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Theorem utop2nei 22860
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 22844 . . . . . . . 8 (𝑈 ∈ (UnifOn‘𝑋) → (unifTop‘𝑈) ∈ Top)
31, 2eqeltrid 2897 . . . . . . 7 (𝑈 ∈ (UnifOn‘𝑋) → 𝐽 ∈ Top)
4 txtop 22178 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝐽 ∈ Top) → (𝐽 ×t 𝐽) ∈ Top)
53, 3, 4syl2anc 587 . . . . . 6 (𝑈 ∈ (UnifOn‘𝑋) → (𝐽 ×t 𝐽) ∈ Top)
653ad2ant1 1130 . . . . 5 ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) → (𝐽 ×t 𝐽) ∈ Top)
76adantr 484 . . . 4 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑀 = ∅) → (𝐽 ×t 𝐽) ∈ Top)
8 0nei 21737 . . . 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 5696 . . . . . . 7 (𝑀 = ∅ → (𝑀𝑉) = (∅ ∘ 𝑉))
11 co01 6085 . . . . . . 7 (∅ ∘ 𝑉) = ∅
1210, 11eqtrdi 2852 . . . . . 6 (𝑀 = ∅ → (𝑀𝑉) = ∅)
1312coeq2d 5701 . . . . 5 (𝑀 = ∅ → (𝑉 ∘ (𝑀𝑉)) = (𝑉 ∘ ∅))
14 co02 6084 . . . . 5 (𝑉 ∘ ∅) = ∅
1513, 14eqtrdi 2852 . . . 4 (𝑀 = ∅ → (𝑉 ∘ (𝑀𝑉)) = ∅)
1615adantl 485 . . 3 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑀 = ∅) → (𝑉 ∘ (𝑀𝑉)) = ∅)
17 simpr 488 . . . 4 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑀 = ∅) → 𝑀 = ∅)
1817fveq2d 6653 . . 3 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑀 = ∅) → ((nei‘(𝐽 ×t 𝐽))‘𝑀) = ((nei‘(𝐽 ×t 𝐽))‘∅))
199, 16, 183eltr4d 2908 . 2 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑀 = ∅) → (𝑉 ∘ (𝑀𝑉)) ∈ ((nei‘(𝐽 ×t 𝐽))‘𝑀))
206adantr 484 . . . . . 6 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟𝑀) → (𝐽 ×t 𝐽) ∈ Top)
21 simpl1 1188 . . . . . . . . . 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 1223 . . . . . . . . . 10 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟𝑀) → 𝑉𝑈)
24 simp3 1135 . . . . . . . . . . . 12 ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) → 𝑀 ⊆ (𝑋 × 𝑋))
2524sselda 3918 . . . . . . . . . . 11 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟𝑀) → 𝑟 ∈ (𝑋 × 𝑋))
26 xp1st 7707 . . . . . . . . . . 11 (𝑟 ∈ (𝑋 × 𝑋) → (1st𝑟) ∈ 𝑋)
2725, 26syl 17 . . . . . . . . . 10 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟𝑀) → (1st𝑟) ∈ 𝑋)
281utopsnnei 22859 . . . . . . . . . 10 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈 ∧ (1st𝑟) ∈ 𝑋) → (𝑉 “ {(1st𝑟)}) ∈ ((nei‘𝐽)‘{(1st𝑟)}))
2921, 23, 27, 28syl3anc 1368 . . . . . . . . 9 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟𝑀) → (𝑉 “ {(1st𝑟)}) ∈ ((nei‘𝐽)‘{(1st𝑟)}))
30 xp2nd 7708 . . . . . . . . . . 11 (𝑟 ∈ (𝑋 × 𝑋) → (2nd𝑟) ∈ 𝑋)
3125, 30syl 17 . . . . . . . . . 10 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟𝑀) → (2nd𝑟) ∈ 𝑋)
321utopsnnei 22859 . . . . . . . . . 10 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈 ∧ (2nd𝑟) ∈ 𝑋) → (𝑉 “ {(2nd𝑟)}) ∈ ((nei‘𝐽)‘{(2nd𝑟)}))
3321, 23, 31, 32syl3anc 1368 . . . . . . . . 9 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟𝑀) → (𝑉 “ {(2nd𝑟)}) ∈ ((nei‘𝐽)‘{(2nd𝑟)}))
