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Theorem utop3cls 21865
 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 5150 . . . . 5 Rel (𝑋 × 𝑋)
2 utoptop.1 . . . . . . . . . . 11 𝐽 = (unifTop‘𝑈)
3 utoptop 21848 . . . . . . . . . . 11 (𝑈 ∈ (UnifOn‘𝑋) → (unifTop‘𝑈) ∈ Top)
42, 3syl5eqel 2692 . . . . . . . . . 10 (𝑈 ∈ (UnifOn‘𝑋) → 𝐽 ∈ Top)
5 txtop 21182 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ 𝐽 ∈ Top) → (𝐽 ×t 𝐽) ∈ Top)
64, 4, 5syl2anc 691 . . . . . . . . 9 (𝑈 ∈ (UnifOn‘𝑋) → (𝐽 ×t 𝐽) ∈ Top)
76ad3antrrr 762 . . . . . . . 8 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ (𝑉𝑈𝑉 = 𝑉)) ∧ 𝑧 ∈ ((cls‘(𝐽 ×t 𝐽))‘𝑀)) → (𝐽 ×t 𝐽) ∈ Top)
8 simpllr 795 . . . . . . . . 9 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ (𝑉𝑈𝑉 = 𝑉)) ∧ 𝑧 ∈ ((cls‘(𝐽 ×t 𝐽))‘𝑀)) → 𝑀 ⊆ (𝑋 × 𝑋))
9 utoptopon 21850 . . . . . . . . . . . . . 14 (𝑈 ∈ (UnifOn‘𝑋) → (unifTop‘𝑈) ∈ (TopOn‘𝑋))
102, 9syl5eqel 2692 . . . . . . . . . . . . 13 (𝑈 ∈ (UnifOn‘𝑋) → 𝐽 ∈ (TopOn‘𝑋))
11 toponuni 20542 . . . . . . . . . . . . 13 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
1210, 11syl 17 . . . . . . . . . . . 12 (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = 𝐽)
1312sqxpeqd 5065 . . . . . . . . . . 11 (𝑈 ∈ (UnifOn‘𝑋) → (𝑋 × 𝑋) = ( 𝐽 × 𝐽))
14 eqid 2610 . . . . . . . . . . . . 13 𝐽 = 𝐽
1514, 14txuni 21205 . . . . . . . . . . . 12 ((𝐽 ∈ Top ∧ 𝐽 ∈ Top) → ( 𝐽 × 𝐽) = (𝐽 ×t 𝐽))
164, 4, 15syl2anc 691 . . . . . . . . . . 11 (𝑈 ∈ (UnifOn‘𝑋) → ( 𝐽 × 𝐽) = (𝐽 ×t 𝐽))
1713, 16eqtrd 2644 . . . . . . . . . 10 (𝑈 ∈ (UnifOn‘𝑋) → (𝑋 × 𝑋) = (𝐽 ×t 𝐽))
1817ad3antrrr 762 . . . . . . . . 9 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ (𝑉𝑈𝑉 = 𝑉)) ∧ 𝑧 ∈ ((cls‘(𝐽 ×t 𝐽))‘𝑀)) → (𝑋 × 𝑋) = (𝐽 ×t 𝐽))
198, 18sseqtrd 3604 . . . . . . . 8 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ (𝑉𝑈𝑉 = 𝑉)) ∧ 𝑧 ∈ ((cls‘(𝐽 ×t 𝐽))‘𝑀)) → 𝑀 (𝐽 ×t 𝐽))
20 eqid 2610 . . . . . . . . 9 (𝐽 ×t 𝐽) = (𝐽 ×t 𝐽)
2120clsss3 20673 . . . . . . . 8 (((𝐽 ×t 𝐽) ∈ Top ∧ 𝑀 (𝐽 ×t 𝐽)) → ((cls‘(𝐽 ×t 𝐽))‘𝑀) ⊆ (𝐽 ×t 𝐽))
227, 19, 21syl2anc 691 . . . . . . 7 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ (𝑉𝑈𝑉 = 𝑉)) ∧ 𝑧 ∈ ((cls‘(𝐽 ×t 𝐽))‘𝑀)) → ((cls‘(𝐽 ×t 𝐽))‘𝑀) ⊆ (𝐽 ×t 𝐽))
2322, 18sseqtr4d 3605 . . . . . 6 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ (𝑉𝑈𝑉 = 𝑉)) ∧ 𝑧 ∈ ((cls‘(𝐽 ×t 𝐽))‘𝑀)) → ((cls‘(𝐽 ×t 𝐽))‘𝑀) ⊆ (𝑋 × 𝑋))
