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Theorem utop2nei 22101
Description: For any symmetrical entourage 𝑉 and any relation 𝑀, build a neighborhood of 𝑀. First part of proposition 2 of [BourbakiTop1] p. II.4. (Contributed by Thierry Arnoux, 14-Jan-2018.)
Hypothesis
Ref Expression
utoptop.1 𝐽 = (unifTop‘𝑈)
Assertion
Ref Expression
utop2nei ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) → (𝑉 ∘ (𝑀𝑉)) ∈ ((nei‘(𝐽 ×t 𝐽))‘𝑀))

Proof of Theorem utop2nei
Dummy variables 𝑟 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 utoptop.1 . . . . . . . 8 𝐽 = (unifTop‘𝑈)
2 utoptop 22085 . . . . . . . 8 (𝑈 ∈ (UnifOn‘𝑋) → (unifTop‘𝑈) ∈ Top)
31, 2syl5eqel 2734 . . . . . . 7 (𝑈 ∈ (UnifOn‘𝑋) → 𝐽 ∈ Top)
4 txtop 21420 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝐽 ∈ Top) → (𝐽 ×t 𝐽) ∈ Top)
53, 3, 4syl2anc 694 . . . . . 6 (𝑈 ∈ (UnifOn‘𝑋) → (𝐽 ×t 𝐽) ∈ Top)
653ad2ant1 1102 . . . . 5 ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) → (𝐽 ×t 𝐽) ∈ Top)
76adantr 480 . . . 4 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑀 = ∅) → (𝐽 ×t 𝐽) ∈ Top)
8 0nei 20980 . . . 4 ((𝐽 ×t 𝐽) ∈ Top → ∅ ∈ ((nei‘(𝐽 ×t 𝐽))‘∅))
97, 8syl 17 . . 3 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑀 = ∅) → ∅ ∈ ((nei‘(𝐽 ×t 𝐽))‘∅))
10 coeq1 5312 . . . . . . 7 (𝑀 = ∅ → (𝑀𝑉) = (∅ ∘ 𝑉))
11 co01 5688 . . . . . . 7 (∅ ∘ 𝑉) = ∅
1210, 11syl6eq 2701 . . . . . 6 (𝑀 = ∅ → (𝑀𝑉) = ∅)
1312coeq2d 5317 . . . . 5 (𝑀 = ∅ → (𝑉 ∘ (𝑀𝑉)) = (𝑉 ∘ ∅))
14 co02 5687 . . . . 5 (𝑉 ∘ ∅) = ∅
1513, 14syl6eq 2701 . . . 4 (𝑀 = ∅ → (𝑉 ∘ (𝑀𝑉)) = ∅)
1615adantl 481 . . 3 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑀 = ∅) → (𝑉 ∘ (𝑀𝑉)) = ∅)
17 simpr 476 . . . 4 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑀 = ∅) → 𝑀 = ∅)
1817fveq2d 6233 . . 3 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑀 = ∅) → ((nei‘(𝐽 ×t 𝐽))‘𝑀) = ((nei‘(𝐽 ×t 𝐽))‘∅))
199, 16, 183eltr4d 2745 . 2 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑀 = ∅) → (𝑉 ∘ (𝑀𝑉)) ∈ ((nei‘(𝐽 ×t 𝐽))‘𝑀))
206adantr 480 . . . . . 6 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟𝑀) → (𝐽 ×t 𝐽) ∈ Top)
21 simpl1 1084 . . . . . . . . . 10 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟𝑀) → 𝑈 ∈ (UnifOn‘𝑋))
2221, 3syl 17 . . . . . . . . 9 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟𝑀) → 𝐽 ∈ Top)