34 eqid 2801 . . . . . . . . . 10 𝐽 = 𝐽
3534, 34neitx 22216 . . . . . . . . 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 837 . . . . . . . 8 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟𝑀) → ((𝑉 “ {(1st𝑟)}) × (𝑉 “ {(2nd𝑟)})) ∈ ((nei‘(𝐽 ×t 𝐽))‘({(1st𝑟)} × {(2nd𝑟)})))
37 fvex 6662 . . . . . . . . . 10 (1st𝑟) ∈ V
38 fvex 6662 . . . . . . . . . 10 (2nd𝑟) ∈ V
3937, 38xpsn 6884 . . . . . . . . 9 ({(1st𝑟)} × {(2nd𝑟)}) = {⟨(1st𝑟), (2nd𝑟)⟩}
4039fveq2i 6652 . . . . . . . 8 ((nei‘(𝐽 ×t 𝐽))‘({(1st𝑟)} × {(2nd𝑟)})) = ((nei‘(𝐽 ×t 𝐽))‘{⟨(1st𝑟), (2nd𝑟)⟩})
4136, 40eleqtrdi 2903 . . . . . . 7 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟𝑀) → ((𝑉 “ {(1st𝑟)}) × (𝑉 “ {(2nd𝑟)})) ∈ ((nei‘(𝐽 ×t 𝐽))‘{⟨(1st𝑟), (2nd𝑟)⟩}))
4224adantr 484 . . . . . . . . . . 11 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟𝑀) → 𝑀 ⊆ (𝑋 × 𝑋))
43 xpss 5539 . . . . . . . . . . . . 13 (𝑋 × 𝑋) ⊆ (V × V)
44 sstr 3926 . . . . . . . . . . . . 13 ((𝑀 ⊆ (𝑋 × 𝑋) ∧ (𝑋 × 𝑋) ⊆ (V × V)) → 𝑀 ⊆ (V × V))
4543, 44mpan2 690 . . . . . . . . . . . 12 (𝑀 ⊆ (𝑋 × 𝑋) → 𝑀 ⊆ (V × V))
46 df-rel 5530 . . . . . . . . . . . 12 (Rel 𝑀𝑀 ⊆ (V × V))
4745, 46sylibr 237 . . . . . . . . . . 11 (𝑀 ⊆ (𝑋 × 𝑋) → Rel 𝑀)
4842, 47syl 17 . . . . . . . . . 10 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟𝑀) → Rel 𝑀)
49 1st2nd 7724 . . . . . . . . . 10 ((Rel 𝑀𝑟𝑀) → 𝑟 = ⟨(1st𝑟), (2nd𝑟)⟩)
5048, 49sylancom 591 . . . . . . . . 9 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟𝑀) → 𝑟 = ⟨(1st𝑟), (2nd𝑟)⟩)
5150sneqd 4540 . . . . . . . 8 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟𝑀) → {𝑟} = {⟨(1st𝑟), (2nd𝑟)⟩})
5251fveq2d 6653 . . . . . . 7 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟𝑀) → ((nei‘(𝐽 ×t 𝐽))‘{𝑟}) = ((nei‘(𝐽 ×t 𝐽))‘{⟨(1st𝑟), (2nd𝑟)⟩}))
5341, 52eleqtrrd 2896 . . . . . 6 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟𝑀) → ((𝑉 “ {(1st𝑟)}) × (𝑉 “ {(2nd𝑟)})) ∈ ((nei‘(𝐽 ×t 𝐽))‘{𝑟}))
54 relxp 5541 . . . . . . . . . . 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 7724 . . . . . . . . . 10 ((Rel ((𝑉 “ {(1st𝑟)}) × (𝑉 “ {(2nd𝑟)})) ∧ 𝑧 ∈ ((𝑉 “ {(1st𝑟)}) × (𝑉 “ {(2nd𝑟)}))) → 𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩)
5755, 56sylancom 591 . . . . . . . . 9 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟𝑀) ∧ 𝑧 ∈ ((𝑉 “ {(1st𝑟)}) × (𝑉 “ {(2nd𝑟)}))) → 𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩)
58 simpll2 1210 . . . . . . . . . . . . 13 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟𝑀) ∧ 𝑧 ∈ ((𝑉 “ {(1st𝑟)}) × (𝑉 “ {(2nd𝑟)}))) → (𝑉𝑈𝑉 = 𝑉))
5958simprd 499 . . . . . . . . . . . 12 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟𝑀) ∧ 𝑧 ∈ ((𝑉 “ {(1st𝑟)}) × (𝑉 “ {(2nd𝑟)}))) → 𝑉 = 𝑉)
60 simpll1 1209 . . . . . . . . . . . . . 14 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟𝑀) ∧ 𝑧 ∈ ((𝑉 “ {(1st𝑟)}) × (𝑉 “ {(2nd𝑟)}))) → 𝑈 ∈ (UnifOn‘𝑋))
