MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  utop2nei Structured version   Visualization version   GIF version

Theorem utop2nei 24274
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 24258 . . . . . . . 8 (𝑈 ∈ (UnifOn‘𝑋) → (unifTop‘𝑈) ∈ Top)
31, 2eqeltrid 2842 . . . . . . 7 (𝑈 ∈ (UnifOn‘𝑋) → 𝐽 ∈ Top)
4 txtop 23592 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝐽 ∈ Top) → (𝐽 ×t 𝐽) ∈ Top)
53, 3, 4syl2anc 584 . . . . . 6 (𝑈 ∈ (UnifOn‘𝑋) → (𝐽 ×t 𝐽) ∈ Top)
653ad2ant1 1132 . . . . 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 23151 . . . 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 5870 . . . . . . 7 (𝑀 = ∅ → (𝑀𝑉) = (∅ ∘ 𝑉))
11 co01 6282 . . . . . . 7 (∅ ∘ 𝑉) = ∅
1210, 11eqtrdi 2790 . . . . . 6 (𝑀 = ∅ → (𝑀𝑉) = ∅)
1312coeq2d 5875 . . . . 5 (𝑀 = ∅ → (𝑉 ∘ (𝑀𝑉)) = (𝑉 ∘ ∅))
14 co02 6281 . . . . 5 (𝑉 ∘ ∅) = ∅
1513, 14eqtrdi 2790 . . . 4 (𝑀 = ∅ → (𝑉 ∘ (𝑀𝑉)) = ∅)
1615adantl 481 . . 3 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑀 = ∅) → (𝑉 ∘ (𝑀𝑉)) = ∅)
17 simpr 484 . . . 4 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑀 = ∅) → 𝑀 = ∅)
1817fveq2d 6910 . . 3 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑀 = ∅) → ((nei‘(𝐽 ×t 𝐽))‘𝑀) = ((nei‘(𝐽 ×t 𝐽))‘∅))
199, 16, 183eltr4d 2853 . 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 1190 . . . . . . . . . 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 1225 . . . . . . . . . 10 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟𝑀) → 𝑉𝑈)
24 simp3 1137 . . . . . . . . . . . 12 ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) → 𝑀 ⊆ (𝑋 × 𝑋))
2524sselda 3994 . . . . . . . . . . 11 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟𝑀) → 𝑟 ∈ (𝑋 × 𝑋))
26 xp1st 8044 . . . . . . . . . . 11 (𝑟 ∈ (𝑋 × 𝑋) → (1st𝑟) ∈ 𝑋)
2725, 26syl 17 . . . . . . . . . 10 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟𝑀) → (1st𝑟) ∈ 𝑋)
281utopsnnei 24273 . . . . . . . . . 10 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈 ∧ (1st𝑟) ∈ 𝑋) → (𝑉 “ {(1st𝑟)}) ∈ ((nei‘𝐽)‘{(1st𝑟)}))
2921, 23, 27, 28syl3anc 1370 . . . . . . . . 9 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟𝑀) → (𝑉 “ {(1st𝑟)}) ∈ ((nei‘𝐽)‘{(1st𝑟)}))
30 xp2nd 8045 . . . . . . . . . . 11 (𝑟 ∈ (𝑋 × 𝑋) → (2nd𝑟) ∈ 𝑋)
3125, 30syl 17 . . . . . . . . . 10 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟𝑀) → (2nd𝑟) ∈ 𝑋)
321utopsnnei 24273 . . . . . . . . . 10 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈 ∧ (2nd𝑟) ∈ 𝑋) → (𝑉 “ {(2nd𝑟)}) ∈ ((nei‘𝐽)‘{(2nd𝑟)}))
3321, 23, 31, 32syl3anc 1370 . . . . . . . . 9 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟𝑀) → (𝑉 “ {(2nd𝑟)}) ∈ ((nei‘𝐽)‘{(2nd𝑟)}))
