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Theorem utop2nei 23747
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 23731 . . . . . . . 8 (𝑈 ∈ (UnifOn‘𝑋) → (unifTop‘𝑈) ∈ Top)
31, 2eqeltrid 2838 . . . . . . 7 (𝑈 ∈ (UnifOn‘𝑋) → 𝐽 ∈ Top)
4 txtop 23065 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝐽 ∈ Top) → (𝐽 ×t 𝐽) ∈ Top)
53, 3, 4syl2anc 585 . . . . . 6 (𝑈 ∈ (UnifOn‘𝑋) → (𝐽 ×t 𝐽) ∈ Top)
653ad2ant1 1134 . . . . 5 ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) → (𝐽 ×t 𝐽) ∈ Top)
76adantr 482 . . . 4 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑀 = ∅) → (𝐽 ×t 𝐽) ∈ Top)
8 0nei 22624 . . . 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 5856 . . . . . . 7 (𝑀 = ∅ → (𝑀𝑉) = (∅ ∘ 𝑉))
11 co01 6258 . . . . . . 7 (∅ ∘ 𝑉) = ∅
1210, 11eqtrdi 2789 . . . . . 6 (𝑀 = ∅ → (𝑀𝑉) = ∅)
1312coeq2d 5861 . . . . 5 (𝑀 = ∅ → (𝑉 ∘ (𝑀𝑉)) = (𝑉 ∘ ∅))
14 co02 6257 . . . . 5 (𝑉 ∘ ∅) = ∅
1513, 14eqtrdi 2789 . . . 4 (𝑀 = ∅ → (𝑉 ∘ (𝑀𝑉)) = ∅)
1615adantl 483 . . 3 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑀 = ∅) → (𝑉 ∘ (𝑀𝑉)) = ∅)
17 simpr 486 . . . 4 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑀 = ∅) → 𝑀 = ∅)
1817fveq2d 6893 . . 3 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑀 = ∅) → ((nei‘(𝐽 ×t 𝐽))‘𝑀) = ((nei‘(𝐽 ×t 𝐽))‘∅))
199, 16, 183eltr4d 2849 . 2 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑀 = ∅) → (𝑉 ∘ (𝑀𝑉)) ∈ ((nei‘(𝐽 ×t 𝐽))‘𝑀))
206adantr 482 . . . . . 6 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟𝑀) → (𝐽 ×t 𝐽) ∈ Top)
21 simpl1 1192 . . . . . . . . . 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 1227 . . . . . . . . . 10 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟𝑀) → 𝑉𝑈)
24 simp3 1139 . . . . . . . . . . . 12 ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) → 𝑀 ⊆ (𝑋 × 𝑋))
2524sselda 3982 . . . . . . . . . . 11 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟𝑀) → 𝑟 ∈ (𝑋 × 𝑋))
26 xp1st 8004 . . . . . . . . . . 11 (𝑟 ∈ (𝑋 × 𝑋) → (1st𝑟) ∈ 𝑋)
2725, 26syl 17 . . . . . . . . . 10 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟𝑀) → (1st𝑟) ∈ 𝑋)
281utopsnnei 23746 . . . . . . . . . 10 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈 ∧ (1st𝑟) ∈ 𝑋) → (𝑉 “ {(1st𝑟)}) ∈ ((nei‘𝐽)‘{(1st𝑟)}))
2921, 23, 27, 28syl3anc 1372 . . . . . . . . 9 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟𝑀) → (𝑉 “ {(1st𝑟)}) ∈ ((nei‘𝐽)‘{(1st𝑟)}))
30 xp2nd 8005 . . . . . . . . . . 11 (𝑟 ∈ (𝑋 × 𝑋) → (2nd𝑟) ∈ 𝑋)
