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Theorem utop3cls 24260
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 5703 . . . . 5 Rel (𝑋 × 𝑋)
2 utoptop.1 . . . . . . . . . . 11 𝐽 = (unifTop‘𝑈)
3 utoptop 24243 . . . . . . . . . . 11 (𝑈 ∈ (UnifOn‘𝑋) → (unifTop‘𝑈) ∈ Top)
42, 3eqeltrid 2845 . . . . . . . . . 10 (𝑈 ∈ (UnifOn‘𝑋) → 𝐽 ∈ Top)
5 txtop 23577 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ 𝐽 ∈ Top) → (𝐽 ×t 𝐽) ∈ Top)
64, 4, 5syl2anc 584 . . . . . . . . 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 24245 . . . . . . . . . . . . . 14 (𝑈 ∈ (UnifOn‘𝑋) → (unifTop‘𝑈) ∈ (TopOn‘𝑋))
102, 9eqeltrid 2845 . . . . . . . . . . . . 13 (𝑈 ∈ (UnifOn‘𝑋) → 𝐽 ∈ (TopOn‘𝑋))
11 toponuni 22920 . . . . . . . . . . . . 13 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
1210, 11syl 17 . . . . . . . . . . . 12 (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = 𝐽)
1312sqxpeqd 5717 . . . . . . . . . . 11 (𝑈 ∈ (UnifOn‘𝑋) → (𝑋 × 𝑋) = ( 𝐽 × 𝐽))
14 eqid 2737 . . . . . . . . . . . . 13 𝐽 = 𝐽
1514, 14txuni 23600 . . . . . . . . . . . 12 ((𝐽 ∈ Top ∧ 𝐽 ∈ Top) → ( 𝐽 × 𝐽) = (𝐽 ×t 𝐽))
164, 4, 15syl2anc 584 . . . . . . . . . . 11 (𝑈 ∈ (UnifOn‘𝑋) → ( 𝐽 × 𝐽) = (𝐽 ×t 𝐽))
1713, 16eqtrd 2777 . . . . . . . . . 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 4020 . . . . . . . 8 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ (𝑉𝑈𝑉 = 𝑉)) ∧ 𝑧 ∈ ((cls‘(𝐽 ×t 𝐽))‘𝑀)) → 𝑀 (𝐽 ×t 𝐽))
20 eqid 2737 . . . . . . . . 9 (𝐽 ×t 𝐽) = (𝐽 ×t 𝐽)
2120clsss3 23067 . . . . . . . 8 (((𝐽 ×t 𝐽) ∈ Top ∧ 𝑀 (𝐽 ×t 𝐽)) → ((cls‘(𝐽 ×t 𝐽))‘𝑀) ⊆ (𝐽 ×t 𝐽))
227, 19, 21syl2anc 584 . . . . . . 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 4021 . . . . . 6 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ (𝑉𝑈𝑉 = 𝑉)) ∧ 𝑧 ∈ ((cls‘(𝐽 ×t 𝐽))‘𝑀)) → ((cls‘(𝐽 ×t 𝐽))‘𝑀) ⊆ (𝑋 × 𝑋))
24 simpr 484 . . . . . 6 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ (𝑉𝑈𝑉 = 𝑉)) ∧ 𝑧 ∈ ((cls‘(𝐽 ×t 𝐽))‘𝑀)) → 𝑧 ∈ ((cls‘(𝐽 ×t 𝐽))‘𝑀))
2523, 24sseldd 3984 . . . . 5 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ (𝑉𝑈𝑉 = 𝑉)) ∧ 𝑧 ∈ ((cls‘(𝐽 ×t 𝐽))‘𝑀)) → 𝑧 ∈ (𝑋 × 𝑋))
26 1st2nd 8064 . . . . 5 ((Rel (𝑋 × 𝑋) ∧ 𝑧 ∈ (𝑋 × 𝑋)) → 𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩)
271, 25, 26sylancr 587 . . . 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 1231 . . . . . . . . . . 11 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ ((𝑉𝑈𝑉 = 𝑉) ∧ 𝑧 ∈ ((cls‘(𝐽 ×t 𝐽))‘𝑀) ∧ 𝑟 ∈ (((𝑉 “ {(1st𝑧)}) × (𝑉 “ {(2nd𝑧)})) ∩ 𝑀))) → 𝑉𝑈)
30293anassrs 1361 . . . . . . . . . 10 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ (𝑉𝑈𝑉 = 𝑉)) ∧ 𝑧 ∈ ((cls‘(𝐽 ×t 𝐽))‘𝑀)) ∧ 𝑟 ∈ (((𝑉 “ {(1st𝑧)}) × (𝑉 “ {(2nd𝑧)})) ∩ 𝑀)) → 𝑉𝑈)