23 simpl2l 1134 . . . . . . . . . 10 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟𝑀) → 𝑉𝑈)
24 simp3 1083 . . . . . . . . . . . 12 ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) → 𝑀 ⊆ (𝑋 × 𝑋))
2524sselda 3636 . . . . . . . . . . 11 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟𝑀) → 𝑟 ∈ (𝑋 × 𝑋))
26 xp1st 7242 . . . . . . . . . . 11 (𝑟 ∈ (𝑋 × 𝑋) → (1st𝑟) ∈ 𝑋)
2725, 26syl 17 . . . . . . . . . 10 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟𝑀) → (1st𝑟) ∈ 𝑋)
281utopsnnei 22100 . . . . . . . . . 10 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈 ∧ (1st𝑟) ∈ 𝑋) → (𝑉 “ {(1st𝑟)}) ∈ ((nei‘𝐽)‘{(1st𝑟)}))
2921, 23, 27, 28syl3anc 1366 . . . . . . . . 9 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟𝑀) → (𝑉 “ {(1st𝑟)}) ∈ ((nei‘𝐽)‘{(1st𝑟)}))
30 xp2nd 7243 . . . . . . . . . . 11 (𝑟 ∈ (𝑋 × 𝑋) → (2nd𝑟) ∈ 𝑋)
3125, 30syl 17 . . . . . . . . . 10 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟𝑀) → (2nd𝑟) ∈ 𝑋)
321utopsnnei 22100 . . . . . . . . . 10 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈 ∧ (2nd𝑟) ∈ 𝑋) → (𝑉 “ {(2nd𝑟)}) ∈ ((nei‘𝐽)‘{(2nd𝑟)}))
3321, 23, 31, 32syl3anc 1366 . . . . . . . . 9 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟𝑀) → (𝑉 “ {(2nd𝑟)}) ∈ ((nei‘𝐽)‘{(2nd𝑟)}))
34 eqid 2651 . . . . . . . . . 10 𝐽 = 𝐽
3534, 34neitx 21458 . . . . . . . . 9 (((𝐽 ∈ Top ∧ 𝐽 ∈ Top) ∧ ((𝑉 “ {(1st𝑟)}) ∈ ((nei‘𝐽)‘{(1st𝑟)}) ∧ (𝑉 “ {(2nd𝑟)}) ∈ ((nei‘𝐽)‘{(2nd𝑟)}))) → ((𝑉 “ {(1st𝑟)}) × (𝑉 “ {(2nd𝑟)})) ∈ ((nei‘(𝐽 ×t 𝐽))‘({(1st𝑟)} × {(2nd𝑟)})))
3622, 22, 29, 33, 35syl22anc 1367 . . . . . . . 8 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟𝑀) → ((𝑉 “ {(1st𝑟)}) × (𝑉 “ {(2nd𝑟)})) ∈ ((nei‘(𝐽 ×t 𝐽))‘({(1st𝑟)} × {(2nd𝑟)})))
37 fvex 6239 . . . . . . . . . 10 (1st𝑟) ∈ V
38 fvex 6239 . . . . . . . . . 10 (2nd𝑟) ∈ V
3937, 38xpsn 6447 . . . . . . . . 9 ({(1st𝑟)} × {(2nd𝑟)}) = {⟨(1st𝑟), (2nd𝑟)⟩}
4039fveq2i 6232 . . . . . . . 8 ((nei‘(𝐽 ×t 𝐽))‘({(1st𝑟)} × {(2nd𝑟)})) = ((nei‘(𝐽 ×t 𝐽))‘{⟨(1st𝑟), (2nd𝑟)⟩})
4136, 40syl6eleq 2740 . . . . . . 7 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟𝑀) → ((𝑉 “ {(1st𝑟)}) × (𝑉 “ {(2nd𝑟)})) ∈ ((nei‘(𝐽 ×t 𝐽))‘{⟨(1st𝑟), (2nd𝑟)⟩}))
4224adantr 480 . . . . . . . . . . 11 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟𝑀) → 𝑀 ⊆ (𝑋 × 𝑋))
43 xpss 5159 . . . . . . . . . . . . 13 (𝑋 × 𝑋) ⊆ (V × V)
44 sstr 3644 . . . . . . . . . . . . 13 ((𝑀 ⊆ (𝑋 × 𝑋) ∧ (𝑋 × 𝑋) ⊆ (V × V)) → 𝑀 ⊆ (V × V))