3125, 30syl 17 . . . . . . . . . 10 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟𝑀) → (2nd𝑟) ∈ 𝑋)
321utopsnnei 23746 . . . . . . . . . 10 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈 ∧ (2nd𝑟) ∈ 𝑋) → (𝑉 “ {(2nd𝑟)}) ∈ ((nei‘𝐽)‘{(2nd𝑟)}))
3321, 23, 31, 32syl3anc 1372 . . . . . . . . 9 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟𝑀) → (𝑉 “ {(2nd𝑟)}) ∈ ((nei‘𝐽)‘{(2nd𝑟)}))
34 eqid 2733 . . . . . . . . . 10 𝐽 = 𝐽
3534, 34neitx 23103 . . . . . . . . 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 838 . . . . . . . 8 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟𝑀) → ((𝑉 “ {(1st𝑟)}) × (𝑉 “ {(2nd𝑟)})) ∈ ((nei‘(𝐽 ×t 𝐽))‘({(1st𝑟)} × {(2nd𝑟)})))
37 fvex 6902 . . . . . . . . . 10 (1st𝑟) ∈ V
38 fvex 6902 . . . . . . . . . 10 (2nd𝑟) ∈ V
3937, 38xpsn 7136 . . . . . . . . 9 ({(1st𝑟)} × {(2nd𝑟)}) = {⟨(1st𝑟), (2nd𝑟)⟩}
4039fveq2i 6892 . . . . . . . 8 ((nei‘(𝐽 ×t 𝐽))‘({(1st𝑟)} × {(2nd𝑟)})) = ((nei‘(𝐽 ×t 𝐽))‘{⟨(1st𝑟), (2nd𝑟)⟩})
4136, 40eleqtrdi 2844 . . . . . . 7 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟𝑀) → ((𝑉 “ {(1st𝑟)}) × (𝑉 “ {(2nd𝑟)})) ∈ ((nei‘(𝐽 ×t 𝐽))‘{⟨(1st𝑟), (2nd𝑟)⟩}))
4224adantr 482 . . . . . . . . . . 11 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟𝑀) → 𝑀 ⊆ (𝑋 × 𝑋))
43 xpss 5692 . . . . . . . . . . . . 13 (𝑋 × 𝑋) ⊆ (V × V)
44 sstr 3990 . . . . . . . . . . . . 13 ((𝑀 ⊆ (𝑋 × 𝑋) ∧ (𝑋 × 𝑋) ⊆ (V × V)) → 𝑀 ⊆ (V × V))
4543, 44mpan2 690 . . . . . . . . . . . 12 (𝑀 ⊆ (𝑋 × 𝑋) → 𝑀 ⊆ (V × V))
46 df-rel 5683 . . . . . . . . . . . 12 (Rel 𝑀𝑀 ⊆ (V × V))
4745, 46sylibr 233 . . . . . . . . . . 11 (𝑀 ⊆ (𝑋 × 𝑋) → Rel 𝑀)
4842, 47syl 17 . . . . . . . . . 10 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟𝑀) → Rel 𝑀)
49 1st2nd 8022 . . . . . . . . . 10 ((Rel 𝑀𝑟𝑀) → 𝑟 = ⟨(1st𝑟), (2nd𝑟)⟩)
5048, 49sylancom 589 . . . . . . . . 9 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟𝑀) → 𝑟 = ⟨(1st𝑟), (2nd𝑟)⟩)
5150sneqd 4640 . . . . . . . 8 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟𝑀) → {𝑟} = {⟨(1st𝑟), (2nd𝑟)⟩})
5251fveq2d 6893 . . . . . . 7 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟𝑀) → ((nei‘(𝐽 ×t 𝐽))‘{𝑟}) = ((nei‘(𝐽 ×t 𝐽))‘{⟨(1st𝑟), (2nd𝑟)⟩}))
5341, 52eleqtrrd 2837 . . . . . 6 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟𝑀) → ((𝑉 “ {(1st𝑟)}) × (𝑉 “ {(2nd𝑟)})) ∈ ((nei‘(𝐽 ×t 𝐽))‘{𝑟}))
54 relxp 5694 . . . . . . . . . . 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 8022 . . . . . . . . . 10 ((Rel ((𝑉 “ {(1st𝑟)}) × (𝑉 “ {(2nd𝑟)})) ∧ 𝑧 ∈ ((𝑉 “ {(1st𝑟)}) × (𝑉 “ {(2nd𝑟)}))) → 𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩)
5755, 56sylancom 589 . . . . . . . . 9 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟𝑀) ∧ 𝑧 ∈ ((𝑉 “ {(1st𝑟)}) × (𝑉 “ {(2nd𝑟)}))) → 𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩)
58 simpll2 1214 . . . . . . . . . . . . 13 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟𝑀) ∧ 𝑧 ∈ ((𝑉 “ {(1st𝑟)}) × (𝑉 “ {(2nd𝑟)}))) → (𝑉𝑈𝑉 = 𝑉))
5958simprd 497 . . . . . . . . . . . 12 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟𝑀) ∧ 𝑧 ∈ ((𝑉 “ {(1st𝑟)}) × (𝑉 “ {(2nd𝑟)}))) → 𝑉 = 𝑉)
60 simpll1 1213 . . . . . . . . . . . . . 14 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟𝑀) ∧ 𝑧 ∈ ((𝑉 “ {(1st𝑟)}) × (𝑉 “ {(2nd𝑟)}))) → 𝑈 ∈ (UnifOn‘𝑋))
6158simpld 496 . . . . . . . . . . . . . 14 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟𝑀) ∧ 𝑧 ∈ ((𝑉 “ {(1st𝑟)}) × (𝑉 “ {(2nd𝑟)}))) → 𝑉𝑈)
62 ustrel 23708 . . . . . . . . . . . . . 14 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) → Rel 𝑉)
6360, 61, 62syl2anc 585 . . . . . . . . . . . . 13 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟𝑀) ∧ 𝑧 ∈ ((𝑉 “ {(1st𝑟)}) × (𝑉 “ {(2nd𝑟)}))) → Rel 𝑉)
64 xp1st 8004 . . . . . . . . . . . . . 14 (𝑧 ∈ ((𝑉 “ {(1st𝑟)}) × (𝑉 “ {(2nd𝑟)})) → (1st𝑧) ∈ (𝑉 “ {(1st𝑟)}))
6564adantl 483 . . . . . . . . . . . . 13 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟𝑀) ∧ 𝑧 ∈ ((𝑉 “ {(1st𝑟)}) × (𝑉 “ {(2nd𝑟)}))) → (1st𝑧) ∈ (𝑉 “ {(1st𝑟)}))
66 elrelimasn 6082 . . . . . . . . . . . . . 14 (Rel 𝑉 → ((1st𝑧) ∈ (𝑉 “ {(1st𝑟)}) ↔ (1st𝑟)𝑉(1st𝑧)))
6766biimpa 478 . . . . . . . . . . . . 13 ((Rel 𝑉 ∧ (1st𝑧) ∈ (𝑉 “ {(1st𝑟)})) → (1st𝑟)𝑉(1st𝑧))
6863, 65, 67syl2anc 585 . . . . . . . . . . . 12 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟𝑀) ∧ 𝑧 ∈ ((𝑉 “ {(1st𝑟)}) × (𝑉 “ {(2nd𝑟)}))) → (1st𝑟)𝑉(1st𝑧))
69 fvex 6902 . . . . . . . . . . . . . . 15 (1st𝑧) ∈ V
7037, 69brcnv 5881 . . . . . . . . . . . . . 14 ((1st𝑟)𝑉(1st𝑧) ↔ (1st𝑧)𝑉(1st𝑟))
71 breq 5150 . . . . . . . . . . . . . 14 (𝑉 = 𝑉 → ((1st𝑟)𝑉(1st𝑧) ↔ (1st𝑟)𝑉(1st𝑧)))
7270, 71bitr3id 285 . . . . . . . . . . . . 13 (𝑉 = 𝑉 → ((1st𝑧)𝑉(1st𝑟) ↔ (1st𝑟)𝑉(1st𝑧)))
7372biimpar 479 . . . . . . . . . . . 12 ((𝑉 = 𝑉 ∧ (1st𝑟)𝑉(1st𝑧)) → (1st𝑧)𝑉(1st𝑟))
7459, 68, 73syl2anc 585 . . . . . . . . . . 11 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟𝑀) ∧ 𝑧 ∈ ((𝑉 “ {(1st𝑟)}) × (𝑉 “ {(2nd𝑟)}))) → (1st𝑧)𝑉(1st𝑟))
75 simpll3 1215 . . . . . . . . . . . 12 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟𝑀) ∧ 𝑧 ∈ ((𝑉 “ {(1st𝑟)}) × (𝑉 “ {(2nd𝑟)}))) → 𝑀 ⊆ (𝑋 × 𝑋))
76 simplr 768 . . . . . . . . . . . 12 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟𝑀) ∧ 𝑧 ∈ ((𝑉 “ {(1st𝑟)}) × (𝑉 “ {(2nd𝑟)}))) → 𝑟𝑀)
77 1st2ndbr 8025 . . . . . . . . . . . . 13 ((Rel 𝑀𝑟𝑀) → (1st𝑟)𝑀(2nd𝑟))
7847, 77sylan 581 . . . . . . . . . . . 12 ((𝑀 ⊆ (𝑋 × 𝑋) ∧ 𝑟𝑀) → (1st𝑟)𝑀(2nd𝑟))
7975, 76, 78syl2anc 585 . . . . . . . . . . 11 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟𝑀) ∧ 𝑧 ∈ ((𝑉 “ {(1st𝑟)}) × (𝑉 “ {(2nd𝑟)}))) → (1st𝑟)𝑀(2nd𝑟))
80 xp2nd 8005 . . . . . . . . . . . . 13 (𝑧 ∈ ((𝑉 “ {(1st𝑟)}) × (𝑉 “ {(2nd𝑟)})) → (2nd𝑧) ∈ (𝑉 “ {(2nd𝑟)}))
8180adantl 483 . . . . . . . . . . . 12 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟𝑀) ∧ 𝑧 ∈ ((𝑉 “ {(1st𝑟)}) × (𝑉 “ {(2nd𝑟)}))) → (2nd𝑧) ∈ (𝑉 “ {(2nd𝑟)}))
82 elrelimasn 6082 . . . . . . . . . . . . 13 (Rel 𝑉 → ((2nd𝑧) ∈ (𝑉 “ {(2nd𝑟)}) ↔ (2nd𝑟)𝑉(2nd𝑧)))
8382biimpa 478 . . . . . . . . . . . 12 ((Rel 𝑉 ∧ (2nd𝑧) ∈ (𝑉 “ {(2nd𝑟)})) → (2nd𝑟)𝑉(2nd𝑧))
8463, 81, 83syl2anc 585 . . . . . . . . . . 11 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟𝑀) ∧ 𝑧 ∈ ((𝑉 “ {(1st𝑟)}) × (𝑉 “ {(2nd𝑟)}))) → (2nd𝑟)𝑉(2nd𝑧))
8569, 38, 373pm3.2i 1340 . . . . . . . . . . . . 13 ((1st𝑧) ∈ V ∧ (2nd𝑟) ∈ V ∧ (1st𝑟) ∈ V)
86 brcogw 5867 . . . . . . . . . . . . 13 ((((1st𝑧) ∈ V ∧ (2nd𝑟) ∈ V ∧ (1st𝑟) ∈ V) ∧ ((1st𝑧)𝑉(1st𝑟) ∧ (1st𝑟)𝑀(2nd𝑟))) → (1st𝑧)(𝑀𝑉)(2nd𝑟))
8785, 86mpan 689 . . . . . . . . . . . 12 (((1st𝑧)𝑉(1st𝑟) ∧ (1st𝑟)𝑀(2nd𝑟)) → (1st𝑧)(𝑀𝑉)(2nd𝑟))
88 fvex 6902 . . . . . . . . . . . . . 14 (2nd𝑧) ∈ V
8969, 88, 383pm3.2i 1340 . . . . . . . . . . . . 13 ((1st𝑧) ∈ V ∧ (2nd𝑧) ∈ V ∧ (2nd𝑟) ∈ V)
90 brcogw 5867 . . . . . . . . . . . . 13 ((((1st𝑧) ∈ V ∧ (2nd𝑧) ∈ V ∧ (2nd𝑟) ∈ V) ∧ ((1st𝑧)(𝑀𝑉)(2nd𝑟) ∧ (2nd𝑟)𝑉(2nd𝑧))) → (1st𝑧)(𝑉 ∘ (𝑀𝑉))(2nd𝑧))
9189, 90mpan 689 . . . . . . . . . . . 12 (((1st𝑧)(𝑀𝑉)(2nd𝑟) ∧ (2nd𝑟)𝑉(2nd𝑧)) → (1st𝑧)(𝑉 ∘ (𝑀𝑉))(2nd𝑧))
9287, 91sylan 581 . . . . . . . . . . 11 ((((1st𝑧)𝑉(1st𝑟) ∧ (1st𝑟)𝑀(2nd𝑟)) ∧ (2nd𝑟)𝑉(2nd𝑧)) → (1st𝑧)(𝑉 ∘ (𝑀𝑉))(2nd𝑧))
9374, 79, 84, 92syl21anc 837 . . . . . . . . . 10 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟𝑀) ∧ 𝑧 ∈ ((𝑉 “ {(1st𝑟)}) × (𝑉 “ {(2nd𝑟)}))) → (1st𝑧)(𝑉 ∘ (𝑀𝑉))(2nd𝑧))
94 df-br 5149 . . . . . . . . . 10 ((1st𝑧)(𝑉 ∘ (𝑀𝑉))(2nd𝑧) ↔ ⟨(1st𝑧), (2nd𝑧)⟩ ∈ (𝑉 ∘ (𝑀𝑉)))