31 ustrel 24220 . . . . . . . . . 10 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) → Rel 𝑉)
3228, 30, 31syl2anc 584 . . . . . . . . 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 484 . . . . . . . . . . . 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 3967 . . . . . . . . . . . 12 (𝑟 ∈ (((𝑉 “ {(1st𝑧)}) × (𝑉 “ {(2nd𝑧)})) ∩ 𝑀) ↔ (𝑟 ∈ ((𝑉 “ {(1st𝑧)}) × (𝑉 “ {(2nd𝑧)})) ∧ 𝑟𝑀))
3533, 34sylib 218 . . . . . . . . . . 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 494 . . . . . . . . . 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 8046 . . . . . . . . . 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 6104 . . . . . . . . . 10 (Rel 𝑉 → ((1st𝑟) ∈ (𝑉 “ {(1st𝑧)}) ↔ (1st𝑧)𝑉(1st𝑟)))
4039biimpa 476 . . . . . . . . 9 ((Rel 𝑉 ∧ (1st𝑟) ∈ (𝑉 “ {(1st𝑧)})) → (1st𝑧)𝑉(1st𝑟))
4132, 38, 40syl2anc 584 . . . . . . . 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 5701 . . . . . . . . . . 11 (𝑋 × 𝑋) ⊆ (V × V)
4442, 43sstrdi 3996 . . . . . . . . . 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 5692 . . . . . . . . . 10 (Rel 𝑀𝑀 ⊆ (V × V))
4644, 45sylibr 234 . . . . . . . . 9 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ (𝑉𝑈𝑉 = 𝑉)) ∧ 𝑧 ∈ ((cls‘(𝐽 ×t 𝐽))‘𝑀)) ∧ 𝑟 ∈ (((𝑉 “ {(1st𝑧)}) × (𝑉 “ {(2nd𝑧)})) ∩ 𝑀)) → Rel 𝑀)
4735simprd 495 . . . . . . . . 9 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ (𝑉𝑈𝑉 = 𝑉)) ∧ 𝑧 ∈ ((cls‘(𝐽 ×t 𝐽))‘𝑀)) ∧ 𝑟 ∈ (((𝑉 “ {(1st𝑧)}) × (𝑉 “ {(2nd𝑧)})) ∩ 𝑀)) → 𝑟𝑀)
48 1st2ndbr 8067 . . . . . . . . 9 ((Rel 𝑀𝑟𝑀) → (1st𝑟)𝑀(2nd𝑟))
4946, 47, 48syl2anc 584 . . . . . . . 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 8047 . . . . . . . . . . 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 6104 . . . . . . . . . . 11 (Rel 𝑉 → ((2nd𝑟) ∈ (𝑉 “ {(2nd𝑧)}) ↔ (2nd𝑧)𝑉(2nd𝑟)))
5352biimpa 476 . . . . . . . . . 10 ((Rel 𝑉 ∧ (2nd𝑟) ∈ (𝑉 “ {(2nd𝑧)})) → (2nd𝑧)𝑉(2nd𝑟))
5432, 51, 53syl2anc 584 . . . . . . . . 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 1232 . . . . . . . . . . 11 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ ((𝑉𝑈𝑉 = 𝑉) ∧ 𝑧 ∈ ((cls‘(𝐽 ×t 𝐽))‘𝑀) ∧ 𝑟 ∈ (((𝑉 “ {(1st𝑧)}) × (𝑉 “ {(2nd𝑧)})) ∩ 𝑀))) → 𝑉 = 𝑉)
56553anassrs 1361 . . . . . . . . . 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 5145 . . . . . . . . . . 11 (𝑉 = 𝑉 → ((2nd𝑟)𝑉(2nd𝑧) ↔ (2nd𝑟)𝑉(2nd𝑧)))
58 fvex 6919 . . . . . . . . . . . 12 (2nd𝑟) ∈ V
59 fvex 6919 . . . . . . . . . . . 12 (2nd𝑧) ∈ V
6058, 59brcnv 5893 . . . . . . . . . . 11 ((2nd𝑟)𝑉(2nd𝑧) ↔ (2nd𝑧)𝑉(2nd𝑟))