4543, 44mpan2 707 . . . . . . . . . . . 12 (𝑀 ⊆ (𝑋 × 𝑋) → 𝑀 ⊆ (V × V))
46 df-rel 5150 . . . . . . . . . . . 12 (Rel 𝑀𝑀 ⊆ (V × V))
4745, 46sylibr 224 . . . . . . . . . . 11 (𝑀 ⊆ (𝑋 × 𝑋) → Rel 𝑀)
4842, 47syl 17 . . . . . . . . . 10 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟𝑀) → Rel 𝑀)
49 1st2nd 7258 . . . . . . . . . 10 ((Rel 𝑀𝑟𝑀) → 𝑟 = ⟨(1st𝑟), (2nd𝑟)⟩)
5048, 49sylancom 702 . . . . . . . . 9 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟𝑀) → 𝑟 = ⟨(1st𝑟), (2nd𝑟)⟩)
5150sneqd 4222 . . . . . . . 8 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟𝑀) → {𝑟} = {⟨(1st𝑟), (2nd𝑟)⟩})
5251fveq2d 6233 . . . . . . 7 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟𝑀) → ((nei‘(𝐽 ×t 𝐽))‘{𝑟}) = ((nei‘(𝐽 ×t 𝐽))‘{⟨(1st𝑟), (2nd𝑟)⟩}))
5341, 52eleqtrrd 2733 . . . . . 6 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟𝑀) → ((𝑉 “ {(1st𝑟)}) × (𝑉 “ {(2nd𝑟)})) ∈ ((nei‘(𝐽 ×t 𝐽))‘{𝑟}))
54 relxp 5160 . . . . . . . . . . 11 Rel ((𝑉 “ {(1st𝑟)}) × (𝑉 “ {(2nd𝑟)}))
5554a1i 11 . . . . . . . . . 10 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟𝑀) ∧ 𝑧 ∈ ((𝑉 “ {(1st𝑟)}) × (𝑉 “ {(2nd𝑟)}))) → Rel ((𝑉 “ {(1st𝑟)}) × (𝑉 “ {(2nd𝑟)})))
56 1st2nd 7258 . . . . . . . . . 10 ((Rel ((𝑉 “ {(1st𝑟)}) × (𝑉 “ {(2nd𝑟)})) ∧ 𝑧 ∈ ((𝑉 “ {(1st𝑟)}) × (𝑉 “ {(2nd𝑟)}))) → 𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩)
5755, 56sylancom 702 . . . . . . . . 9 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟𝑀) ∧ 𝑧 ∈ ((𝑉 “ {(1st𝑟)}) × (𝑉 “ {(2nd𝑟)}))) → 𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩)
58 simpll2 1121 . . . . . . . . . . . . 13 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟𝑀) ∧ 𝑧 ∈ ((𝑉 “ {(1st𝑟)}) × (𝑉 “ {(2nd𝑟)}))) → (𝑉𝑈𝑉 = 𝑉))
5958simprd 478 . . . . . . . . . . . 12 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟𝑀) ∧ 𝑧 ∈ ((𝑉 “ {(1st𝑟)}) × (𝑉 “ {(2nd𝑟)}))) → 𝑉 = 𝑉)
60 simpll1 1120 . . . . . . . . . . . . . 14 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟𝑀) ∧ 𝑧 ∈ ((𝑉 “ {(1st𝑟)}) × (𝑉 “ {(2nd𝑟)}))) → 𝑈 ∈ (UnifOn‘𝑋))
6158simpld 474 . . . . . . . . . . . . . 14 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟𝑀) ∧ 𝑧 ∈ ((𝑉 “ {(1st𝑟)}) × (𝑉 “ {(2nd𝑟)}))) → 𝑉𝑈)
62 ustrel 22062 . . . . . . . . . . . . . 14 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) → Rel 𝑉)
