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Theorem utop3cls 22945
 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)) ∧ (𝑉𝑈𝑉 = 𝑉)) → ((cls‘(𝐽 ×t 𝐽))‘𝑀) ⊆ (𝑉 ∘ (𝑀𝑉)))

Proof of Theorem utop3cls
Dummy variables 𝑟 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relxp 5543 . . . . 5 Rel (𝑋 × 𝑋)
2 utoptop.1 . . . . . . . . . . 11 𝐽 = (unifTop‘𝑈)
3 utoptop 22928 . . . . . . . . . . 11 (𝑈 ∈ (UnifOn‘𝑋) → (unifTop‘𝑈) ∈ Top)
42, 3eqeltrid 2857 . . . . . . . . . 10 (𝑈 ∈ (UnifOn‘𝑋) → 𝐽 ∈ Top)
5 txtop 22262 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ 𝐽 ∈ Top) → (𝐽 ×t 𝐽) ∈ Top)
64, 4, 5syl2anc 588 . . . . . . . . 9 (𝑈 ∈ (UnifOn‘𝑋) → (𝐽 ×t 𝐽) ∈ Top)
76ad3antrrr 730 . . . . . . . 8 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ (𝑉𝑈𝑉 = 𝑉)) ∧ 𝑧 ∈ ((cls‘(𝐽 ×t 𝐽))‘𝑀)) → (𝐽 ×t 𝐽) ∈ Top)
8 simpllr 776 . . . . . . . . 9 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ (𝑉𝑈𝑉 = 𝑉)) ∧ 𝑧 ∈ ((cls‘(𝐽 ×t 𝐽))‘𝑀)) → 𝑀 ⊆ (𝑋 × 𝑋))
9 utoptopon 22930 . . . . . . . . . . . . . 14 (𝑈 ∈ (UnifOn‘𝑋) → (unifTop‘𝑈) ∈ (TopOn‘𝑋))
102, 9eqeltrid 2857 . . . . . . . . . . . . 13 (𝑈 ∈ (UnifOn‘𝑋) → 𝐽 ∈ (TopOn‘𝑋))
11 toponuni 21607 . . . . . . . . . . . . 13 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
1210, 11syl 17 . . . . . . . . . . . 12 (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = 𝐽)
1312sqxpeqd 5557 . . . . . . . . . . 11 (𝑈 ∈ (UnifOn‘𝑋) → (𝑋 × 𝑋) = ( 𝐽 × 𝐽))
14 eqid 2759 . . . . . . . . . . . . 13 𝐽 = 𝐽
1514, 14txuni 22285 . . . . . . . . . . . 12 ((𝐽 ∈ Top ∧ 𝐽 ∈ Top) → ( 𝐽 × 𝐽) = (𝐽 ×t 𝐽))
164, 4, 15syl2anc 588 . . . . . . . . . . 11 (𝑈 ∈ (UnifOn‘𝑋) → ( 𝐽 × 𝐽) = (𝐽 ×t 𝐽))
1713, 16eqtrd 2794 . . . . . . . . . 10 (𝑈 ∈ (UnifOn‘𝑋) → (𝑋 × 𝑋) = (𝐽 ×t 𝐽))
1817ad3antrrr 730 . . . . . . . . 9 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ (𝑉𝑈𝑉 = 𝑉)) ∧ 𝑧 ∈ ((cls‘(𝐽 ×t 𝐽))‘𝑀)) → (𝑋 × 𝑋) = (𝐽 ×t 𝐽))
198, 18sseqtrd 3933 . . . . . . . 8 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ (𝑉𝑈𝑉 = 𝑉)) ∧ 𝑧 ∈ ((cls‘(𝐽 ×t 𝐽))‘𝑀)) → 𝑀 (𝐽 ×t 𝐽))
20 eqid 2759 . . . . . . . . 9 (𝐽 ×t 𝐽) = (𝐽 ×t 𝐽)
2120clsss3 21752 . . . . . . . 8 (((𝐽 ×t 𝐽) ∈ Top ∧ 𝑀 (𝐽 ×t 𝐽)) → ((cls‘(𝐽 ×t 𝐽))‘𝑀) ⊆ (𝐽 ×t 𝐽))
227, 19, 21syl2anc 588 . . . . . . 7 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ (𝑉𝑈𝑉 = 𝑉)) ∧ 𝑧 ∈ ((cls‘(𝐽 ×t 𝐽))‘𝑀)) → ((cls‘(𝐽 ×t 𝐽))‘𝑀) ⊆ (𝐽 ×t 𝐽))
2322, 18sseqtrrd 3934 . . . . . 6 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ (𝑉𝑈𝑉 = 𝑉)) ∧ 𝑧 ∈ ((cls‘(𝐽 ×t 𝐽))‘𝑀)) → ((cls‘(𝐽 ×t 𝐽))‘𝑀) ⊆ (𝑋 × 𝑋))
24 simpr 489 . . . . . 6 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ (𝑉𝑈𝑉 = 𝑉)) ∧ 𝑧 ∈ ((cls‘(𝐽 ×t 𝐽))‘𝑀)) → 𝑧 ∈ ((cls‘(𝐽 ×t 𝐽))‘𝑀))
2523, 24sseldd 3894 . . . . 5 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ (𝑉𝑈𝑉 = 𝑉)) ∧ 𝑧 ∈ ((cls‘(𝐽 ×t 𝐽))‘𝑀)) → 𝑧 ∈ (𝑋 × 𝑋))
26 1st2nd 7743 . . . . 5 ((Rel (𝑋 × 𝑋) ∧ 𝑧 ∈ (𝑋 × 𝑋)) → 𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩)
271, 25, 26sylancr 591 . . . 4 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ (𝑉𝑈𝑉 = 𝑉)) ∧ 𝑧 ∈ ((cls‘(𝐽 ×t 𝐽))‘𝑀)) → 𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩)
28 simp-4l 783 . . . . . . . . . 10 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ (𝑉𝑈𝑉 = 𝑉)) ∧ 𝑧 ∈ ((cls‘(𝐽 ×t 𝐽))‘𝑀)) ∧ 𝑟 ∈ (((𝑉 “ {(1st𝑧)}) × (𝑉 “ {(2nd𝑧)})) ∩ 𝑀)) → 𝑈 ∈ (UnifOn‘𝑋))
29 simpr1l 1228 . . . . . . . . . . 11 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ ((𝑉𝑈𝑉 = 𝑉) ∧ 𝑧 ∈ ((cls‘(𝐽 ×t 𝐽))‘𝑀) ∧ 𝑟 ∈ (((𝑉 “ {(1st𝑧)}) × (𝑉 “ {(2nd𝑧)})) ∩ 𝑀))) → 𝑉𝑈)
30293anassrs 1358 . . . . . . . . . 10 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ (𝑉𝑈𝑉 = 𝑉)) ∧ 𝑧 ∈ ((cls‘(𝐽 ×t 𝐽))‘𝑀)) ∧ 𝑟 ∈ (((𝑉 “ {(1st𝑧)}) × (𝑉 “ {(2nd𝑧)})) ∩ 𝑀)) → 𝑉𝑈)
31 ustrel 22905 . . . . . . . . . 10 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) → Rel 𝑉)
3228, 30, 31syl2anc 588 . . . . . . . . 9 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ (𝑉𝑈𝑉 = 𝑉)) ∧ 𝑧 ∈ ((cls‘(𝐽 ×t 𝐽))‘𝑀)) ∧ 𝑟 ∈ (((𝑉 “ {(1st𝑧)}) × (𝑉 “ {(2nd𝑧)})) ∩ 𝑀)) → Rel 𝑉)
33 simpr 489 . . . . . . . . . . . 12 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ (𝑉𝑈𝑉 = 𝑉)) ∧ 𝑧 ∈ ((cls‘(𝐽 ×t 𝐽))‘𝑀)) ∧ 𝑟 ∈ (((𝑉 “ {(1st𝑧)}) × (𝑉 “ {(2nd𝑧)})) ∩ 𝑀)) → 𝑟 ∈ (((𝑉 “ {(1st𝑧)}) × (𝑉 “ {(2nd𝑧)})) ∩ 𝑀))
34 elin 3875 . . . . . . . . . . . 12 (𝑟 ∈ (((𝑉 “ {(1st𝑧)}) × (𝑉 “ {(2nd𝑧)})) ∩ 𝑀) ↔ (𝑟 ∈ ((𝑉 “ {(1st𝑧)}) × (𝑉 “ {(2nd𝑧)})) ∧ 𝑟𝑀))
3533, 34sylib 221 . . . . . . . . . . 11 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ (𝑉𝑈𝑉 = 𝑉)) ∧ 𝑧 ∈ ((cls‘(𝐽 ×t 𝐽))‘𝑀)) ∧ 𝑟 ∈ (((𝑉 “ {(1st𝑧)}) × (𝑉 “ {(2nd𝑧)})) ∩ 𝑀)) → (𝑟 ∈ ((𝑉 “ {(1st𝑧)}) × (𝑉 “ {(2nd𝑧)})) ∧ 𝑟𝑀))
3635simpld 499 . . . . . . . . . 10 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ (𝑉𝑈𝑉 = 𝑉)) ∧ 𝑧 ∈ ((cls‘(𝐽 ×t 𝐽))‘𝑀)) ∧ 𝑟 ∈ (((𝑉 “ {(1st𝑧)}) × (𝑉 “ {(2nd𝑧)})) ∩ 𝑀)) → 𝑟 ∈ ((𝑉 “ {(1st𝑧)}) × (𝑉 “ {(2nd𝑧)})))
37 xp1st 7726 . . . . . . . . . 10 (𝑟 ∈ ((𝑉 “ {(1st𝑧)}) × (𝑉 “ {(2nd𝑧)})) → (1st𝑟) ∈ (𝑉 “ {(1st𝑧)}))
3836, 37syl 17 . . . . . . . . 9 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ (𝑉𝑈𝑉 = 𝑉)) ∧ 𝑧 ∈ ((cls‘(𝐽 ×t 𝐽))‘𝑀)) ∧ 𝑟 ∈ (((𝑉 “ {(1st𝑧)}) × (𝑉 “ {(2nd𝑧)})) ∩ 𝑀)) → (1st𝑟) ∈ (𝑉 “ {(1st𝑧)}))
39 elrelimasn 5926 . . . . . . . . . 10 (Rel 𝑉 → ((1st𝑟) ∈ (𝑉 “ {(1st𝑧)}) ↔ (1st𝑧)𝑉(1st𝑟)))
4039biimpa 481 . . . . . . . . 9 ((Rel 𝑉 ∧ (1st𝑟) ∈ (𝑉 “ {(1st𝑧)})) → (1st𝑧)𝑉(1st𝑟))
4132, 38, 40syl2anc 588 . . . . . . . 8 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ (𝑉𝑈𝑉 = 𝑉)) ∧ 𝑧 ∈ ((cls‘(𝐽 ×t 𝐽))‘𝑀)) ∧ 𝑟 ∈ (((𝑉 “ {(1st𝑧)}) × (𝑉 “ {(2nd𝑧)})) ∩ 𝑀)) → (1st𝑧)𝑉(1st𝑟))
42 simp-4r 784 . . . . . . . . . . 11 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ (𝑉𝑈𝑉 = 𝑉)) ∧ 𝑧 ∈ ((cls‘(𝐽 ×t 𝐽))‘𝑀)) ∧ 𝑟 ∈ (((𝑉 “ {(1st𝑧)}) × (𝑉 “ {(2nd𝑧)})) ∩ 𝑀)) → 𝑀 ⊆ (𝑋 × 𝑋))
43 xpss 5541 . . . . . . . . . . 11 (𝑋 × 𝑋) ⊆ (V × V)
4442, 43sstrdi 3905 . . . . . . . . . 10 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ (𝑉𝑈𝑉 = 𝑉)) ∧ 𝑧 ∈ ((cls‘(𝐽 ×t 𝐽))‘𝑀)) ∧ 𝑟 ∈ (((𝑉 “ {(1st𝑧)}) × (𝑉 “ {(2nd𝑧)})) ∩ 𝑀)) → 𝑀 ⊆ (V × V))
45 df-rel 5532 . . . . . . . . . 10 (Rel 𝑀𝑀 ⊆ (V × V))
4644, 45sylibr 237 . . . . . . . . 9 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ (𝑉𝑈𝑉 = 𝑉)) ∧ 𝑧 ∈ ((cls‘(𝐽 ×t 𝐽))‘𝑀)) ∧ 𝑟 ∈ (((𝑉 “ {(1st𝑧)}) × (𝑉 “ {(2nd𝑧)})) ∩ 𝑀)) → Rel 𝑀)
4735simprd 500 . . . . . . . . 9 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ (𝑉𝑈𝑉 = 𝑉)) ∧ 𝑧 ∈ ((cls‘(𝐽 ×t 𝐽))‘𝑀)) ∧ 𝑟 ∈ (((𝑉 “ {(1st𝑧)}) × (𝑉 “ {(2nd𝑧)})) ∩ 𝑀)) → 𝑟𝑀)
48 1st2ndbr 7746 . . . . . . . . 9 ((Rel 𝑀𝑟𝑀) → (1st𝑟)𝑀(2nd𝑟))
4946, 47, 48syl2anc 588 . . . . . . . 8 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ (𝑉𝑈𝑉 = 𝑉)) ∧ 𝑧 ∈ ((cls‘(𝐽 ×t 𝐽))‘𝑀)) ∧ 𝑟 ∈ (((𝑉 “ {(1st𝑧)}) × (𝑉 “ {(2nd𝑧)})) ∩ 𝑀)) → (1st𝑟)𝑀(2nd𝑟))
50 xp2nd 7727 . . . . . . . . . . 11 (𝑟 ∈ ((𝑉 “ {(1st𝑧)}) × (𝑉 “ {(2nd𝑧)})) → (2nd𝑟) ∈ (𝑉 “ {(2nd𝑧)}))
5136, 50syl 17 . . . . . . . . . 10 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ (𝑉𝑈𝑉 = 𝑉)) ∧ 𝑧 ∈ ((cls‘(𝐽 ×t 𝐽))‘𝑀)) ∧ 𝑟 ∈ (((𝑉 “ {(1st𝑧)}) × (𝑉 “ {(2nd𝑧)})) ∩ 𝑀)) → (2nd𝑟) ∈ (𝑉 “ {(2nd𝑧)}))
52 elrelimasn 5926 . . . . . . . . . . 11 (Rel 𝑉 → ((2nd𝑟) ∈ (𝑉 “ {(2nd𝑧)}) ↔ (2nd𝑧)𝑉(2nd𝑟)))
5352biimpa 481 . . . . . . . . . 10 ((Rel 𝑉 ∧ (2nd𝑟) ∈ (𝑉 “ {(2nd𝑧)})) → (2nd𝑧)𝑉(2nd𝑟))
5432, 51, 53syl2anc 588 . . . . . . . . 9 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ (𝑉𝑈𝑉 = 𝑉)) ∧ 𝑧 ∈ ((cls‘(𝐽 ×t 𝐽))‘𝑀)) ∧ 𝑟 ∈ (((𝑉 “ {(1st𝑧)}) × (𝑉 “ {(2nd𝑧)})) ∩ 𝑀)) → (2nd𝑧)𝑉(2nd𝑟))
55 simpr1r 1229 . . . . . . . . . . 11 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ ((𝑉𝑈𝑉 = 𝑉) ∧ 𝑧 ∈ ((cls‘(𝐽 ×t 𝐽))‘𝑀) ∧ 𝑟 ∈ (((𝑉 “ {(1st𝑧)}) × (𝑉 “ {(2nd𝑧)})) ∩ 𝑀))) → 𝑉 = 𝑉)
56553anassrs 1358 . . . . . . . . . 10 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ (𝑉𝑈𝑉 = 𝑉)) ∧ 𝑧 ∈ ((cls‘(𝐽 ×t 𝐽))‘𝑀)) ∧ 𝑟 ∈ (((𝑉 “ {(1st𝑧)}) × (𝑉 “ {(2nd𝑧)})) ∩ 𝑀)) → 𝑉 = 𝑉)
57 breq 5035 . . . . . . . . . . 11 (𝑉 = 𝑉 → ((2nd𝑟)𝑉(2nd𝑧) ↔ (2nd𝑟)𝑉(2nd𝑧)))
58 fvex 6672 . . . . . . . . . . . 12 (2nd𝑟) ∈ V
59 fvex 6672 . . . . . . . . . . . 12 (2nd𝑧) ∈ V
6058, 59brcnv 5723 . . . . . . . . . . 11 ((2nd𝑟)𝑉(2nd𝑧) ↔ (2nd𝑧)𝑉(2nd𝑟))
6157, 60bitr3di 289 . . . . . . . . . 10 (𝑉 = 𝑉 → ((2nd𝑟)𝑉(2nd𝑧) ↔ (2nd𝑧)𝑉(2nd𝑟)))
6256, 61syl 17 . . . . . . . . 9 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ (𝑉𝑈𝑉 = 𝑉)) ∧ 𝑧 ∈ ((cls‘(𝐽 ×t 𝐽))‘𝑀)) ∧ 𝑟 ∈ (((𝑉 “ {(1st𝑧)}) × (𝑉 “ {(2nd𝑧)})) ∩ 𝑀)) → ((2nd𝑟)𝑉(2nd𝑧) ↔ (2nd𝑧)𝑉(2nd𝑟)))
6354, 62mpbird 260 . . . . . . . 8 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ (𝑉𝑈𝑉 = 𝑉)) ∧ 𝑧 ∈ ((cls‘(𝐽 ×t 𝐽))‘𝑀)) ∧ 𝑟 ∈ (((𝑉 “ {(1st𝑧)}) × (𝑉 “ {(2nd𝑧)})) ∩ 𝑀)) → (2nd𝑟)𝑉(2nd𝑧))
64 fvex 6672 . . . . . . . . . 10 (1st𝑧) ∈ V
65 fvex 6672 . . . . . . . . . 10 (1st𝑟) ∈ V
66 brcogw 5709 . . . . . . . . . . 11 ((((1st𝑧) ∈ V ∧ (2nd𝑟) ∈ V ∧ (1st𝑟) ∈ V) ∧ ((1st𝑧)𝑉(1st𝑟) ∧ (1st𝑟)𝑀(2nd𝑟))) → (1st𝑧)(𝑀𝑉)(2nd𝑟))
6766ex 417 . . . . . . . . . 10 (((1st𝑧) ∈ V ∧ (2nd𝑟) ∈ V ∧ (1st𝑟) ∈ V) → (((1st𝑧)𝑉(1st𝑟) ∧ (1st𝑟)𝑀(2nd𝑟)) → (1st𝑧)(𝑀𝑉)(2nd𝑟)))
6864, 58, 65, 67mp3an 1459 . . . . . . . . 9 (((1st𝑧)𝑉(1st𝑟) ∧ (1st𝑟)𝑀(2nd𝑟)) → (1st𝑧)(𝑀𝑉)(2nd𝑟))
69 brcogw 5709 . . . . . . . . . . 11 ((((1st𝑧) ∈ V ∧ (2nd𝑧) ∈ V ∧ (2nd𝑟) ∈ V) ∧ ((1st𝑧)(𝑀𝑉)(2nd𝑟) ∧ (2nd𝑟)𝑉(2nd𝑧))) → (1st𝑧)(𝑉 ∘ (𝑀𝑉))(2nd𝑧))
7069ex 417 . . . . . . . . . 10 (((1st𝑧) ∈ V ∧ (2nd𝑧) ∈ V ∧ (2nd𝑟) ∈ V) → (((1st𝑧)(𝑀𝑉)(2nd𝑟) ∧ (2nd𝑟)𝑉(2nd𝑧)) → (1st𝑧)(𝑉 ∘ (𝑀𝑉))(2nd𝑧)))
7164, 59, 58, 70mp3an 1459 . . . . . . . . 9 (((1st𝑧)(𝑀𝑉)(2nd𝑟) ∧ (2nd𝑟)𝑉(2nd𝑧)) → (1st𝑧)(𝑉 ∘ (𝑀𝑉))(2nd𝑧))
7268, 71sylan 584 . . . . . . . 8 ((((1st𝑧)𝑉(1st𝑟) ∧ (1st𝑟)𝑀(2nd𝑟)) ∧ (2nd𝑟)𝑉(2nd𝑧)) → (1st𝑧)(𝑉 ∘ (𝑀𝑉))(2nd𝑧))
7341, 49, 63, 72syl21anc 837 . . . . . . 7 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ (𝑉𝑈𝑉 = 𝑉)) ∧ 𝑧 ∈ ((cls‘(𝐽 ×t 𝐽))‘𝑀)) ∧ 𝑟 ∈ (((𝑉 “ {(1st𝑧)}) × (𝑉 “ {(2nd𝑧)})) ∩ 𝑀)) → (1st𝑧)(𝑉 ∘ (𝑀𝑉))(2nd𝑧))
7473ralrimiva 3114 . . . . . 6 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ (𝑉𝑈𝑉 = 𝑉)) ∧ 𝑧 ∈ ((cls‘(𝐽 ×t 𝐽))‘𝑀)) → ∀𝑟 ∈ (((𝑉 “ {(1st𝑧)}) × (𝑉 “ {(2nd𝑧)})) ∩ 𝑀)(1st𝑧)(𝑉 ∘ (𝑀𝑉))(2nd𝑧))
75 simplll 775 . . . . . . . . 9 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ (𝑉𝑈𝑉 = 𝑉)) ∧ 𝑧 ∈ ((cls‘(𝐽 ×t 𝐽))‘𝑀)) → 𝑈 ∈ (UnifOn‘𝑋))
76 simplrl 777 . . . . . . . . 9 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ (𝑉𝑈𝑉 = 𝑉)) ∧ 𝑧 ∈ ((cls‘(𝐽 ×t 𝐽))‘𝑀)) → 𝑉𝑈)
7743ad2ant1 1131 . . . . . . . . . . 11 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈𝑧 ∈ (𝑋 × 𝑋)) → 𝐽 ∈ Top)
78 xp1st 7726 . . . . . . . . . . . 12 (𝑧 ∈ (𝑋 × 𝑋) → (1st𝑧) ∈ 𝑋)
792utopsnnei 22943 . . . . . . . . . . . 12 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈 ∧ (1st𝑧) ∈ 𝑋) → (𝑉 “ {(1st𝑧)}) ∈ ((nei‘𝐽)‘{(1st𝑧)}))
8078, 79syl3an3 1163 . . . . . . . . . . 11 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈𝑧 ∈ (𝑋 × 𝑋)) → (𝑉 “ {(1st𝑧)}) ∈ ((nei‘𝐽)‘{(1st𝑧)}))
81 xp2nd 7727 . . . . . . . . . . . 12 (𝑧 ∈ (𝑋 × 𝑋) → (2nd𝑧) ∈ 𝑋)
822utopsnnei 22943 . . . . . . . . . . . 12 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈 ∧ (2nd𝑧) ∈ 𝑋) → (𝑉 “ {(2nd𝑧)}) ∈ ((nei‘𝐽)‘{(2nd𝑧)}))
8381, 82syl3an3 1163 . . . . . . . . . . 11 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈𝑧 ∈ (𝑋 × 𝑋)) → (𝑉 “ {(2nd𝑧)}) ∈ ((nei‘𝐽)‘{(2nd𝑧)}))
8414, 14neitx 22300 . . . . . . . . . . 11 (((𝐽 ∈ Top ∧ 𝐽 ∈ Top) ∧ ((𝑉 “ {(1st𝑧)}) ∈ ((nei‘𝐽)‘{(1st𝑧)}) ∧ (𝑉 “ {(2nd𝑧)}) ∈ ((nei‘𝐽)‘{(2nd𝑧)}))) → ((𝑉 “ {(1st𝑧)}) × (𝑉 “ {(2nd𝑧)})) ∈ ((nei‘(𝐽 ×t 𝐽))‘({(1st𝑧)} × {(2nd𝑧)})))
8577, 77, 80, 83, 84syl22anc 838 . . . . . . . . . 10 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈𝑧 ∈ (𝑋 × 𝑋)) → ((𝑉 “ {(1st𝑧)}) × (𝑉 “ {(2nd𝑧)})) ∈ ((nei‘(𝐽 ×t 𝐽))‘({(1st𝑧)} × {(2nd𝑧)})))
86 1st2nd2 7733 . . . . . . . . . . . . . 14 (𝑧 ∈ (𝑋 × 𝑋) → 𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩)
8786sneqd 4535 . . . . . . . . . . . . 13 (𝑧 ∈ (𝑋 × 𝑋) → {𝑧} = {⟨(1st𝑧), (2nd𝑧)⟩})
8864, 59xpsn 6895 . . . . . . . . . . . . 13 ({(1st𝑧)} × {(2nd𝑧)}) = {⟨(1st𝑧), (2nd𝑧)⟩}
8987, 88eqtr4di 2812 . . . . . . . . . . . 12 (𝑧 ∈ (𝑋 × 𝑋) → {𝑧} = ({(1st𝑧)} × {(2nd𝑧)}))
9089fveq2d 6663 . . . . . . . . . . 11 (𝑧 ∈ (𝑋 × 𝑋) → ((nei‘(𝐽 ×t 𝐽))‘{𝑧}) = ((nei‘(𝐽 ×t 𝐽))‘({(1st𝑧)} × {(2nd𝑧)})))
91903ad2ant3 1133 . . . . . . . . . 10 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈𝑧 ∈ (𝑋 × 𝑋)) → ((nei‘(𝐽 ×t 𝐽))‘{𝑧}) = ((nei‘(𝐽 ×t 𝐽))‘({(1st𝑧)} × {(2nd𝑧)})))
9285, 91eleqtrrd 2856 . . . . . . . . 9 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈𝑧 ∈ (𝑋 × 𝑋)) → ((𝑉 “ {(1st𝑧)}) × (𝑉 “ {(2nd𝑧)})) ∈ ((nei‘(𝐽 ×t 𝐽))‘{𝑧}))
9375, 76, 25, 92syl3anc 1369 . . . . . . . 8 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ (𝑉𝑈𝑉 = 𝑉)) ∧ 𝑧 ∈ ((cls‘(𝐽 ×t 𝐽))‘𝑀)) → ((𝑉 “ {(1st𝑧)}) × (𝑉 “ {(2nd𝑧)})) ∈ ((nei‘(𝐽 ×t 𝐽))‘{𝑧}))
9420neindisj 21810 . . . . . . . 8 ((((𝐽 ×t 𝐽) ∈ Top ∧ 𝑀 (𝐽 ×t 𝐽)) ∧ (𝑧 ∈ ((cls‘(𝐽 ×t 𝐽))‘𝑀) ∧ ((𝑉 “ {(1st𝑧)}) × (𝑉 “ {(2nd𝑧)})) ∈ ((nei‘(𝐽 ×t 𝐽))‘{𝑧}))) → (((𝑉 “ {(1st𝑧)}) × (𝑉 “ {(2nd𝑧)})) ∩ 𝑀) ≠ ∅)
957, 19, 24, 93, 94syl22anc 838 . . . . . . 7 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ (𝑉𝑈𝑉 = 𝑉)) ∧ 𝑧 ∈ ((cls‘(𝐽 ×t 𝐽))‘𝑀)) → (((𝑉 “ {(1st𝑧)}) × (𝑉 “ {(2nd𝑧)})) ∩ 𝑀) ≠ ∅)
96 r19.3rzv 4393 . . . . . . 7 ((((𝑉 “ {(1st𝑧)}) × (𝑉 “ {(2nd𝑧)})) ∩ 𝑀) ≠ ∅ → ((1st𝑧)(𝑉 ∘ (𝑀𝑉))(2nd𝑧) ↔ ∀𝑟 ∈ (((𝑉 “ {(1st𝑧)}) × (𝑉 “ {(2nd𝑧)})) ∩ 𝑀)(1st𝑧)(𝑉 ∘ (𝑀𝑉))(2nd𝑧)))
9795, 96syl 17 . . . . . 6 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ (𝑉𝑈𝑉 = 𝑉)) ∧ 𝑧 ∈ ((cls‘(𝐽 ×t 𝐽))‘𝑀)) → ((1st𝑧)(𝑉 ∘ (𝑀𝑉))(2nd𝑧) ↔ ∀𝑟 ∈ (((𝑉 “ {(1st𝑧)}) × (𝑉 “ {(2nd𝑧)})) ∩ 𝑀)(1st𝑧)(𝑉 ∘ (𝑀𝑉))(2nd𝑧)))
9874, 97mpbird 260 . . . . 5 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ (𝑉𝑈𝑉 = 𝑉)) ∧ 𝑧 ∈ ((cls‘(𝐽 ×t 𝐽))‘𝑀)) → (1st𝑧)(𝑉 ∘ (𝑀𝑉))(2nd𝑧))
99 df-br 5034 . . . . 5 ((1st𝑧)(𝑉 ∘ (𝑀𝑉))(2nd𝑧) ↔ ⟨(1st𝑧), (2nd𝑧)⟩ ∈ (𝑉 ∘ (𝑀𝑉)))
10098, 99sylib 221 . . . 4 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ (𝑉𝑈𝑉 = 𝑉)) ∧ 𝑧 ∈ ((cls‘(𝐽 ×t 𝐽))‘𝑀)) → ⟨(1st𝑧), (2nd𝑧)⟩ ∈ (𝑉 ∘ (𝑀𝑉)))
10127, 100eqeltrd 2853 . . 3 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ (𝑉𝑈𝑉 = 𝑉)) ∧ 𝑧 ∈ ((cls‘(𝐽 ×t 𝐽))‘𝑀)) → 𝑧 ∈ (𝑉 ∘ (𝑀𝑉)))