6157, 60bitr3di 286 . . . . . . . . . 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 257 . . . . . . . 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 6919 . . . . . . . . . 10 (1st𝑧) ∈ V
65 fvex 6919 . . . . . . . . . 10 (1st𝑟) ∈ V
66 brcogw 5879 . . . . . . . . . . 11 ((((1st𝑧) ∈ V ∧ (2nd𝑟) ∈ V ∧ (1st𝑟) ∈ V) ∧ ((1st𝑧)𝑉(1st𝑟) ∧ (1st𝑟)𝑀(2nd𝑟))) → (1st𝑧)(𝑀𝑉)(2nd𝑟))
6766ex 412 . . . . . . . . . 10 (((1st𝑧) ∈ V ∧ (2nd𝑟) ∈ V ∧ (1st𝑟) ∈ V) → (((1st𝑧)𝑉(1st𝑟) ∧ (1st𝑟)𝑀(2nd𝑟)) → (1st𝑧)(𝑀𝑉)(2nd𝑟)))
6864, 58, 65, 67mp3an 1463 . . . . . . . . 9 (((1st𝑧)𝑉(1st𝑟) ∧ (1st𝑟)𝑀(2nd𝑟)) → (1st𝑧)(𝑀𝑉)(2nd𝑟))
69 brcogw 5879 . . . . . . . . . . 11 ((((1st𝑧) ∈ V ∧ (2nd𝑧) ∈ V ∧ (2nd𝑟) ∈ V) ∧ ((1st𝑧)(𝑀𝑉)(2nd𝑟) ∧ (2nd𝑟)𝑉(2nd𝑧))) → (1st𝑧)(𝑉 ∘ (𝑀𝑉))(2nd𝑧))
7069ex 412 . . . . . . . . . 10 (((1st𝑧) ∈ V ∧ (2nd𝑧) ∈ V ∧ (2nd𝑟) ∈ V) → (((1st𝑧)(𝑀𝑉)(2nd𝑟) ∧ (2nd𝑟)𝑉(2nd𝑧)) → (1st𝑧)(𝑉 ∘ (𝑀𝑉))(2nd𝑧)))
7164, 59, 58, 70mp3an 1463 . . . . . . . . 9 (((1st𝑧)(𝑀𝑉)(2nd𝑟) ∧ (2nd𝑟)𝑉(2nd𝑧)) → (1st𝑧)(𝑉 ∘ (𝑀𝑉))(2nd𝑧))
7268, 71sylan 580 . . . . . . . 8 ((((1st𝑧)𝑉(1st𝑟) ∧ (1st𝑟)𝑀(2nd𝑟)) ∧ (2nd𝑟)𝑉(2nd𝑧)) → (1st𝑧)(𝑉 ∘ (𝑀𝑉))(2nd𝑧))
7341, 49, 63, 72syl21anc 838 . . . . . . 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 3146 . . . . . 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 1134 . . . . . . . . . . 11 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈𝑧 ∈ (𝑋 × 𝑋)) → 𝐽 ∈ Top)
78 xp1st 8046 . . . . . . . . . . . 12 (𝑧 ∈ (𝑋 × 𝑋) → (1st𝑧) ∈ 𝑋)
792utopsnnei 24258 . . . . . . . . . . . 12 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈 ∧ (1st𝑧) ∈ 𝑋) → (𝑉 “ {(1st𝑧)}) ∈ ((nei‘𝐽)‘{(1st𝑧)}))
8078, 79syl3an3 1166 . . . . . . . . . . 11 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈𝑧 ∈ (𝑋 × 𝑋)) → (𝑉 “ {(1st𝑧)}) ∈ ((nei‘𝐽)‘{(1st𝑧)}))
81 xp2nd 8047 . . . . . . . . . . . 12 (𝑧 ∈ (𝑋 × 𝑋) → (2nd𝑧) ∈ 𝑋)
822utopsnnei 24258 . . . . . . . . . . . 12 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈 ∧ (2nd𝑧) ∈ 𝑋) → (𝑉 “ {(2nd𝑧)}) ∈ ((nei‘𝐽)‘{(2nd𝑧)}))
8381, 82syl3an3 1166 . . . . . . . . . . 11 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈𝑧 ∈ (𝑋 × 𝑋)) → (𝑉 “ {(2nd𝑧)}) ∈ ((nei‘𝐽)‘{(2nd𝑧)}))
8414, 14neitx 23615 . . . . . . . . . . 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 839 . . . . . . . . . 10 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈𝑧 ∈ (𝑋 × 𝑋)) → ((𝑉 “ {(1st𝑧)}) × (𝑉 “ {(2nd𝑧)})) ∈ ((nei‘(𝐽 ×t 𝐽))‘({(1st𝑧)} × {(2nd𝑧)})))
86 1st2nd2 8053 . . . . . . . . . . . . . 14 (𝑧 ∈ (𝑋 × 𝑋) → 𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩)
8786sneqd 4638 . . . . . . . . . . . . 13 (𝑧 ∈ (𝑋 × 𝑋) → {𝑧} = {⟨(1st𝑧), (2nd𝑧)⟩})