6360, 61, 62syl2anc 694 . . . . . . . . . . . . 13 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟𝑀) ∧ 𝑧 ∈ ((𝑉 “ {(1st𝑟)}) × (𝑉 “ {(2nd𝑟)}))) → Rel 𝑉)
64 xp1st 7242 . . . . . . . . . . . . . 14 (𝑧 ∈ ((𝑉 “ {(1st𝑟)}) × (𝑉 “ {(2nd𝑟)})) → (1st𝑧) ∈ (𝑉 “ {(1st𝑟)}))
6564adantl 481 . . . . . . . . . . . . 13 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟𝑀) ∧ 𝑧 ∈ ((𝑉 “ {(1st𝑟)}) × (𝑉 “ {(2nd𝑟)}))) → (1st𝑧) ∈ (𝑉 “ {(1st𝑟)}))
66 elrelimasn 5524 . . . . . . . . . . . . . 14 (Rel 𝑉 → ((1st𝑧) ∈ (𝑉 “ {(1st𝑟)}) ↔ (1st𝑟)𝑉(1st𝑧)))
6766biimpa 500 . . . . . . . . . . . . 13 ((Rel 𝑉 ∧ (1st𝑧) ∈ (𝑉 “ {(1st𝑟)})) → (1st𝑟)𝑉(1st𝑧))
6863, 65, 67syl2anc 694 . . . . . . . . . . . 12 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟𝑀) ∧ 𝑧 ∈ ((𝑉 “ {(1st𝑟)}) × (𝑉 “ {(2nd𝑟)}))) → (1st𝑟)𝑉(1st𝑧))
69 fvex 6239 . . . . . . . . . . . . . . 15 (1st𝑧) ∈ V
7037, 69brcnv 5337 . . . . . . . . . . . . . 14 ((1st𝑟)𝑉(1st𝑧) ↔ (1st𝑧)𝑉(1st𝑟))
71 breq 4687 . . . . . . . . . . . . . 14 (𝑉 = 𝑉 → ((1st𝑟)𝑉(1st𝑧) ↔ (1st𝑟)𝑉(1st𝑧)))
7270, 71syl5bbr 274 . . . . . . . . . . . . 13 (𝑉 = 𝑉 → ((1st𝑧)𝑉(1st𝑟) ↔ (1st𝑟)𝑉(1st𝑧)))
7372biimpar 501 . . . . . . . . . . . 12 ((𝑉 = 𝑉 ∧ (1st𝑟)𝑉(1st𝑧)) → (1st𝑧)𝑉(1st𝑟))
7459, 68, 73syl2anc 694 . . . . . . . . . . 11 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟𝑀) ∧ 𝑧 ∈ ((𝑉 “ {(1st𝑟)}) × (𝑉 “ {(2nd𝑟)}))) → (1st𝑧)𝑉(1st𝑟))
75 simpll3 1122 . . . . . . . . . . . 12 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟𝑀) ∧ 𝑧 ∈ ((𝑉 “ {(1st𝑟)}) × (𝑉 “ {(2nd𝑟)}))) → 𝑀 ⊆ (𝑋 × 𝑋))
76 simplr 807 . . . . . . . . . . . 12 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟𝑀) ∧ 𝑧 ∈ ((𝑉 “ {(1st𝑟)}) × (𝑉 “ {(2nd𝑟)}))) → 𝑟𝑀)
77 1st2ndbr 7261 . . . . . . . . . . . . 13 ((Rel 𝑀𝑟𝑀) → (1st𝑟)𝑀(2nd𝑟))
7847, 77sylan 487 . . . . . . . . . . . 12 ((𝑀 ⊆ (𝑋 × 𝑋) ∧ 𝑟𝑀) → (1st𝑟)𝑀(2nd𝑟))
7975, 76, 78syl2anc 694 . . . . . . . . . . 11 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟𝑀) ∧ 𝑧 ∈ ((𝑉 “ {(1st𝑟)}) × (𝑉 “ {(2nd𝑟)}))) → (1st𝑟)𝑀(2nd𝑟))
80 xp2nd 7243 . . . . . . . . . . . . 13 (𝑧 ∈ ((𝑉 “ {(1st𝑟)}) × (𝑉 “ {(2nd𝑟)})) → (2nd𝑧) ∈ (𝑉 “ {(2nd𝑟)}))
8180adantl 481 . . . . . . . . . . . 12 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟𝑀) ∧ 𝑧 ∈ ((𝑉 “ {(1st𝑟)}) × (𝑉 “ {(2nd𝑟)}))) → (2nd𝑧) ∈ (𝑉 “ {(2nd𝑟)}))
82 elrelimasn 5524 . . . . . . . . . . . . 13 (Rel 𝑉 → ((2nd𝑧) ∈ (𝑉 “ {(2nd𝑟)}) ↔ (2nd𝑟)𝑉(2nd𝑧)))
8382biimpa 500 . . . . . . . . . . . 12 ((Rel 𝑉 ∧ (2nd𝑧) ∈ (𝑉 “ {(2nd𝑟)})) → (2nd𝑟)𝑉(2nd𝑧))
8463, 81, 83syl2anc 694 . . . . . . . . . . 11 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟𝑀) ∧ 𝑧 ∈ ((𝑉 “ {(1st𝑟)}) × (𝑉 “ {(2nd𝑟)}))) → (2nd𝑟)𝑉(2nd𝑧))
8569, 38, 373pm3.2i 1259 . . . . . . . . . . . . 13 ((1st𝑧) ∈ V ∧ (2nd𝑟) ∈ V ∧ (1st𝑟) ∈ V)
86 brcogw 5323 . . . . . . . . . . . . 13 ((((1st𝑧) ∈ V ∧ (2nd𝑟) ∈ V ∧ (1st𝑟) ∈ V) ∧ ((1st𝑧)𝑉(1st𝑟) ∧ (1st𝑟)𝑀(2nd𝑟))) → (1st𝑧)(𝑀𝑉)(2nd𝑟))
8785, 86mpan 706 . . . . . . . . . . . 12 (((1st𝑧)𝑉(1st𝑟) ∧ (1st𝑟)𝑀(2nd𝑟)) → (1st𝑧)(𝑀𝑉)(2nd𝑟))
88 fvex 6239 . . . . . . . . . . . . . 14 (2nd𝑧) ∈ V
8969, 88, 383pm3.2i 1259 . . . . . . . . . . . . 13 ((1st𝑧) ∈ V ∧ (2nd𝑧) ∈ V ∧ (2nd𝑟) ∈ V)
90 brcogw 5323 . . . . . . . . . . . . 13 ((((1st𝑧) ∈ V ∧ (2nd𝑧) ∈ V ∧ (2nd𝑟) ∈ V) ∧ ((1st𝑧)(𝑀𝑉)(2nd𝑟) ∧ (2nd𝑟)𝑉(2nd𝑧))) → (1st𝑧)(𝑉 ∘ (𝑀𝑉))(2nd𝑧))
9189, 90mpan 706 . . . . . . . . . . . 12 (((1st𝑧)(𝑀𝑉)(2nd𝑟) ∧ (2nd𝑟)𝑉(2nd𝑧)) → (1st𝑧)(𝑉 ∘ (𝑀𝑉))(2nd𝑧))
9287, 91sylan 487 . . . . . . . . . . 11 ((((1st𝑧)𝑉(1st𝑟) ∧ (1st𝑟)𝑀(2nd𝑟)) ∧ (2nd𝑟)𝑉(2nd𝑧)) → (1st𝑧)(𝑉 ∘ (𝑀𝑉))(2nd𝑧))
9374, 79, 84, 92syl21anc 1365 . . . . . . . . . 10 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟𝑀) ∧ 𝑧 ∈ ((𝑉 “ {(1st𝑟)}) × (𝑉 “ {(2nd𝑟)}))) → (1st𝑧)(𝑉 ∘ (𝑀𝑉))(2nd𝑧))
94 df-br 4686 . . . . . . . . . 10 ((1st𝑧)(𝑉 ∘ (𝑀𝑉))(2nd𝑧) ↔ ⟨(1st𝑧), (2nd𝑧)⟩ ∈ (𝑉 ∘ (𝑀𝑉)))
9593, 94sylib 208 . . . . . . . . 9 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟𝑀) ∧ 𝑧 ∈ ((𝑉 “ {(1st𝑟)}) × (𝑉 “ {(2nd𝑟)}))) → ⟨(1st𝑧), (2nd𝑧)⟩ ∈ (𝑉 ∘ (𝑀𝑉)))
9657, 95eqeltrd 2730 . . . . . . . 8 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟𝑀) ∧ 𝑧 ∈ ((𝑉 “ {(1st𝑟)}) × (𝑉 “ {(2nd𝑟)}))) → 𝑧 ∈ (𝑉 ∘ (𝑀𝑉)))
9796ex 449 . . . . . . 7 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟𝑀) → (𝑧 ∈ ((𝑉 “ {(1st𝑟)}) × (𝑉 “ {(2nd𝑟)})) → 𝑧 ∈ (𝑉 ∘ (𝑀𝑉))))
9897ssrdv 3642 . . . . . 6 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟𝑀) → ((𝑉 “ {(1st𝑟)}) × (𝑉 “ {(2nd𝑟)})) ⊆ (𝑉 ∘ (𝑀𝑉)))
99 simp1 1081 . . . . . . . . . . 11 ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) → 𝑈 ∈ (UnifOn‘𝑋))
100 simp2l 1107 . . . . . . . . . . 11 ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) → 𝑉𝑈)
101 ustssxp 22055 . . . . . . . . . . 11 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) → 𝑉 ⊆ (𝑋 × 𝑋))
10299, 100, 101syl2anc 694 . . . . . . . . . 10 ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) → 𝑉 ⊆ (𝑋 × 𝑋))
103 coss1 5310 . . . . . . . . . 10 (𝑉 ⊆ (𝑋 × 𝑋) → (𝑉 ∘ (𝑀𝑉)) ⊆ ((𝑋 × 𝑋) ∘ (𝑀𝑉)))
104102, 103syl 17 . . . . . . . . 9 ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) → (𝑉 ∘ (𝑀𝑉)) ⊆ ((𝑋 × 𝑋) ∘ (𝑀𝑉)))
105 coss1 5310 . . . . . . . . . . . 12 (𝑀 ⊆ (𝑋 × 𝑋) → (𝑀𝑉) ⊆ ((𝑋 × 𝑋) ∘ 𝑉))
10624, 105syl 17 . . . . . . . . . . 11 ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) → (𝑀𝑉) ⊆ ((𝑋 × 𝑋) ∘ 𝑉))
107 coss2 5311 . . . . . . . . . . . . 13 (𝑉 ⊆ (𝑋 × 𝑋) → ((𝑋 × 𝑋) ∘ 𝑉) ⊆ ((𝑋 × 𝑋) ∘ (𝑋 × 𝑋)))
108 xpcoid 5714 . . . . . . . . . . . . 13 ((𝑋 × 𝑋) ∘ (𝑋 × 𝑋)) = (𝑋 × 𝑋)
109107, 108syl6sseq 3684 . . . . . . . . . . . 12 (𝑉 ⊆ (𝑋 × 𝑋) → ((𝑋 × 𝑋) ∘ 𝑉) ⊆ (𝑋 × 𝑋))
110102, 109syl 17 . . . . . . . . . . 11 ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) → ((𝑋 × 𝑋) ∘ 𝑉) ⊆ (𝑋 × 𝑋))
111106, 110sstrd 3646 . . . . . . . . . 10 ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) → (𝑀𝑉) ⊆ (𝑋 × 𝑋))
112 coss2 5311 . . . . . . . . . . 11 ((𝑀𝑉) ⊆ (𝑋 × 𝑋) → ((𝑋 × 𝑋) ∘ (𝑀𝑉)) ⊆ ((𝑋 × 𝑋) ∘ (𝑋 × 𝑋)))
113112, 108syl6sseq 3684 . . . . . . . . . 10 ((𝑀𝑉) ⊆ (𝑋 × 𝑋) → ((𝑋 × 𝑋) ∘ (𝑀𝑉)) ⊆ (𝑋 × 𝑋))
114111, 113syl 17 . . . . . . . . 9 ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) → ((𝑋 × 𝑋) ∘ (𝑀𝑉)) ⊆ (𝑋 × 𝑋))
115104, 114sstrd 3646 . . . . . . . 8 ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) → (𝑉 ∘ (𝑀𝑉)) ⊆ (𝑋 × 𝑋))
116 utopbas 22086 . . . . . . . . . . . 12 (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = (unifTop‘𝑈))
1171unieqi 4477 . . . . . . . . . . . 12 𝐽 = (unifTop‘𝑈)
118116, 117syl6eqr 2703 . . . . . . . . . . 11 (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = 𝐽)
119118sqxpeqd 5175 . . . . . . . . . 10 (𝑈 ∈ (UnifOn‘𝑋) → (𝑋 × 𝑋) = ( 𝐽 × 𝐽))
12034, 34txuni 21443 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ 𝐽 ∈ Top) → ( 𝐽 × 𝐽) = (𝐽 ×t 𝐽))
1213, 3, 120syl2anc 694 . . . . . . . . . 10 (𝑈 ∈ (UnifOn‘𝑋) → ( 𝐽 × 𝐽) = (𝐽 ×t 𝐽))
122119, 121eqtrd 2685 . . . . . . . . 9 (𝑈 ∈ (UnifOn‘𝑋) → (𝑋 × 𝑋) = (𝐽 ×t 𝐽))
1231223ad2ant1 1102 . . . . . . . 8 ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) → (𝑋 × 𝑋) = (𝐽 ×t 𝐽))
124115, 123sseqtrd 3674 . . . . . . 7 ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) → (𝑉 ∘ (𝑀𝑉)) ⊆ (𝐽 ×t 𝐽))
125124adantr 480 . . . . . 6 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟𝑀) → (𝑉 ∘ (𝑀𝑉)) ⊆ (𝐽 ×t 𝐽))