24 simpr 476 . . . . . 6 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ (𝑉𝑈𝑉 = 𝑉)) ∧ 𝑧 ∈ ((cls‘(𝐽 ×t 𝐽))‘𝑀)) → 𝑧 ∈ ((cls‘(𝐽 ×t 𝐽))‘𝑀))
2523, 24sseldd 3569 . . . . 5 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ (𝑉𝑈𝑉 = 𝑉)) ∧ 𝑧 ∈ ((cls‘(𝐽 ×t 𝐽))‘𝑀)) → 𝑧 ∈ (𝑋 × 𝑋))
26 1st2nd 7105 . . . . 5 ((Rel (𝑋 × 𝑋) ∧ 𝑧 ∈ (𝑋 × 𝑋)) → 𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩)
271, 25, 26sylancr 694 . . . 4 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ (𝑉𝑈𝑉 = 𝑉)) ∧ 𝑧 ∈ ((cls‘(𝐽 ×t 𝐽))‘𝑀)) → 𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩)
28 simp-4l 802 . . . . . . . . . 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 1111 . . . . . . . . . . 11 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ ((𝑉𝑈𝑉 = 𝑉) ∧ 𝑧 ∈ ((cls‘(𝐽 ×t 𝐽))‘𝑀) ∧ 𝑟 ∈ (((𝑉 “ {(1st𝑧)}) × (𝑉 “ {(2nd𝑧)})) ∩ 𝑀))) → 𝑉𝑈)
30293anassrs 1282 . . . . . . . . . 10 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ (𝑉𝑈𝑉 = 𝑉)) ∧ 𝑧 ∈ ((cls‘(𝐽 ×t 𝐽))‘𝑀)) ∧ 𝑟 ∈ (((𝑉 “ {(1st𝑧)}) × (𝑉 “ {(2nd𝑧)})) ∩ 𝑀)) → 𝑉𝑈)
31 ustrel 21825 . . . . . . . . . 10 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) → Rel 𝑉)
3228, 30, 31syl2anc 691 . . . . . . . . 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 476 . . . . . . . . . . . 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 3758 . . . . . . . . . . . 12 (𝑟 ∈ (((𝑉 “ {(1st𝑧)}) × (𝑉 “ {(2nd𝑧)})) ∩ 𝑀) ↔ (𝑟 ∈ ((𝑉 “ {(1st𝑧)}) × (𝑉 “ {(2nd𝑧)})) ∧ 𝑟𝑀))
3533, 34sylib 207 . . . . . . . . . . 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 474 . . . . . . . . . 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 7089 . . . . . . . . . 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 5408 . . . . . . . . . 10 (Rel 𝑉 → ((1st𝑟) ∈ (𝑉 “ {(1st𝑧)}) ↔ (1st𝑧)𝑉(1st𝑟)))
4039biimpa 500 . . . . . . . . 9 ((Rel 𝑉 ∧ (1st𝑟) ∈ (𝑉 “ {(1st𝑧)})) → (1st𝑧)𝑉(1st𝑟))
4132, 38, 40syl2anc 691 . . . . . . . 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 803 . . . . . . . . . . 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 5149 . . . . . . . . . . 11 (𝑋 × 𝑋) ⊆ (V × V)
4442, 43syl6ss 3580 . . . . . . . . . 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 5045 . . . . . . . . . 10 (Rel 𝑀𝑀 ⊆ (V × V))