6158simpld 498 . . . . . . . . . . . . . 14 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟𝑀) ∧ 𝑧 ∈ ((𝑉 “ {(1st𝑟)}) × (𝑉 “ {(2nd𝑟)}))) → 𝑉𝑈)
62 ustrel 22821 . . . . . . . . . . . . . 14 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) → Rel 𝑉)
6360, 61, 62syl2anc 587 . . . . . . . . . . . . 13 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟𝑀) ∧ 𝑧 ∈ ((𝑉 “ {(1st𝑟)}) × (𝑉 “ {(2nd𝑟)}))) → Rel 𝑉)
64 xp1st 7707 . . . . . . . . . . . . . 14 (𝑧 ∈ ((𝑉 “ {(1st𝑟)}) × (𝑉 “ {(2nd𝑟)})) → (1st𝑧) ∈ (𝑉 “ {(1st𝑟)}))
6564adantl 485 . . . . . . . . . . . . 13 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟𝑀) ∧ 𝑧 ∈ ((𝑉 “ {(1st𝑟)}) × (𝑉 “ {(2nd𝑟)}))) → (1st𝑧) ∈ (𝑉 “ {(1st𝑟)}))
66 elrelimasn 5924 . . . . . . . . . . . . . 14 (Rel 𝑉 → ((1st𝑧) ∈ (𝑉 “ {(1st𝑟)}) ↔ (1st𝑟)𝑉(1st𝑧)))
6766biimpa 480 . . . . . . . . . . . . 13 ((Rel 𝑉 ∧ (1st𝑧) ∈ (𝑉 “ {(1st𝑟)})) → (1st𝑟)𝑉(1st𝑧))
6863, 65, 67syl2anc 587 . . . . . . . . . . . 12 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟𝑀) ∧ 𝑧 ∈ ((𝑉 “ {(1st𝑟)}) × (𝑉 “ {(2nd𝑟)}))) → (1st𝑟)𝑉(1st𝑧))
69 fvex 6662 . . . . . . . . . . . . . . 15 (1st𝑧) ∈ V
7037, 69brcnv 5721 . . . . . . . . . . . . . 14 ((1st𝑟)𝑉(1st𝑧) ↔ (1st𝑧)𝑉(1st𝑟))
71 breq 5035 . . . . . . . . . . . . . 14 (𝑉 = 𝑉 → ((1st𝑟)𝑉(1st𝑧) ↔ (1st𝑟)𝑉(1st𝑧)))
7270, 71bitr3id 288 . . . . . . . . . . . . 13 (𝑉 = 𝑉 → ((1st𝑧)𝑉(1st𝑟) ↔ (1st𝑟)𝑉(1st𝑧)))
7372biimpar 481 . . . . . . . . . . . 12 ((𝑉 = 𝑉 ∧ (1st𝑟)𝑉(1st𝑧)) → (1st𝑧)𝑉(1st𝑟))
7459, 68, 73syl2anc 587 . . . . . . . . . . 11 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟𝑀) ∧ 𝑧 ∈ ((𝑉 “ {(1st𝑟)}) × (𝑉 “ {(2nd𝑟)}))) → (1st𝑧)𝑉(1st𝑟))
75 simpll3 1211 . . . . . . . . . . . 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 7727 . . . . . . . . . . . . 13 ((Rel 𝑀𝑟𝑀) → (1st𝑟)𝑀(2nd𝑟))
7847, 77sylan 583 . . . . . . . . . . . 12 ((𝑀 ⊆ (𝑋 × 𝑋) ∧ 𝑟𝑀) → (1st𝑟)𝑀(2nd𝑟))
7975, 76, 78syl2anc 587 . . . . . . . . . . 11 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟𝑀) ∧ 𝑧 ∈ ((𝑉 “ {(1st𝑟)}) × (𝑉 “ {(2nd𝑟)}))) → (1st𝑟)𝑀(2nd𝑟))
80 xp2nd 7708 . . . . . . . . . . . . 13 (𝑧 ∈ ((𝑉 “ {(1st𝑟)}) × (𝑉 “ {(2nd𝑟)})) → (2nd𝑧) ∈ (𝑉 “ {(2nd𝑟)}))
8180adantl 485 . . . . . . . . . . . 12 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟𝑀) ∧ 𝑧 ∈ ((𝑉 “ {(1st𝑟)}) × (𝑉 “ {(2nd𝑟)}))) → (2nd𝑧) ∈ (𝑉 “ {(2nd𝑟)}))
82 elrelimasn 5924 . . . . . . . . . . . . 13 (Rel 𝑉 → ((2nd𝑧) ∈ (𝑉 “ {(2nd𝑟)}) ↔ (2nd𝑟)𝑉(2nd𝑧)))
8382biimpa 480 . . . . . . . . . . . 12 ((Rel 𝑉 ∧ (2nd𝑧) ∈ (𝑉 “ {(2nd𝑟)})) → (2nd𝑟)𝑉(2nd𝑧))
8463, 81, 83syl2anc 587 . . . . . . . . . . 11 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟𝑀) ∧ 𝑧 ∈ ((𝑉 “ {(1st𝑟)}) × (𝑉 “ {(2nd𝑟)}))) → (2nd𝑟)𝑉(2nd𝑧))
8569, 38, 373pm3.2i 1336 . . . . . . . . . . . . 13 ((1st𝑧) ∈ V ∧ (2nd𝑟) ∈ V ∧ (1st𝑟) ∈ V)
86 brcogw 5707 . . . . . . . . . . . . 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 6662 . . . . . . . . . . . . . 14 (2nd𝑧) ∈ V
8969, 88, 383pm3.2i 1336 . . . . . . . . . . . . 13 ((1st𝑧) ∈ V ∧ (2nd𝑧) ∈ V ∧ (2nd𝑟) ∈ V)
90 brcogw 5707 . . . . . . . . . . . . 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 583 . . . . . . . . . . 11 ((((1st𝑧)𝑉(1st𝑟) ∧ (1st𝑟)𝑀(2nd𝑟)) ∧ (2nd𝑟)𝑉(2nd𝑧)) → (1st𝑧)(𝑉 ∘ (𝑀𝑉))(2nd𝑧))
9374, 79, 84, 92syl21anc 836 . . . . . . . . . 10 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟𝑀) ∧ 𝑧 ∈ ((𝑉 “ {(1st𝑟)}) × (𝑉 “ {(2nd𝑟)}))) → (1st𝑧)(𝑉 ∘ (𝑀𝑉))(2nd𝑧))
94 df-br 5034 . . . . . . . . . 10 ((1st𝑧)(𝑉 ∘ (𝑀𝑉))(2nd𝑧) ↔ ⟨(1st𝑧), (2nd𝑧)⟩ ∈ (𝑉 ∘ (𝑀𝑉)))
9593, 94sylib 221 . . . . . . . . 9 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟𝑀) ∧ 𝑧 ∈ ((𝑉 “ {(1st𝑟)}) × (𝑉 “ {(2nd𝑟)}))) → ⟨(1st𝑧), (2nd𝑧)⟩ ∈ (𝑉 ∘ (𝑀𝑉)))
9657, 95eqeltrd 2893 . . . . . . . 8 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟𝑀) ∧ 𝑧 ∈ ((𝑉 “ {(1st𝑟)}) × (𝑉 “ {(2nd𝑟)}))) → 𝑧 ∈ (𝑉 ∘ (𝑀𝑉)))
9796ex 416 . . . . . . 7 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟𝑀) → (𝑧 ∈ ((𝑉 “ {(1st𝑟)}) × (𝑉 “ {(2nd𝑟)})) → 𝑧 ∈ (𝑉 ∘ (𝑀𝑉))))
9897ssrdv 3924 . . . . . 6 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟𝑀) → ((𝑉 “ {(1st𝑟)}) × (𝑉 “ {(2nd𝑟)})) ⊆ (𝑉 ∘ (𝑀𝑉)))
99 simp1 1133 . . . . . . . . . . 11 ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) → 𝑈 ∈ (UnifOn‘𝑋))
100 simp2l 1196 . . . . . . . . . . 11 ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) → 𝑉𝑈)
101 ustssxp 22814 . . . . . . . . . . 11 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) → 𝑉 ⊆ (𝑋 × 𝑋))
10299, 100, 101syl2anc 587 . . . . . . . . . 10 ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) → 𝑉 ⊆ (𝑋 × 𝑋))
103 coss1 5694 . . . . . . . . . 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 5694 . . . . . . . . . . . 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 5695 . . . . . . . . . . . . 13 (𝑉 ⊆ (𝑋 × 𝑋) → ((𝑋 × 𝑋) ∘ 𝑉) ⊆ ((𝑋 × 𝑋) ∘ (𝑋 × 𝑋)))
108 xpcoid 6113 . . . . . . . . . . . . 13 ((𝑋 × 𝑋) ∘ (𝑋 × 𝑋)) = (𝑋 × 𝑋)
109107, 108sseqtrdi 3968 . . . . . . . . . . . 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 3928 . . . . . . . . . 10 ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) → (𝑀𝑉) ⊆ (𝑋 × 𝑋))
112 coss2 5695 . . . . . . . . . . 11 ((𝑀𝑉) ⊆ (𝑋 × 𝑋) → ((𝑋 × 𝑋) ∘ (𝑀𝑉)) ⊆ ((𝑋 × 𝑋) ∘ (𝑋 × 𝑋)))
113112, 108sseqtrdi 3968 . . . . . . . . . 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 3928 . . . . . . . 8 ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) → (𝑉 ∘ (𝑀𝑉)) ⊆ (𝑋 × 𝑋))
116 utopbas 22845 . . . . . . . . . . . 12 (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = (unifTop‘𝑈))
1171unieqi 4816 . . . . . . . . . . . 12 𝐽 = (unifTop‘𝑈)
118116, 117eqtr4di 2854 . . . . . . . . . . 11 (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = 𝐽)
119118sqxpeqd 5555 . . . . . . . . . 10 (𝑈 ∈ (UnifOn‘𝑋) → (𝑋 × 𝑋) = ( 𝐽 × 𝐽))
12034, 34txuni 22201 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ 𝐽 ∈ Top) → ( 𝐽 × 𝐽) = (𝐽 ×t 𝐽))
1213, 3, 120syl2anc 587 . . . . . . . . . 10 (𝑈 ∈ (UnifOn‘𝑋) → ( 𝐽 × 𝐽) = (𝐽 ×t 𝐽))
122119, 121eqtrd 2836 . . . . . . . . 9 (𝑈 ∈ (UnifOn‘𝑋) → (𝑋 × 𝑋) = (𝐽 ×t 𝐽))
1231223ad2ant1 1130 . . . . . . . 8 ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) → (𝑋 × 𝑋) = (𝐽 ×t 𝐽))
124115, 123sseqtrd 3958 . . . . . . 7 ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) → (𝑉 ∘ (𝑀𝑉)) ⊆ (𝐽 ×t 𝐽))
125124adantr 484 . . . . . 6 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟𝑀) → (𝑉 ∘ (𝑀𝑉)) ⊆ (𝐽 ×t 𝐽))
126 eqid 2801 . . . . . . 7 (𝐽 ×t 𝐽) = (𝐽 ×t 𝐽)
127126ssnei2 21725 . . . . . 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 837 . . . . 5 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟𝑀) → (𝑉 ∘ (𝑀𝑉)) ∈ ((nei‘(𝐽 ×t 𝐽))‘{𝑟}))
129128ralrimiva 3152 . . . 4 ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) → ∀𝑟𝑀 (𝑉 ∘ (𝑀𝑉)) ∈ ((nei‘(𝐽 ×t 𝐽))‘{𝑟}))
130129adantr 484 . . 3 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑀 ≠ ∅) → ∀𝑟𝑀 (𝑉 ∘ (𝑀𝑉)) ∈ ((nei‘(𝐽 ×t 𝐽))‘{𝑟}))
1316adantr 484 . . . 4 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑀 ≠ ∅) → (𝐽 ×t 𝐽) ∈ Top)
13224, 123sseqtrd 3958 . . . . 5 ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) → 𝑀 (𝐽 ×t 𝐽))
133132adantr 484 . . . 4 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑀 ≠ ∅) → 𝑀 (𝐽 ×t 𝐽))
134 simpr 488 . . . 4 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑀 ≠ ∅) → 𝑀 ≠ ∅)
135126neips 21722 . . . 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 1368 . . 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 260 . 2 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑀 ≠ ∅) → (𝑉 ∘ (𝑀𝑉)) ∈ ((nei‘(𝐽 ×t 𝐽))‘𝑀))
13819, 137pm2.61dane 3077 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 209  wa 399  w3a 1084   = wceq 1538  wcel 2112  wne 2990  wral 3109  Vcvv 3444  wss 3884  c0 4246  {csn 4528  cop 4534   cuni 4803   class class class wbr 5033   × cxp 5521  ccnv 5522  cima 5526  ccom 5527  Rel wrel 5528  cfv 6328  (class class class)co 7139  1st c1st 7673  2nd c2nd 7674  Topctop 21502  neicnei 21706   ×t ctx 22169  UnifOncust 22809  unifTopcutop 22840
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-int 4842  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7142  df-oprab 7143  df-mpo 7144  df-om 7565  df-1st 7675  df-2nd 7676  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-1o 8089  df-oadd 8093  df-er 8276  df-en 8497  df-fin 8500  df-fi 8863  df-topgen 16713  df-top 21503  df-topon 21520  df-bases 21555  df-nei 21707  df-tx 22171  df-ust 22810  df-utop 22841
This theorem is referenced by: (None)
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