126 eqid 2651 . . . . . . 7 (𝐽 ×t 𝐽) = (𝐽 ×t 𝐽)
127126ssnei2 20968 . . . . . 6 ((((𝐽 ×t 𝐽) ∈ Top ∧ ((𝑉 “ {(1st𝑟)}) × (𝑉 “ {(2nd𝑟)})) ∈ ((nei‘(𝐽 ×t 𝐽))‘{𝑟})) ∧ (((𝑉 “ {(1st𝑟)}) × (𝑉 “ {(2nd𝑟)})) ⊆ (𝑉 ∘ (𝑀𝑉)) ∧ (𝑉 ∘ (𝑀𝑉)) ⊆ (𝐽 ×t 𝐽))) → (𝑉 ∘ (𝑀𝑉)) ∈ ((nei‘(𝐽 ×t 𝐽))‘{𝑟}))
12820, 53, 98, 125, 127syl22anc 1367 . . . . 5 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟𝑀) → (𝑉 ∘ (𝑀𝑉)) ∈ ((nei‘(𝐽 ×t 𝐽))‘{𝑟}))
129128ralrimiva 2995 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) → ∀𝑟𝑀 (𝑉 ∘ (𝑀𝑉)) ∈ ((nei‘(𝐽 ×t 𝐽))‘{𝑟}))
130129adantr 480 . . 3 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑀 ≠ ∅) → ∀𝑟𝑀 (𝑉 ∘ (𝑀𝑉)) ∈ ((nei‘(𝐽 ×t 𝐽))‘{𝑟}))
1316adantr 480 . . . 4 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑀 ≠ ∅) → (𝐽 ×t 𝐽) ∈ Top)
13224, 123sseqtrd 3674 . . . . 5 ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) → 𝑀 (𝐽 ×t 𝐽))
133132adantr 480 . . . 4 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑀 ≠ ∅) → 𝑀 (𝐽 ×t 𝐽))
134 simpr 476 . . . 4 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑀 ≠ ∅) → 𝑀 ≠ ∅)
135126neips 20965 . . . 4 (((𝐽 ×t 𝐽) ∈ Top ∧ 𝑀 (𝐽 ×t 𝐽) ∧ 𝑀 ≠ ∅) → ((𝑉 ∘ (𝑀𝑉)) ∈ ((nei‘(𝐽 ×t 𝐽))‘𝑀) ↔ ∀𝑟𝑀 (𝑉 ∘ (𝑀𝑉)) ∈ ((nei‘(𝐽 ×t 𝐽))‘{𝑟})))
136131, 133, 134, 135syl3anc 1366 . . 3 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑀 ≠ ∅) → ((𝑉 ∘ (𝑀𝑉)) ∈ ((nei‘(𝐽 ×t 𝐽))‘𝑀) ↔ ∀𝑟𝑀 (𝑉 ∘ (𝑀𝑉)) ∈ ((nei‘(𝐽 ×t 𝐽))‘{𝑟})))
137130, 136mpbird 247 . 2 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑀 ≠ ∅) → (𝑉 ∘ (𝑀𝑉)) ∈ ((nei‘(𝐽 ×t 𝐽))‘𝑀))
13819, 137pm2.61dane 2910 1 ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) → (𝑉 ∘ (𝑀𝑉)) ∈ ((nei‘(𝐽 ×t 𝐽))‘𝑀))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  wne 2823  wral 2941  Vcvv 3231  wss 3607  c0 3948  {csn 4210  cop 4216   cuni 4468   class class class wbr 4685   × cxp 5141  ccnv 5142  cima 5146  ccom 5147  Rel wrel 5148  cfv 5926  (class class class)co 6690  1st c1st 7208  2nd c2nd 7209  Topctop 20746  neicnei 20949   ×t ctx 21411  UnifOncust 22050  unifTopcutop 22081
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-en 7998  df-fin 8001  df-fi 8358  df-topgen 16151  df-top 20747  df-topon 20764  df-bases 20798  df-nei 20950  df-tx 21413  df-ust 22051  df-utop 22082
This theorem is referenced by: (None)
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