9593, 94sylib 217 . . . . . . . . 9 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟𝑀) ∧ 𝑧 ∈ ((𝑉 “ {(1st𝑟)}) × (𝑉 “ {(2nd𝑟)}))) → ⟨(1st𝑧), (2nd𝑧)⟩ ∈ (𝑉 ∘ (𝑀𝑉)))
9657, 95eqeltrd 2834 . . . . . . . 8 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟𝑀) ∧ 𝑧 ∈ ((𝑉 “ {(1st𝑟)}) × (𝑉 “ {(2nd𝑟)}))) → 𝑧 ∈ (𝑉 ∘ (𝑀𝑉)))
9796ex 414 . . . . . . 7 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟𝑀) → (𝑧 ∈ ((𝑉 “ {(1st𝑟)}) × (𝑉 “ {(2nd𝑟)})) → 𝑧 ∈ (𝑉 ∘ (𝑀𝑉))))
9897ssrdv 3988 . . . . . 6 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟𝑀) → ((𝑉 “ {(1st𝑟)}) × (𝑉 “ {(2nd𝑟)})) ⊆ (𝑉 ∘ (𝑀𝑉)))
99 simp1 1137 . . . . . . . . . . 11 ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) → 𝑈 ∈ (UnifOn‘𝑋))
100 simp2l 1200 . . . . . . . . . . 11 ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) → 𝑉𝑈)
101 ustssxp 23701 . . . . . . . . . . 11 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) → 𝑉 ⊆ (𝑋 × 𝑋))
10299, 100, 101syl2anc 585 . . . . . . . . . 10 ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) → 𝑉 ⊆ (𝑋 × 𝑋))
103 coss1 5854 . . . . . . . . . 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 5854 . . . . . . . . . . . 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 5855 . . . . . . . . . . . . 13 (𝑉 ⊆ (𝑋 × 𝑋) → ((𝑋 × 𝑋) ∘ 𝑉) ⊆ ((𝑋 × 𝑋) ∘ (𝑋 × 𝑋)))
108 xpcoid 6287 . . . . . . . . . . . . 13 ((𝑋 × 𝑋) ∘ (𝑋 × 𝑋)) = (𝑋 × 𝑋)
109107, 108sseqtrdi 4032 . . . . . . . . . . . 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 3992 . . . . . . . . . 10 ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) → (𝑀𝑉) ⊆ (𝑋 × 𝑋))
112 coss2 5855 . . . . . . . . . . 11 ((𝑀𝑉) ⊆ (𝑋 × 𝑋) → ((𝑋 × 𝑋) ∘ (𝑀𝑉)) ⊆ ((𝑋 × 𝑋) ∘ (𝑋 × 𝑋)))
113112, 108sseqtrdi 4032 . . . . . . . . . 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 3992 . . . . . . . 8 ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) → (𝑉 ∘ (𝑀𝑉)) ⊆ (𝑋 × 𝑋))
116 utopbas 23732 . . . . . . . . . . . 12 (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = (unifTop‘𝑈))
1171unieqi 4921 . . . . . . . . . . . 12 𝐽 = (unifTop‘𝑈)
118116, 117eqtr4di 2791 . . . . . . . . . . 11 (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = 𝐽)
119118sqxpeqd 5708 . . . . . . . . . 10 (𝑈 ∈ (UnifOn‘𝑋) → (𝑋 × 𝑋) = ( 𝐽 × 𝐽))
12034, 34txuni 23088 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ 𝐽 ∈ Top) → ( 𝐽 × 𝐽) = (𝐽 ×t 𝐽))
1213, 3, 120syl2anc 585 . . . . . . . . . 10 (𝑈 ∈ (UnifOn‘𝑋) → ( 𝐽 × 𝐽) = (𝐽 ×t 𝐽))
122119, 121eqtrd 2773 . . . . . . . . 9 (𝑈 ∈ (UnifOn‘𝑋) → (𝑋 × 𝑋) = (𝐽 ×t 𝐽))
1231223ad2ant1 1134 . . . . . . . 8 ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) → (𝑋 × 𝑋) = (𝐽 ×t 𝐽))
124115, 123sseqtrd 4022 . . . . . . 7 ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) → (𝑉 ∘ (𝑀𝑉)) ⊆ (𝐽 ×t 𝐽))
125124adantr 482 . . . . . 6 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟𝑀) → (𝑉 ∘ (𝑀𝑉)) ⊆ (𝐽 ×t 𝐽))
126 eqid 2733 . . . . . . 7 (𝐽 ×t 𝐽) = (𝐽 ×t 𝐽)
127126ssnei2 22612 . . . . . 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 838 . . . . 5 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟𝑀) → (𝑉 ∘ (𝑀𝑉)) ∈ ((nei‘(𝐽 ×t 𝐽))‘{𝑟}))
129128ralrimiva 3147 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) → ∀𝑟𝑀 (𝑉 ∘ (𝑀𝑉)) ∈ ((nei‘(𝐽 ×t 𝐽))‘{𝑟}))
130129adantr 482 . . 3 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑀 ≠ ∅) → ∀𝑟𝑀 (𝑉 ∘ (𝑀𝑉)) ∈ ((nei‘(𝐽 ×t 𝐽))‘{𝑟}))
1316adantr 482 . . . 4 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑀 ≠ ∅) → (𝐽 ×t 𝐽) ∈ Top)
13224, 123sseqtrd 4022 . . . . 5 ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) → 𝑀 (𝐽 ×t 𝐽))
133132adantr 482 . . . 4 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑀 ≠ ∅) → 𝑀 (𝐽 ×t 𝐽))
134 simpr 486 . . . 4 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑀 ≠ ∅) → 𝑀 ≠ ∅)
135126neips 22609 . . . 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 1372 . . 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 257 . 2 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑀 ≠ ∅) → (𝑉 ∘ (𝑀𝑉)) ∈ ((nei‘(𝐽 ×t 𝐽))‘𝑀))
13819, 137pm2.61dane 3030 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 205  wa 397  w3a 1088   = wceq 1542  wcel 2107  wne 2941  wral 3062  Vcvv 3475  wss 3948  c0 4322  {csn 4628  cop 4634   cuni 4908   class class class wbr 5148   × cxp 5674  ccnv 5675  cima 5679  ccom 5680  Rel wrel 5681  cfv 6541  (class class class)co 7406  1st c1st 7970  2nd c2nd 7971  Topctop 22387  neicnei 22593   ×t ctx 23056  UnifOncust 23696  unifTopcutop 23727
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6493  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-ov 7409  df-oprab 7410  df-mpo 7411  df-om 7853  df-1st 7972  df-2nd 7973  df-1o 8463  df-er 8700  df-en 8937  df-fin 8940  df-fi 9403  df-topgen 17386  df-top 22388  df-topon 22405  df-bases 22441  df-nei 22594  df-tx 23058  df-ust 23697  df-utop 23728
This theorem is referenced by: (None)
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