4644, 45sylibr 223 . . . . . . . . 9 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ (𝑉𝑈𝑉 = 𝑉)) ∧ 𝑧 ∈ ((cls‘(𝐽 ×t 𝐽))‘𝑀)) ∧ 𝑟 ∈ (((𝑉 “ {(1st𝑧)}) × (𝑉 “ {(2nd𝑧)})) ∩ 𝑀)) → Rel 𝑀)
4735simprd 478 . . . . . . . . 9 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ (𝑉𝑈𝑉 = 𝑉)) ∧ 𝑧 ∈ ((cls‘(𝐽 ×t 𝐽))‘𝑀)) ∧ 𝑟 ∈ (((𝑉 “ {(1st𝑧)}) × (𝑉 “ {(2nd𝑧)})) ∩ 𝑀)) → 𝑟𝑀)
48 1st2ndbr 7108 . . . . . . . . 9 ((Rel 𝑀𝑟𝑀) → (1st𝑟)𝑀(2nd𝑟))
4946, 47, 48syl2anc 691 . . . . . . . 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 7090 . . . . . . . . . . 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 5408 . . . . . . . . . . 11 (Rel 𝑉 → ((2nd𝑟) ∈ (𝑉 “ {(2nd𝑧)}) ↔ (2nd𝑧)𝑉(2nd𝑟)))
5352biimpa 500 . . . . . . . . . 10 ((Rel 𝑉 ∧ (2nd𝑟) ∈ (𝑉 “ {(2nd𝑧)})) → (2nd𝑧)𝑉(2nd𝑟))
5432, 51, 53syl2anc 691 . . . . . . . . 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 1112 . . . . . . . . . . 11 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ ((𝑉𝑈𝑉 = 𝑉) ∧ 𝑧 ∈ ((cls‘(𝐽 ×t 𝐽))‘𝑀) ∧ 𝑟 ∈ (((𝑉 “ {(1st𝑧)}) × (𝑉 “ {(2nd𝑧)})) ∩ 𝑀))) → 𝑉 = 𝑉)
56553anassrs 1282 . . . . . . . . . 10 (((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ (𝑉𝑈𝑉 = 𝑉)) ∧ 𝑧 ∈ ((cls‘(𝐽 ×t 𝐽))‘𝑀)) ∧ 𝑟 ∈ (((𝑉 “ {(1st𝑧)}) × (𝑉 “ {(2nd𝑧)})) ∩ 𝑀)) → 𝑉 = 𝑉)
57 fvex 6113 . . . . . . . . . . . 12 (2nd𝑟) ∈ V
58 fvex 6113 . . . . . . . . . . . 12 (2nd𝑧) ∈ V
5957, 58brcnv 5227 . . . . . . . . . . 11 ((2nd𝑟)𝑉(2nd𝑧) ↔ (2nd𝑧)𝑉(2nd𝑟))
60 breq 4585 . . . . . . . . . . 11 (𝑉 = 𝑉 → ((2nd𝑟)𝑉(2nd𝑧) ↔ (2nd𝑟)𝑉(2nd𝑧)))
6159, 60syl5rbbr 274 . . . . . . . . . 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 246 . . . . . . . 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 6113 . . . . . . . . . 10 (1st𝑧) ∈ V
65 fvex 6113 . . . . . . . . . 10 (1st𝑟) ∈ V
66 brcogw 5212 . . . . . . . . . . 11 ((((1st𝑧) ∈ V ∧ (2nd𝑟) ∈ V ∧ (1st𝑟) ∈ V) ∧ ((1st𝑧)𝑉(1st𝑟) ∧ (1st𝑟)𝑀(2nd𝑟))) → (1st𝑧)(𝑀𝑉)(2nd𝑟))
6766ex 449 . . . . . . . . . 10 (((1st𝑧) ∈ V ∧ (2nd𝑟) ∈ V ∧ (1st𝑟) ∈ V) → (((1st𝑧)𝑉(1st𝑟) ∧ (1st𝑟)𝑀(2nd𝑟)) → (1st𝑧)(𝑀𝑉)(2nd𝑟)))
6864, 57, 65, 67mp3an 1416 . . . . . . . . 9 (((1st𝑧)𝑉(1st𝑟) ∧ (1st𝑟)𝑀(2nd𝑟)) → (1st𝑧)(𝑀𝑉)(2nd𝑟))
69 brcogw 5212 . . . . . . . . . . 11 ((((1st𝑧) ∈ V ∧ (2nd𝑧) ∈ V ∧ (2nd𝑟) ∈ V) ∧ ((1st𝑧)(𝑀𝑉)(2nd𝑟) ∧ (2nd𝑟)𝑉(2nd𝑧))) → (1st𝑧)(𝑉 ∘ (𝑀𝑉))(2nd𝑧))
7069ex 449 . . . . . . . . . 10 (((1st𝑧) ∈ V ∧ (2nd𝑧) ∈ V ∧ (2nd𝑟) ∈ V) → (((1st𝑧)(𝑀𝑉)(2nd𝑟) ∧ (2nd𝑟)𝑉(2nd𝑧)) → (1st𝑧)(𝑉 ∘ (𝑀𝑉))(2nd𝑧)))
7164, 58, 57, 70mp3an 1416 . . . . . . . . 9 (((1st𝑧)(𝑀𝑉)(2nd𝑟) ∧ (2nd𝑟)𝑉(2nd𝑧)) → (1st𝑧)(𝑉 ∘ (𝑀𝑉))(2nd𝑧))
7268, 71sylan 487 . . . . . . . 8 ((((1st𝑧)𝑉(1st𝑟) ∧ (1st𝑟)𝑀(2nd𝑟)) ∧ (2nd𝑟)𝑉(2nd𝑧)) → (1st𝑧)(𝑉 ∘ (𝑀𝑉))(2nd𝑧))
7341, 49, 63, 72syl21anc 1317 . . . . . . 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 2949 . . . . . 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 794 . . . . . . . . 9 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ (𝑉𝑈𝑉 = 𝑉)) ∧ 𝑧 ∈ ((cls‘(𝐽 ×t 𝐽))‘𝑀)) → 𝑈 ∈ (UnifOn‘𝑋))
76 simplrl 796 . . . . . . . . 9 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ (𝑉𝑈𝑉 = 𝑉)) ∧ 𝑧 ∈ ((cls‘(𝐽 ×t 𝐽))‘𝑀)) → 𝑉𝑈)
7743ad2ant1 1075 . . . . . . . . . . 11 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈𝑧 ∈ (𝑋 × 𝑋)) → 𝐽 ∈ Top)
78 xp1st 7089 . . . . . . . . . . . 12 (𝑧 ∈ (𝑋 × 𝑋) → (1st𝑧) ∈ 𝑋)
792utopsnnei 21863 . . . . . . . . . . . 12 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈 ∧ (1st𝑧) ∈ 𝑋) → (𝑉 “ {(1st𝑧)}) ∈ ((nei‘𝐽)‘{(1st𝑧)}))
8078, 79syl3an3 1353 . . . . . . . . . . 11 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈𝑧 ∈ (𝑋 × 𝑋)) → (𝑉 “ {(1st𝑧)}) ∈ ((nei‘𝐽)‘{(1st𝑧)}))
81 xp2nd 7090 . . . . . . . . . . . 12 (𝑧 ∈ (𝑋 × 𝑋) → (2nd𝑧) ∈ 𝑋)
822utopsnnei 21863 . . . . . . . . . . . 12 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈 ∧ (2nd𝑧) ∈ 𝑋) → (𝑉 “ {(2nd𝑧)}) ∈ ((nei‘𝐽)‘{(2nd𝑧)}))
8381, 82syl3an3 1353 . . . . . . . . . . 11 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈𝑧 ∈ (𝑋 × 𝑋)) → (𝑉 “ {(2nd𝑧)}) ∈ ((nei‘𝐽)‘{(2nd𝑧)}))
8414, 14neitx 21220 . . . . . . . . . . 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 1319 . . . . . . . . . 10 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈𝑧 ∈ (𝑋 × 𝑋)) → ((𝑉 “ {(1st𝑧)}) × (𝑉 “ {(2nd𝑧)})) ∈ ((nei‘(𝐽 ×t 𝐽))‘({(1st𝑧)} × {(2nd𝑧)})))
86 1st2nd2 7096 . . . . . . . . . . . . . 14 (𝑧 ∈ (𝑋 × 𝑋) → 𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩)
8786sneqd 4137 . . . . . . . . . . . . 13 (𝑧 ∈ (𝑋 × 𝑋) → {𝑧} = {⟨(1st𝑧), (2nd𝑧)⟩})
8864, 58xpsn 6313 . . . . . . . . . . . . 13 ({(1st𝑧)} × {(2nd𝑧)}) = {⟨(1st𝑧), (2nd𝑧)⟩}
8987, 88syl6eqr 2662 . . . . . . . . . . . 12 (𝑧 ∈ (𝑋 × 𝑋) → {𝑧} = ({(1st𝑧)} × {(2nd𝑧)}))
9089fveq2d 6107 . . . . . . . . . . 11 (𝑧 ∈ (𝑋 × 𝑋) → ((nei‘(𝐽 ×t 𝐽))‘{𝑧}) = ((nei‘(𝐽 ×t 𝐽))‘({(1st𝑧)} × {(2nd𝑧)})))
91903ad2ant3 1077 . . . . . . . . . 10 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈𝑧 ∈ (𝑋 × 𝑋)) → ((nei‘(𝐽 ×t 𝐽))‘{𝑧}) = ((nei‘(𝐽 ×t 𝐽))‘({(1st𝑧)} × {(2nd𝑧)})))
9285, 91eleqtrrd 2691 . . . . . . . . 9 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈𝑧 ∈ (𝑋 × 𝑋)) → ((𝑉 “ {(1st𝑧)}) × (𝑉 “ {(2nd𝑧)})) ∈ ((nei‘(𝐽 ×t 𝐽))‘{𝑧}))
9375, 76, 25, 92syl3anc 1318 . . . . . . . 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 20731 . . . . . . . 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 1319 . . . . . . 7 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ (𝑉𝑈𝑉 = 𝑉)) ∧ 𝑧 ∈ ((cls‘(𝐽 ×t 𝐽))‘𝑀)) → (((𝑉 “ {(1st𝑧)}) × (𝑉 “ {(2nd𝑧)})) ∩ 𝑀) ≠ ∅)
96 r19.3rzv 4016 . . . . . . 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 246 . . . . 5 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ (𝑉𝑈𝑉 = 𝑉)) ∧ 𝑧 ∈ ((cls‘(𝐽 ×t 𝐽))‘𝑀)) → (1st𝑧)(𝑉 ∘ (𝑀𝑉))(2nd𝑧))
99 df-br 4584 . . . . 5 ((1st𝑧)(𝑉 ∘ (𝑀𝑉))(2nd𝑧) ↔ ⟨(1st𝑧), (2nd𝑧)⟩ ∈ (𝑉 ∘ (𝑀𝑉)))
10098, 99sylib 207 . . . 4 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ (𝑉𝑈𝑉 = 𝑉)) ∧ 𝑧 ∈ ((cls‘(𝐽 ×t 𝐽))‘𝑀)) → ⟨(1st𝑧), (2nd𝑧)⟩ ∈ (𝑉 ∘ (𝑀𝑉)))
10127, 100eqeltrd 2688 . . 3 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ (𝑉𝑈𝑉 = 𝑉)) ∧ 𝑧 ∈ ((cls‘(𝐽 ×t 𝐽))‘𝑀)) → 𝑧 ∈ (𝑉 ∘ (𝑀𝑉)))
102101ex 449 . 2 (((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ (𝑉𝑈𝑉 = 𝑉)) → (𝑧 ∈ ((cls‘(𝐽 ×t 𝐽))‘𝑀) → 𝑧 ∈ (𝑉 ∘ (𝑀𝑉))))
103102ssrdv 3574 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 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  Vcvv 3173   ∩ cin 3539   ⊆ wss 3540  ∅c0 3874  {csn 4125  ⟨cop 4131  ∪ cuni 4372   class class class wbr 4583   × cxp 5036  ◡ccnv 5037   “ cima 5041   ∘ ccom 5042  Rel wrel 5043  ‘cfv 5804  (class class class)co 6549  1st c1st 7057  2nd c2nd 7058  Topctop 20517  TopOnctopon 20518  clsccl 20632  neicnei 20711   ×t ctx 21173  UnifOncust 21813  unifTopcutop 21844 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-en 7842  df-fin 7845  df-fi 8200  df-topgen 15927  df-top 20521  df-bases 20522  df-topon 20523  df-cld 20633  df-ntr 20634  df-cls 20635  df-nei 20712  df-tx 21175  df-ust 21814  df-utop 21845 This theorem is referenced by:  utopreg  21866
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