34 eqid 2734 . . . . . . . . . 10 𝐽 = 𝐽
3534, 34neitx 23630 . . . . . . . . 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 839 . . . . . . . 8 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟𝑀) → ((𝑉 “ {(1st𝑟)}) × (𝑉 “ {(2nd𝑟)})) ∈ ((nei‘(𝐽 ×t 𝐽))‘({(1st𝑟)} × {(2nd𝑟)})))
37 fvex 6919 . . . . . . . . . 10 (1st𝑟) ∈ V
38 fvex 6919 . . . . . . . . . 10 (2nd𝑟) ∈ V
3937, 38xpsn 7160 . . . . . . . . 9 ({(1st𝑟)} × {(2nd𝑟)}) = {⟨(1st𝑟), (2nd𝑟)⟩}
4039fveq2i 6909 . . . . . . . 8 ((nei‘(𝐽 ×t 𝐽))‘({(1st𝑟)} × {(2nd𝑟)})) = ((nei‘(𝐽 ×t 𝐽))‘{⟨(1st𝑟), (2nd𝑟)⟩})
4136, 40eleqtrdi 2848 . . . . . . 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 5704 . . . . . . . . . . . . 13 (𝑋 × 𝑋) ⊆ (V × V)
44 sstr 4003 . . . . . . . . . . . . 13 ((𝑀 ⊆ (𝑋 × 𝑋) ∧ (𝑋 × 𝑋) ⊆ (V × V)) → 𝑀 ⊆ (V × V))
4543, 44mpan2 691 . . . . . . . . . . . 12 (𝑀 ⊆ (𝑋 × 𝑋) → 𝑀 ⊆ (V × V))
46 df-rel 5695 . . . . . . . . . . . 12 (Rel 𝑀𝑀 ⊆ (V × V))
4745, 46sylibr 234 . . . . . . . . . . 11 (𝑀 ⊆ (𝑋 × 𝑋) → Rel 𝑀)
4842, 47syl 17 . . . . . . . . . 10 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟𝑀) → Rel 𝑀)
49 1st2nd 8062 . . . . . . . . . 10 ((Rel 𝑀𝑟𝑀) → 𝑟 = ⟨(1st𝑟), (2nd𝑟)⟩)
5048, 49sylancom 588 . . . . . . . . 9 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟𝑀) → 𝑟 = ⟨(1st𝑟), (2nd𝑟)⟩)
5150sneqd 4642 . . . . . . . 8 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟𝑀) → {𝑟} = {⟨(1st𝑟), (2nd𝑟)⟩})
5251fveq2d 6910 . . . . . . 7 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟𝑀) → ((nei‘(𝐽 ×t 𝐽))‘{𝑟}) = ((nei‘(𝐽 ×t 𝐽))‘{⟨(1st𝑟), (2nd𝑟)⟩}))
5341, 52eleqtrrd 2841 . . . . . 6 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟𝑀) → ((𝑉 “ {(1st𝑟)}) × (𝑉 “ {(2nd𝑟)})) ∈ ((nei‘(𝐽 ×t 𝐽))‘{𝑟}))
54 relxp 5706 . . . . . . . . . . 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 8062 . . . . . . . . . 10 ((Rel ((𝑉 “ {(1st𝑟)}) × (𝑉 “ {(2nd𝑟)})) ∧ 𝑧 ∈ ((𝑉 “ {(1st𝑟)}) × (𝑉 “ {(2nd𝑟)}))) → 𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩)
5755, 56sylancom 588 . . . . . . . . 9 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟𝑀) ∧ 𝑧 ∈ ((𝑉 “ {(1st𝑟)}) × (𝑉 “ {(2nd𝑟)}))) → 𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩)
58 simpll2 1212 . . . . . . . . . . . . 13 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟𝑀) ∧ 𝑧 ∈ ((𝑉 “ {(1st𝑟)}) × (𝑉 “ {(2nd𝑟)}))) → (𝑉𝑈𝑉 = 𝑉))
5958simprd 495 . . . . . . . . . . . 12 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟𝑀) ∧ 𝑧 ∈ ((𝑉 “ {(1st𝑟)}) × (𝑉 “ {(2nd𝑟)}))) → 𝑉 = 𝑉)