102101ex 417 . 2 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ (𝑉𝑈𝑉 = 𝑉)) → (𝑧 ∈ ((cls‘(𝐽 ×t 𝐽))‘𝑀) → 𝑧 ∈ (𝑉 ∘ (𝑀𝑉))))
103102ssrdv 3899 1 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ (𝑉𝑈𝑉 = 𝑉)) → ((cls‘(𝐽 ×t 𝐽))‘𝑀) ⊆ (𝑉 ∘ (𝑀𝑉)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1085   = wceq 1539   ∈ wcel 2112   ≠ wne 2952  ∀wral 3071  Vcvv 3410   ∩ cin 3858   ⊆ wss 3859  ∅c0 4226  {csn 4523  ⟨cop 4529  ∪ cuni 4799   class class class wbr 5033   × cxp 5523  ◡ccnv 5524   “ cima 5528   ∘ ccom 5529  Rel wrel 5530  ‘cfv 6336  (class class class)co 7151  1st c1st 7692  2nd c2nd 7693  Topctop 21586  TopOnctopon 21603  clsccl 21711  neicnei 21790   ×t ctx 22253  UnifOncust 22893  unifTopcutop 22924 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 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5235  ax-pr 5299  ax-un 7460 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-ral 3076  df-rex 3077  df-reu 3078  df-rab 3080  df-v 3412  df-sbc 3698  df-csb 3807  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-pss 3878  df-nul 4227  df-if 4422  df-pw 4497  df-sn 4524  df-pr 4526  df-tp 4528  df-op 4530  df-uni 4800  df-int 4840  df-iun 4886  df-iin 4887  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5431  df-eprel 5436  df-po 5444  df-so 5445  df-fr 5484  df-we 5486  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-pred 6127  df-ord 6173  df-on 6174  df-lim 6175  df-suc 6176  df-iota 6295  df-fun 6338  df-fn 6339  df-f 6340  df-f1 6341  df-fo 6342  df-f1o 6343  df-fv 6344  df-ov 7154  df-oprab 7155  df-mpo 7156  df-om 7581  df-1st 7694  df-2nd 7695  df-wrecs 7958  df-recs 8019  df-rdg 8057  df-1o 8113  df-oadd 8117  df-er 8300  df-en 8529  df-fin 8532  df-fi 8901  df-topgen 16768  df-top 21587  df-topon 21604  df-bases 21639  df-cld 21712  df-ntr 21713  df-cls 21714  df-nei 21791  df-tx 22255  df-ust 22894  df-utop 22925 This theorem is referenced by:  utopreg  22946
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