8864, 59xpsn 7161 . . . . . . . . . . . . 13 ({(1st𝑧)} × {(2nd𝑧)}) = {⟨(1st𝑧), (2nd𝑧)⟩}
8987, 88eqtr4di 2795 . . . . . . . . . . . 12 (𝑧 ∈ (𝑋 × 𝑋) → {𝑧} = ({(1st𝑧)} × {(2nd𝑧)}))
9089fveq2d 6910 . . . . . . . . . . 11 (𝑧 ∈ (𝑋 × 𝑋) → ((nei‘(𝐽 ×t 𝐽))‘{𝑧}) = ((nei‘(𝐽 ×t 𝐽))‘({(1st𝑧)} × {(2nd𝑧)})))
91903ad2ant3 1136 . . . . . . . . . 10 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈𝑧 ∈ (𝑋 × 𝑋)) → ((nei‘(𝐽 ×t 𝐽))‘{𝑧}) = ((nei‘(𝐽 ×t 𝐽))‘({(1st𝑧)} × {(2nd𝑧)})))
9285, 91eleqtrrd 2844 . . . . . . . . 9 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈𝑧 ∈ (𝑋 × 𝑋)) → ((𝑉 “ {(1st𝑧)}) × (𝑉 “ {(2nd𝑧)})) ∈ ((nei‘(𝐽 ×t 𝐽))‘{𝑧}))
9375, 76, 25, 92syl3anc 1373 . . . . . . . 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 23125 . . . . . . . 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 839 . . . . . . 7 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ (𝑉𝑈𝑉 = 𝑉)) ∧ 𝑧 ∈ ((cls‘(𝐽 ×t 𝐽))‘𝑀)) → (((𝑉 “ {(1st𝑧)}) × (𝑉 “ {(2nd𝑧)})) ∩ 𝑀) ≠ ∅)
96 r19.3rzv 4499 . . . . . . 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 257 . . . . 5 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ (𝑉𝑈𝑉 = 𝑉)) ∧ 𝑧 ∈ ((cls‘(𝐽 ×t 𝐽))‘𝑀)) → (1st𝑧)(𝑉 ∘ (𝑀𝑉))(2nd𝑧))
99 df-br 5144 . . . . 5 ((1st𝑧)(𝑉 ∘ (𝑀𝑉))(2nd𝑧) ↔ ⟨(1st𝑧), (2nd𝑧)⟩ ∈ (𝑉 ∘ (𝑀𝑉)))
10098, 99sylib 218 . . . 4 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ (𝑉𝑈𝑉 = 𝑉)) ∧ 𝑧 ∈ ((cls‘(𝐽 ×t 𝐽))‘𝑀)) → ⟨(1st𝑧), (2nd𝑧)⟩ ∈ (𝑉 ∘ (𝑀𝑉)))
10127, 100eqeltrd 2841 . . 3 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ (𝑉𝑈𝑉 = 𝑉)) ∧ 𝑧 ∈ ((cls‘(𝐽 ×t 𝐽))‘𝑀)) → 𝑧 ∈ (𝑉 ∘ (𝑀𝑉)))
102101ex 412 . 2 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ (𝑉𝑈𝑉 = 𝑉)) → (𝑧 ∈ ((cls‘(𝐽 ×t 𝐽))‘𝑀) → 𝑧 ∈ (𝑉 ∘ (𝑀𝑉))))
103102ssrdv 3989 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 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  wne 2940  wral 3061  Vcvv 3480  cin 3950  wss 3951  c0 4333  {csn 4626  cop 4632   cuni 4907   class class class wbr 5143   × cxp 5683  ccnv 5684  cima 5688  ccom 5689  Rel wrel 5690  cfv 6561  (class class class)co 7431  1st c1st 8012  2nd c2nd 8013  Topctop 22899  TopOnctopon 22916  clsccl 23026  neicnei 23105   ×t ctx 23568  UnifOncust 24208  unifTopcutop 24239
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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-1o 8506  df-2o 8507  df-en 8986  df-fin 8989  df-fi 9451  df-topgen 17488  df-top 22900  df-topon 22917  df-bases 22953  df-cld 23027  df-ntr 23028  df-cls 23029  df-nei 23106  df-tx 23570  df-ust 24209  df-utop 24240
This theorem is referenced by:  utopreg  24261
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