60 simpll1 1211 . . . . . . . . . . . . . 14 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟𝑀) ∧ 𝑧 ∈ ((𝑉 “ {(1st𝑟)}) × (𝑉 “ {(2nd𝑟)}))) → 𝑈 ∈ (UnifOn‘𝑋))
6158simpld 494 . . . . . . . . . . . . . 14 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟𝑀) ∧ 𝑧 ∈ ((𝑉 “ {(1st𝑟)}) × (𝑉 “ {(2nd𝑟)}))) → 𝑉𝑈)
62 ustrel 24235 . . . . . . . . . . . . . 14 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) → Rel 𝑉)
6360, 61, 62syl2anc 584 . . . . . . . . . . . . 13 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟𝑀) ∧ 𝑧 ∈ ((𝑉 “ {(1st𝑟)}) × (𝑉 “ {(2nd𝑟)}))) → Rel 𝑉)
64 xp1st 8044 . . . . . . . . . . . . . 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 6105 . . . . . . . . . . . . . 14 (Rel 𝑉 → ((1st𝑧) ∈ (𝑉 “ {(1st𝑟)}) ↔ (1st𝑟)𝑉(1st𝑧)))
6766biimpa 476 . . . . . . . . . . . . 13 ((Rel 𝑉 ∧ (1st𝑧) ∈ (𝑉 “ {(1st𝑟)})) → (1st𝑟)𝑉(1st𝑧))
6863, 65, 67syl2anc 584 . . . . . . . . . . . 12 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟𝑀) ∧ 𝑧 ∈ ((𝑉 “ {(1st𝑟)}) × (𝑉 “ {(2nd𝑟)}))) → (1st𝑟)𝑉(1st𝑧))
69 fvex 6919 . . . . . . . . . . . . . . 15 (1st𝑧) ∈ V
7037, 69brcnv 5895 . . . . . . . . . . . . . 14 ((1st𝑟)𝑉(1st𝑧) ↔ (1st𝑧)𝑉(1st𝑟))
71 breq 5149 . . . . . . . . . . . . . 14 (𝑉 = 𝑉 → ((1st𝑟)𝑉(1st𝑧) ↔ (1st𝑟)𝑉(1st𝑧)))
7270, 71bitr3id 285 . . . . . . . . . . . . 13 (𝑉 = 𝑉 → ((1st𝑧)𝑉(1st𝑟) ↔ (1st𝑟)𝑉(1st𝑧)))
7372biimpar 477 . . . . . . . . . . . 12 ((𝑉 = 𝑉 ∧ (1st𝑟)𝑉(1st𝑧)) → (1st𝑧)𝑉(1st𝑟))
7459, 68, 73syl2anc 584 . . . . . . . . . . 11 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟𝑀) ∧ 𝑧 ∈ ((𝑉 “ {(1st𝑟)}) × (𝑉 “ {(2nd𝑟)}))) → (1st𝑧)𝑉(1st𝑟))
75 simpll3 1213 . . . . . . . . . . . 12 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟𝑀) ∧ 𝑧 ∈ ((𝑉 “ {(1st𝑟)}) × (𝑉 “ {(2nd𝑟)}))) → 𝑀 ⊆ (𝑋 × 𝑋))
76 simplr 769 . . . . . . . . . . . 12 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟𝑀) ∧ 𝑧 ∈ ((𝑉 “ {(1st𝑟)}) × (𝑉 “ {(2nd𝑟)}))) → 𝑟𝑀)
77 1st2ndbr 8065 . . . . . . . . . . . . 13 ((Rel 𝑀𝑟𝑀) → (1st𝑟)𝑀(2nd𝑟))
7847, 77sylan 580 . . . . . . . . . . . 12 ((𝑀 ⊆ (𝑋 × 𝑋) ∧ 𝑟𝑀) → (1st𝑟)𝑀(2nd𝑟))
7975, 76, 78syl2anc 584 . . . . . . . . . . 11 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟𝑀) ∧ 𝑧 ∈ ((𝑉 “ {(1st𝑟)}) × (𝑉 “ {(2nd𝑟)}))) → (1st𝑟)𝑀(2nd𝑟))
80 xp2nd 8045 . . . . . . . . . . . . 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 6105 . . . . . . . . . . . . 13 (Rel 𝑉 → ((2nd𝑧) ∈ (𝑉 “ {(2nd𝑟)}) ↔ (2nd𝑟)𝑉(2nd𝑧)))
8382biimpa 476 . . . . . . . . . . . 12 ((Rel 𝑉 ∧ (2nd𝑧) ∈ (𝑉 “ {(2nd𝑟)})) → (2nd𝑟)𝑉(2nd𝑧))
8463, 81, 83syl2anc 584 . . . . . . . . . . 11 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟𝑀) ∧ 𝑧 ∈ ((𝑉 “ {(1st𝑟)}) × (𝑉 “ {(2nd𝑟)}))) → (2nd𝑟)𝑉(2nd𝑧))
