↔ 𝑀 ⊆ (V × V))
46	44, 45	sylibr 234	. . . . . . . . 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 495	. . . . . . . . 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 8000	. . . . . . . . 9 ⊢ ((Rel 𝑀 ∧ 𝑟 ∈ 𝑀) → (1st ‘𝑟)𝑀(2nd ‘𝑟))
49	46, 47, 48	syl2anc 584	. . . . . . . 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 7980	. . . . . . . . . . 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 6046	. . . . . . . . . . 11 ⊢ (Rel 𝑉 → ((2nd ‘𝑟) ∈ (𝑉 “ {(2nd ‘𝑧)}) ↔ (2nd ‘𝑧)𝑉(2nd ‘𝑟)))
53	52	biimpa 476	. . . . . . . . . 10 ⊢ ((Rel 𝑉 ∧ (2nd ‘𝑟) ∈ (𝑉 “ {(2nd ‘𝑧)})) → (2nd ‘𝑧)𝑉(2nd ‘𝑟))
54	32, 51, 53	syl2anc 584	. . . . . . . . 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 1232	. . . . . . . . . . 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 1361	. . . . . . . . . 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		breq 5104	. . . . . . . . . . 11 ⊢ (◡𝑉 = 𝑉 → ((2nd ‘𝑟)◡𝑉(2nd ‘𝑧) ↔ (2nd ‘𝑟)𝑉(2nd ‘𝑧)))
58		fvex 6853	. . . . . . . . . . . 12 ⊢ (2nd ‘𝑟) ∈ V
59		fvex 6853	. . . . . . . . . . . 12 ⊢ (2nd ‘𝑧) ∈ V
60	58, 59	brcnv 5836	. . . . . . . . . . 11 ⊢ ((2nd ‘𝑟)◡𝑉(2nd ‘𝑧) ↔ (2nd ‘𝑧)𝑉(2nd ‘𝑟))
61	57, 60	bitr3di 286	. . . . . . . . . 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 257	. . . . . . . 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 6853	. . . . . . . . . 10 ⊢ (1st ‘𝑧) ∈ V
65		fvex 6853	. . . . . . . . . 10 ⊢ (1st ‘𝑟) ∈ V
66		brcogw 5822	. . . . . . . . . . 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 412	. . . . . . . . . 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, 58, 65, 67	mp3an 1463	. . . . . . . . 9 ⊢ (((1st ‘𝑧)𝑉(1st ‘𝑟) ∧ (1st ‘𝑟)𝑀(2nd ‘𝑟)) → (1st ‘𝑧)(𝑀 ∘ 𝑉)(2nd ‘𝑟))
69		brcogw 5822	. . . . . . . . . . 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 412	. . . . . . . . . 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, 59, 58, 70	mp3an 1463	. . . . . . . . 9 ⊢ (((1st ‘𝑧)(𝑀 ∘ 𝑉)(2nd ‘𝑟) ∧ (2nd ‘𝑟)𝑉(2nd ‘𝑧)) → (1st ‘𝑧)(𝑉 ∘ (𝑀 ∘ 𝑉))(2nd ‘𝑧))
72	68, 71	sylan 580	. . . . . . . 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 837	. . . . . . 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 3125	. . . . . 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 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 1133	. . . . . . . . . . 11 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈 ∧ 𝑧 ∈ (𝑋 × 𝑋)) → 𝐽 ∈ Top)
78		xp1st 7979	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ (𝑋 × 𝑋) → (1st ‘𝑧) ∈ 𝑋)
79	2	utopsnnei 24113	. . . . . . . . . . . 12 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈 ∧ (1st ‘𝑧) ∈ 𝑋) → (𝑉 “ {(1st ‘𝑧)}) ∈ ((nei‘𝐽)‘{(1st ‘𝑧)}))
80	78, 79	syl3an3 1165	. . . . . . . . . . 11 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈 ∧ 𝑧 ∈ (𝑋 × 𝑋)) → (𝑉 “ {(1st ‘𝑧)}) ∈ ((nei‘𝐽)‘{(1st ‘𝑧)}))
81		xp2nd 7980	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ (𝑋 × 𝑋) → (2nd ‘𝑧) ∈ 𝑋)
82	2	utopsnnei 24113	. . . . . . . . . . . 12 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈 ∧ (2nd ‘𝑧) ∈ 𝑋) → (𝑉 “ {(2nd ‘𝑧)}) ∈ ((nei‘𝐽)‘{(2nd ‘𝑧)}))
83	81, 82	syl3an3 1165	. . . . . . . . . . 11 ⊢ ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉 ∈ 𝑈 ∧ 𝑧 ∈ (𝑋 × 𝑋)) → (𝑉 “ {(2nd ‘𝑧)}) ∈ ((nei‘𝐽)‘{(2nd ‘𝑧)}))
84	14, 14	neitx 23470	. . . . . . . . . . 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 838	. . . . . . . . . 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 7986	. . . . . . . . . . . . . 14 ⊢ (𝑧 ∈ (𝑋 × 𝑋) → 𝑧 = ⟨(1st ‘𝑧), (2nd ‘𝑧)⟩)
87	86	sneqd 4597	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ (𝑋 × 𝑋) → {𝑧} = {⟨(1st ‘𝑧), (2nd ‘𝑧)⟩})
88	64, 59	xpsn 7095	. . . . . . . . . . . . 13 ⊢ ({(1st ‘𝑧)} × {(2nd ‘𝑧)}) = {⟨(1st ‘𝑧), (2nd ‘𝑧)⟩}
89	87, 88	eqtr4di 2782	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ (𝑋 × 𝑋) → {𝑧} = ({(1st ‘𝑧)} × {(2nd ‘𝑧)}))
90	89	fveq2d 6844	. . . . . . . . . . 11 ⊢ (𝑧 ∈ (𝑋 × 𝑋) → ((nei‘(𝐽 ×t 𝐽))‘{𝑧}) = ((nei‘(𝐽 ×t 𝐽))‘({(1st ‘𝑧)} × {(2nd ‘𝑧)})))
91	90	3ad2ant3 1135	. . . . . . . . . 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 2831	. . . . . . . . 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 1373	. . . . . . . 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 22980	. . . . . . . 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 838	. . . . . . 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 4458	. . . . . . 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 257	. . . . 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 5103	. . . . 5 ⊢ ((1st ‘𝑧)(𝑉 ∘ (𝑀 ∘ 𝑉))(2nd ‘𝑧) ↔ ⟨(1st ‘𝑧), (2nd ‘𝑧)⟩ ∈ (𝑉 ∘ (𝑀 ∘ 𝑉)))
100	98, 99	sylib 218	. . . 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 2828	. . 3 ⊢ ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ (𝑉 ∈ 𝑈 ∧ ◡𝑉 = 𝑉)) ∧ 𝑧 ∈ ((cls‘(𝐽 ×t 𝐽))‘𝑀)) → 𝑧 ∈ (𝑉 ∘ (𝑀 ∘ 𝑉)))
102	101	ex 412	. 2 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ (𝑉 ∈ 𝑈 ∧ ◡𝑉 = 𝑉)) → (𝑧 ∈ ((cls‘(𝐽 ×t 𝐽))‘𝑀) → 𝑧 ∈ (𝑉 ∘ (𝑀 ∘ 𝑉))))
103	102	ssrdv 3949	1 ⊢ (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ (𝑉 ∈ 𝑈 ∧ ◡𝑉 = 𝑉)) → ((cls‘(𝐽 ×t 𝐽))‘𝑀) ⊆ (𝑉 ∘ (𝑀 ∘ 𝑉)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  ∀wral 3044  Vcvv 3444   ∩ cin 3910   ⊆ wss 3911  ∅c0 4292  {csn 4585  ⟨cop 4591  ∪ cuni 4867   class class class wbr 5102   × cxp 5629  ^◡ccnv 5630   “ cima 5634   ∘ ccom 5635  Rel wrel 5636  ‘cfv 6499  (class class class)co 7369  1^st c1st 7945  2^nd c2nd 7946  Topctop 22756  TopOnctopon 22773  clsccl 22881  neicnei 22960   ×_t ctx 23423  UnifOncust 24063  unifTopcutop 24094

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-int 4907  df-iun 4953  df-iin 4954  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-ord 6323  df-on 6324  df-lim 6325  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-ov 7372  df-oprab 7373  df-mpo 7374  df-om 7823  df-1st 7947  df-2nd 7948  df-1o 8411  df-2o 8412  df-en 8896  df-fin 8899  df-fi 9338  df-topgen 17382  df-top 22757  df-topon 22774  df-bases 22809  df-cld 22882  df-ntr 22883  df-cls 22884  df-nei 22961  df-tx 23425  df-ust 24064  df-utop 24095

This theorem is referenced by:  utopreg  24116