8569, 38, 373pm3.2i 1338 . . . . . . . . . . . . 13 ((1st𝑧) ∈ V ∧ (2nd𝑟) ∈ V ∧ (1st𝑟) ∈ V)
86 brcogw 5881 . . . . . . . . . . . . 13 ((((1st𝑧) ∈ V ∧ (2nd𝑟) ∈ V ∧ (1st𝑟) ∈ V) ∧ ((1st𝑧)𝑉(1st𝑟) ∧ (1st𝑟)𝑀(2nd𝑟))) → (1st𝑧)(𝑀𝑉)(2nd𝑟))
8785, 86mpan 690 . . . . . . . . . . . 12 (((1st𝑧)𝑉(1st𝑟) ∧ (1st𝑟)𝑀(2nd𝑟)) → (1st𝑧)(𝑀𝑉)(2nd𝑟))
88 fvex 6919 . . . . . . . . . . . . . 14 (2nd𝑧) ∈ V
8969, 88, 383pm3.2i 1338 . . . . . . . . . . . . 13 ((1st𝑧) ∈ V ∧ (2nd𝑧) ∈ V ∧ (2nd𝑟) ∈ V)
90 brcogw 5881 . . . . . . . . . . . . 13 ((((1st𝑧) ∈ V ∧ (2nd𝑧) ∈ V ∧ (2nd𝑟) ∈ V) ∧ ((1st𝑧)(𝑀𝑉)(2nd𝑟) ∧ (2nd𝑟)𝑉(2nd𝑧))) → (1st𝑧)(𝑉 ∘ (𝑀𝑉))(2nd𝑧))
9189, 90mpan 690 . . . . . . . . . . . 12 (((1st𝑧)(𝑀𝑉)(2nd𝑟) ∧ (2nd𝑟)𝑉(2nd𝑧)) → (1st𝑧)(𝑉 ∘ (𝑀𝑉))(2nd𝑧))
9287, 91sylan 580 . . . . . . . . . . 11 ((((1st𝑧)𝑉(1st𝑟) ∧ (1st𝑟)𝑀(2nd𝑟)) ∧ (2nd𝑟)𝑉(2nd𝑧)) → (1st𝑧)(𝑉 ∘ (𝑀𝑉))(2nd𝑧))
9374, 79, 84, 92syl21anc 838 . . . . . . . . . 10 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟𝑀) ∧ 𝑧 ∈ ((𝑉 “ {(1st𝑟)}) × (𝑉 “ {(2nd𝑟)}))) → (1st𝑧)(𝑉 ∘ (𝑀𝑉))(2nd𝑧))
94 df-br 5148 . . . . . . . . . 10 ((1st𝑧)(𝑉 ∘ (𝑀𝑉))(2nd𝑧) ↔ ⟨(1st𝑧), (2nd𝑧)⟩ ∈ (𝑉 ∘ (𝑀𝑉)))
9593, 94sylib 218 . . . . . . . . 9 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟𝑀) ∧ 𝑧 ∈ ((𝑉 “ {(1st𝑟)}) × (𝑉 “ {(2nd𝑟)}))) → ⟨(1st𝑧), (2nd𝑧)⟩ ∈ (𝑉 ∘ (𝑀𝑉)))
9657, 95eqeltrd 2838 . . . . . . . 8 ((((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟𝑀) ∧ 𝑧 ∈ ((𝑉 “ {(1st𝑟)}) × (𝑉 “ {(2nd𝑟)}))) → 𝑧 ∈ (𝑉 ∘ (𝑀𝑉)))
9796ex 412 . . . . . . 7 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟𝑀) → (𝑧 ∈ ((𝑉 “ {(1st𝑟)}) × (𝑉 “ {(2nd𝑟)})) → 𝑧 ∈ (𝑉 ∘ (𝑀𝑉))))
9897ssrdv 4000 . . . . . 6 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟𝑀) → ((𝑉 “ {(1st𝑟)}) × (𝑉 “ {(2nd𝑟)})) ⊆ (𝑉 ∘ (𝑀𝑉)))
99 simp1 1135 . . . . . . . . . . 11 ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) → 𝑈 ∈ (UnifOn‘𝑋))
100 simp2l 1198 . . . . . . . . . . 11 ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) → 𝑉𝑈)
101 ustssxp 24228 . . . . . . . . . . 11 ((𝑈 ∈ (UnifOn‘𝑋) ∧ 𝑉𝑈) → 𝑉 ⊆ (𝑋 × 𝑋))
10299, 100, 101syl2anc 584 . . . . . . . . . 10 ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) → 𝑉 ⊆ (𝑋 × 𝑋))
103 coss1 5868 . . . . . . . . . 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 5868 . . . . . . . . . . . 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 5869 . . . . . . . . . . . . 13 (𝑉 ⊆ (𝑋 × 𝑋) → ((𝑋 × 𝑋) ∘ 𝑉) ⊆ ((𝑋 × 𝑋) ∘ (𝑋 × 𝑋)))
108 xpcoid 6311 . . . . . . . . . . . . 13 ((𝑋 × 𝑋) ∘ (𝑋 × 𝑋)) = (𝑋 × 𝑋)
109107, 108sseqtrdi 4045 . . . . . . . . . . . 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 4005 . . . . . . . . . 10 ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) → (𝑀𝑉) ⊆ (𝑋 × 𝑋))
112 coss2 5869 . . . . . . . . . . 11 ((𝑀𝑉) ⊆ (𝑋 × 𝑋) → ((𝑋 × 𝑋) ∘ (𝑀𝑉)) ⊆ ((𝑋 × 𝑋) ∘ (𝑋 × 𝑋)))
113112, 108sseqtrdi 4045 . . . . . . . . . 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 4005 . . . . . . . 8 ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) → (𝑉 ∘ (𝑀𝑉)) ⊆ (𝑋 × 𝑋))
116 utopbas 24259 . . . . . . . . . . . 12 (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = (unifTop‘𝑈))
1171unieqi 4923 . . . . . . . . . . . 12 𝐽 = (unifTop‘𝑈)
118116, 117eqtr4di 2792 . . . . . . . . . . 11 (𝑈 ∈ (UnifOn‘𝑋) → 𝑋 = 𝐽)
119118sqxpeqd 5720 . . . . . . . . . 10 (𝑈 ∈ (UnifOn‘𝑋) → (𝑋 × 𝑋) = ( 𝐽 × 𝐽))
12034, 34txuni 23615 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ 𝐽 ∈ Top) → ( 𝐽 × 𝐽) = (𝐽 ×t 𝐽))
1213, 3, 120syl2anc 584 . . . . . . . . . 10 (𝑈 ∈ (UnifOn‘𝑋) → ( 𝐽 × 𝐽) = (𝐽 ×t 𝐽))
122119, 121eqtrd 2774 . . . . . . . . 9 (𝑈 ∈ (UnifOn‘𝑋) → (𝑋 × 𝑋) = (𝐽 ×t 𝐽))
1231223ad2ant1 1132 . . . . . . . 8 ((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) → (𝑋 × 𝑋) = (𝐽 ×t 𝐽))
124115, 123sseqtrd 4035 . . . . . . 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 2734 . . . . . . 7 (𝐽 ×t 𝐽) = (𝐽 ×t 𝐽)
127126ssnei2 23139 . . . . . 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 839 . . . . 5 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑟𝑀) → (𝑉 ∘ (𝑀𝑉)) ∈ ((nei‘(𝐽 ×t 𝐽))‘{𝑟}))
129128ralrimiva 3143 . . . 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 4035 . . . . 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 484 . . . 4 (((𝑈 ∈ (UnifOn‘𝑋) ∧ (𝑉𝑈𝑉 = 𝑉) ∧ 𝑀 ⊆ (𝑋 × 𝑋)) ∧ 𝑀 ≠ ∅) → 𝑀 ≠ ∅)
135126neips 23136 . . . 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 1370 . . 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 3026 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 206  wa 395  w3a 1086   = wceq 1536  wcel 2105  wne 2937  wral 3058  Vcvv 3477  wss 3962  c0 4338  {csn 4630  cop 4636   cuni 4911   class class class wbr 5147   × cxp 5686  ccnv 5687  cima 5691  ccom 5692  Rel wrel 5693  cfv 6562  (class class class)co 7430  1st c1st 8010  2nd c2nd 8011  Topctop 22914  neicnei 23120   ×t ctx 23583  UnifOncust 24223  unifTopcutop 24254
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-1st 8012  df-2nd 8013  df-1o 8504  df-2o 8505  df-en 8984  df-fin 8987  df-fi 9448  df-topgen 17489  df-top 22915  df-topon 22932  df-bases 22968  df-nei 23121  df-tx 23585  df-ust 24224  df-utop 24255
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator