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

Theorem hausdiag 21650
Description: A topology is Hausdorff iff the diagonal set is closed in the topology's product with itself. EDITORIAL: very clumsy proof, can probably be shortened substantially. (Contributed by Stefan O'Rear, 25-Jan-2015.)
Hypothesis
Ref Expression
hausdiag.x 𝑋 = 𝐽
Assertion
Ref Expression
hausdiag (𝐽 ∈ Haus ↔ (𝐽 ∈ Top ∧ ( I ↾ 𝑋) ∈ (Clsd‘(𝐽 ×t 𝐽))))

Proof of Theorem hausdiag
Dummy variables 𝑎 𝑏 𝑐 𝑑 𝑒 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 hausdiag.x . . 3 𝑋 = 𝐽
21ishaus 21328 . 2 (𝐽 ∈ Haus ↔ (𝐽 ∈ Top ∧ ∀𝑎𝑋𝑏𝑋 (𝑎𝑏 → ∃𝑐𝐽𝑑𝐽 (𝑎𝑐𝑏𝑑 ∧ (𝑐𝑑) = ∅))))
3 txtop 21574 . . . . . 6 ((𝐽 ∈ Top ∧ 𝐽 ∈ Top) → (𝐽 ×t 𝐽) ∈ Top)
43anidms 680 . . . . 5 (𝐽 ∈ Top → (𝐽 ×t 𝐽) ∈ Top)
5 f1oi 6335 . . . . . . 7 ( I ↾ 𝑋):𝑋1-1-onto𝑋
6 f1of 6298 . . . . . . 7 (( I ↾ 𝑋):𝑋1-1-onto𝑋 → ( I ↾ 𝑋):𝑋𝑋)
7 fssxp 6221 . . . . . . 7 (( I ↾ 𝑋):𝑋𝑋 → ( I ↾ 𝑋) ⊆ (𝑋 × 𝑋))
85, 6, 7mp2b 10 . . . . . 6 ( I ↾ 𝑋) ⊆ (𝑋 × 𝑋)
91, 1txuni 21597 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝐽 ∈ Top) → (𝑋 × 𝑋) = (𝐽 ×t 𝐽))
109anidms 680 . . . . . 6 (𝐽 ∈ Top → (𝑋 × 𝑋) = (𝐽 ×t 𝐽))
118, 10syl5sseq 3794 . . . . 5 (𝐽 ∈ Top → ( I ↾ 𝑋) ⊆ (𝐽 ×t 𝐽))
12 eqid 2760 . . . . . 6 (𝐽 ×t 𝐽) = (𝐽 ×t 𝐽)
1312iscld2 21034 . . . . 5 (((𝐽 ×t 𝐽) ∈ Top ∧ ( I ↾ 𝑋) ⊆ (𝐽 ×t 𝐽)) → (( I ↾ 𝑋) ∈ (Clsd‘(𝐽 ×t 𝐽)) ↔ ( (𝐽 ×t 𝐽) ∖ ( I ↾ 𝑋)) ∈ (𝐽 ×t 𝐽)))
144, 11, 13syl2anc 696 . . . 4 (𝐽 ∈ Top → (( I ↾ 𝑋) ∈ (Clsd‘(𝐽 ×t 𝐽)) ↔ ( (𝐽 ×t 𝐽) ∖ ( I ↾ 𝑋)) ∈ (𝐽 ×t 𝐽)))
15 eltx 21573 . . . . 5 ((𝐽 ∈ Top ∧ 𝐽 ∈ Top) → (( (𝐽 ×t 𝐽) ∖ ( I ↾ 𝑋)) ∈ (𝐽 ×t 𝐽) ↔ ∀𝑒 ∈ ( (𝐽 ×t 𝐽) ∖ ( I ↾ 𝑋))∃𝑐𝐽𝑑𝐽 (𝑒 ∈ (𝑐 × 𝑑) ∧ (𝑐 × 𝑑) ⊆ ( (𝐽 ×t 𝐽) ∖ ( I ↾ 𝑋)))))
1615anidms 680 . . . 4 (𝐽 ∈ Top → (( (𝐽 ×t 𝐽) ∖ ( I ↾ 𝑋)) ∈ (𝐽 ×t 𝐽) ↔ ∀𝑒 ∈ ( (𝐽 ×t 𝐽) ∖ ( I ↾ 𝑋))∃𝑐𝐽𝑑𝐽 (𝑒 ∈ (𝑐 × 𝑑) ∧ (𝑐 × 𝑑) ⊆ ( (𝐽 ×t 𝐽) ∖ ( I ↾ 𝑋)))))
17 eldif 3725 . . . . . . . . . 10 (𝑒 ∈ ( (𝐽 ×t 𝐽) ∖ ( I ↾ 𝑋)) ↔ (𝑒 (𝐽 ×t 𝐽) ∧ ¬ 𝑒 ∈ ( I ↾ 𝑋)))
1810eqcomd 2766 . . . . . . . . . . . 12 (𝐽 ∈ Top → (𝐽 ×t 𝐽) = (𝑋 × 𝑋))
1918eleq2d 2825 . . . . . . . . . . 11 (𝐽 ∈ Top → (𝑒 (𝐽 ×t 𝐽) ↔ 𝑒 ∈ (𝑋 × 𝑋)))
2019anbi1d 743 . . . . . . . . . 10 (𝐽 ∈ Top → ((𝑒 (𝐽 ×t 𝐽) ∧ ¬ 𝑒 ∈ ( I ↾ 𝑋)) ↔ (𝑒 ∈ (𝑋 × 𝑋) ∧ ¬ 𝑒 ∈ ( I ↾ 𝑋))))
2117, 20syl5bb 272 . . . . . . . . 9 (𝐽 ∈ Top → (𝑒 ∈ ( (𝐽 ×t 𝐽) ∖ ( I ↾ 𝑋)) ↔ (𝑒 ∈ (𝑋 × 𝑋) ∧ ¬ 𝑒 ∈ ( I ↾ 𝑋))))
2221imbi1d 330 . . . . . . . 8 (𝐽 ∈ Top → ((𝑒 ∈ ( (𝐽 ×t 𝐽) ∖ ( I ↾ 𝑋)) → ∃𝑐𝐽𝑑𝐽 (𝑒 ∈ (𝑐 × 𝑑) ∧ (𝑐 × 𝑑) ⊆ ( (𝐽 ×t 𝐽) ∖ ( I ↾ 𝑋)))) ↔ ((𝑒 ∈ (𝑋 × 𝑋) ∧ ¬ 𝑒 ∈ ( I ↾ 𝑋)) → ∃𝑐𝐽𝑑𝐽 (𝑒 ∈ (𝑐 × 𝑑) ∧ (𝑐 × 𝑑) ⊆ ( (𝐽 ×t 𝐽) ∖ ( I ↾ 𝑋))))))
23 impexp 461 . . . . . . . 8 (((𝑒 ∈ (𝑋 × 𝑋) ∧ ¬ 𝑒 ∈ ( I ↾ 𝑋)) → ∃𝑐𝐽𝑑𝐽 (𝑒 ∈ (𝑐 × 𝑑) ∧ (𝑐 × 𝑑) ⊆ ( (𝐽 ×t 𝐽) ∖ ( I ↾ 𝑋)))) ↔ (𝑒 ∈ (𝑋 × 𝑋) → (¬ 𝑒 ∈ ( I ↾ 𝑋) → ∃𝑐𝐽𝑑𝐽 (𝑒 ∈ (𝑐 × 𝑑) ∧ (𝑐 × 𝑑) ⊆ ( (𝐽 ×t 𝐽) ∖ ( I ↾ 𝑋))))))
2422, 23syl6bb 276 . . . . . . 7 (𝐽 ∈ Top → ((𝑒 ∈ ( (𝐽 ×t 𝐽) ∖ ( I ↾ 𝑋)) → ∃𝑐𝐽𝑑𝐽 (𝑒 ∈ (𝑐 × 𝑑) ∧ (𝑐 × 𝑑) ⊆ ( (𝐽 ×t 𝐽) ∖ ( I ↾ 𝑋)))) ↔ (𝑒 ∈ (𝑋 × 𝑋) → (¬ 𝑒 ∈ ( I ↾ 𝑋) → ∃𝑐𝐽𝑑𝐽 (𝑒 ∈ (𝑐 × 𝑑) ∧ (𝑐 × 𝑑) ⊆ ( (𝐽 ×t 𝐽) ∖ ( I ↾ 𝑋)))))))
2524ralbidv2 3122 . . . . . 6 (𝐽 ∈ Top → (∀𝑒 ∈ ( (𝐽 ×t 𝐽) ∖ ( I ↾ 𝑋))∃𝑐𝐽𝑑𝐽 (𝑒 ∈ (𝑐 × 𝑑) ∧ (𝑐 × 𝑑) ⊆ ( (𝐽 ×t 𝐽) ∖ ( I ↾ 𝑋))) ↔ ∀𝑒 ∈ (𝑋 × 𝑋)(¬ 𝑒 ∈ ( I ↾ 𝑋) → ∃𝑐𝐽𝑑𝐽 (𝑒 ∈ (𝑐 × 𝑑) ∧ (𝑐 × 𝑑) ⊆ ( (𝐽 ×t 𝐽) ∖ ( I ↾ 𝑋))))))
26 eleq1 2827 . . . . . . . . 9 (𝑒 = ⟨𝑎, 𝑏⟩ → (𝑒 ∈ ( I ↾ 𝑋) ↔ ⟨𝑎, 𝑏⟩ ∈ ( I ↾ 𝑋)))
2726notbid 307 . . . . . . . 8 (𝑒 = ⟨𝑎, 𝑏⟩ → (¬ 𝑒 ∈ ( I ↾ 𝑋) ↔ ¬ ⟨𝑎, 𝑏⟩ ∈ ( I ↾ 𝑋)))
28 eleq1 2827 . . . . . . . . . 10 (𝑒 = ⟨𝑎, 𝑏⟩ → (𝑒 ∈ (𝑐 × 𝑑) ↔ ⟨𝑎, 𝑏⟩ ∈ (𝑐 × 𝑑)))
2928anbi1d 743 . . . . . . . . 9 (𝑒 = ⟨𝑎, 𝑏⟩ → ((𝑒 ∈ (𝑐 × 𝑑) ∧ (𝑐 × 𝑑) ⊆ ( (𝐽 ×t 𝐽) ∖ ( I ↾ 𝑋))) ↔ (⟨𝑎, 𝑏⟩ ∈ (𝑐 × 𝑑) ∧ (𝑐 × 𝑑) ⊆ ( (𝐽 ×t 𝐽) ∖ ( I ↾ 𝑋)))))
30292rexbidv 3195 . . . . . . . 8 (𝑒 = ⟨𝑎, 𝑏⟩ → (∃𝑐𝐽𝑑𝐽 (𝑒 ∈ (𝑐 × 𝑑) ∧ (𝑐 × 𝑑) ⊆ ( (𝐽 ×t 𝐽) ∖ ( I ↾ 𝑋))) ↔ ∃𝑐𝐽𝑑𝐽 (⟨𝑎, 𝑏⟩ ∈ (𝑐 × 𝑑) ∧ (𝑐 × 𝑑) ⊆ ( (𝐽 ×t 𝐽) ∖ ( I ↾ 𝑋)))))
3127, 30imbi12d 333 . . . . . . 7 (𝑒 = ⟨𝑎, 𝑏⟩ → ((¬ 𝑒 ∈ ( I ↾ 𝑋) → ∃𝑐𝐽𝑑𝐽 (𝑒 ∈ (𝑐 × 𝑑) ∧ (𝑐 × 𝑑) ⊆ ( (𝐽 ×t 𝐽) ∖ ( I ↾ 𝑋)))) ↔ (¬ ⟨𝑎, 𝑏⟩ ∈ ( I ↾ 𝑋) → ∃𝑐𝐽𝑑𝐽 (⟨𝑎, 𝑏⟩ ∈ (𝑐 × 𝑑) ∧ (𝑐 × 𝑑) ⊆ ( (𝐽 ×t 𝐽) ∖ ( I ↾ 𝑋))))))
3231ralxp 5419 . . . . . 6 (∀𝑒 ∈ (𝑋 × 𝑋)(¬ 𝑒 ∈ ( I ↾ 𝑋) → ∃𝑐𝐽𝑑𝐽 (𝑒 ∈ (𝑐 × 𝑑) ∧ (𝑐 × 𝑑) ⊆ ( (𝐽 ×t 𝐽) ∖ ( I ↾ 𝑋)))) ↔ ∀𝑎𝑋𝑏𝑋 (¬ ⟨𝑎, 𝑏⟩ ∈ ( I ↾ 𝑋) → ∃𝑐𝐽𝑑𝐽 (⟨𝑎, 𝑏⟩ ∈ (𝑐 × 𝑑) ∧ (𝑐 × 𝑑) ⊆ ( (𝐽 ×t 𝐽) ∖ ( I ↾ 𝑋)))))
3325, 32syl6bb 276 . . . . 5 (𝐽 ∈ Top → (∀𝑒 ∈ ( (𝐽 ×t 𝐽) ∖ ( I ↾ 𝑋))∃𝑐𝐽𝑑𝐽 (𝑒 ∈ (𝑐 × 𝑑) ∧ (𝑐 × 𝑑) ⊆ ( (𝐽 ×t 𝐽) ∖ ( I ↾ 𝑋))) ↔ ∀𝑎𝑋𝑏𝑋 (¬ ⟨𝑎, 𝑏⟩ ∈ ( I ↾ 𝑋) → ∃𝑐𝐽𝑑𝐽 (⟨𝑎, 𝑏⟩ ∈ (𝑐 × 𝑑) ∧ (𝑐 × 𝑑) ⊆ ( (𝐽 ×t 𝐽) ∖ ( I ↾ 𝑋))))))
34 vex 3343 . . . . . . . . . . 11 𝑏 ∈ V
3534opelres 5559 . . . . . . . . . 10 (⟨𝑎, 𝑏⟩ ∈ ( I ↾ 𝑋) ↔ (⟨𝑎, 𝑏⟩ ∈ I ∧ 𝑎𝑋))
36 df-br 4805 . . . . . . . . . . . 12 (𝑎 I 𝑏 ↔ ⟨𝑎, 𝑏⟩ ∈ I )
3734ideq 5430 . . . . . . . . . . . 12 (𝑎 I 𝑏𝑎 = 𝑏)
3836, 37bitr3i 266 . . . . . . . . . . 11 (⟨𝑎, 𝑏⟩ ∈ I ↔ 𝑎 = 𝑏)
39 iba 525 . . . . . . . . . . . 12 (𝑎𝑋 → (⟨𝑎, 𝑏⟩ ∈ I ↔ (⟨𝑎, 𝑏⟩ ∈ I ∧ 𝑎𝑋)))
4039adantr 472 . . . . . . . . . . 11 ((𝑎𝑋𝑏𝑋) → (⟨𝑎, 𝑏⟩ ∈ I ↔ (⟨𝑎, 𝑏⟩ ∈ I ∧ 𝑎𝑋)))
4138, 40syl5rbbr 275 . . . . . . . . . 10 ((𝑎𝑋𝑏𝑋) → ((⟨𝑎, 𝑏⟩ ∈ I ∧ 𝑎𝑋) ↔ 𝑎 = 𝑏))
4235, 41syl5bb 272 . . . . . . . . 9 ((𝑎𝑋𝑏𝑋) → (⟨𝑎, 𝑏⟩ ∈ ( I ↾ 𝑋) ↔ 𝑎 = 𝑏))
4342adantl 473 . . . . . . . 8 ((𝐽 ∈ Top ∧ (𝑎𝑋𝑏𝑋)) → (⟨𝑎, 𝑏⟩ ∈ ( I ↾ 𝑋) ↔ 𝑎 = 𝑏))
4443necon3bbid 2969 . . . . . . 7 ((𝐽 ∈ Top ∧ (𝑎𝑋𝑏𝑋)) → (¬ ⟨𝑎, 𝑏⟩ ∈ ( I ↾ 𝑋) ↔ 𝑎𝑏))
45 elssuni 4619 . . . . . . . . . . . . . . . 16 (𝑐𝐽𝑐 𝐽)
46 elssuni 4619 . . . . . . . . . . . . . . . 16 (𝑑𝐽𝑑 𝐽)
47 xpss12 5281 . . . . . . . . . . . . . . . 16 ((𝑐 𝐽𝑑 𝐽) → (𝑐 × 𝑑) ⊆ ( 𝐽 × 𝐽))
4845, 46, 47syl2an 495 . . . . . . . . . . . . . . 15 ((𝑐𝐽𝑑𝐽) → (𝑐 × 𝑑) ⊆ ( 𝐽 × 𝐽))
491, 1xpeq12i 5294 . . . . . . . . . . . . . . 15 (𝑋 × 𝑋) = ( 𝐽 × 𝐽)
5048, 49syl6sseqr 3793 . . . . . . . . . . . . . 14 ((𝑐𝐽𝑑𝐽) → (𝑐 × 𝑑) ⊆ (𝑋 × 𝑋))
5150adantl 473 . . . . . . . . . . . . 13 (((𝐽 ∈ Top ∧ (𝑎𝑋𝑏𝑋)) ∧ (𝑐𝐽𝑑𝐽)) → (𝑐 × 𝑑) ⊆ (𝑋 × 𝑋))
5210ad2antrr 764 . . . . . . . . . . . . 13 (((𝐽 ∈ Top ∧ (𝑎𝑋𝑏𝑋)) ∧ (𝑐𝐽𝑑𝐽)) → (𝑋 × 𝑋) = (𝐽 ×t 𝐽))
5351, 52sseqtrd 3782 . . . . . . . . . . . 12 (((𝐽 ∈ Top ∧ (𝑎𝑋𝑏𝑋)) ∧ (𝑐𝐽𝑑𝐽)) → (𝑐 × 𝑑) ⊆ (𝐽 ×t 𝐽))
54 reldisj 4163 . . . . . . . . . . . 12 ((𝑐 × 𝑑) ⊆ (𝐽 ×t 𝐽) → (((𝑐 × 𝑑) ∩ ( I ↾ 𝑋)) = ∅ ↔ (𝑐 × 𝑑) ⊆ ( (𝐽 ×t 𝐽) ∖ ( I ↾ 𝑋))))
5553, 54syl 17 . . . . . . . . . . 11 (((𝐽 ∈ Top ∧ (𝑎𝑋𝑏𝑋)) ∧ (𝑐𝐽𝑑𝐽)) → (((𝑐 × 𝑑) ∩ ( I ↾ 𝑋)) = ∅ ↔ (𝑐 × 𝑑) ⊆ ( (𝐽 ×t 𝐽) ∖ ( I ↾ 𝑋))))
56 df-res 5278 . . . . . . . . . . . . . . 15 ( I ↾ 𝑋) = ( I ∩ (𝑋 × V))
5756ineq2i 3954 . . . . . . . . . . . . . 14 ((𝑐 × 𝑑) ∩ ( I ↾ 𝑋)) = ((𝑐 × 𝑑) ∩ ( I ∩ (𝑋 × V)))
58 inass 3966 . . . . . . . . . . . . . . 15 (((𝑐 × 𝑑) ∩ I ) ∩ (𝑋 × V)) = ((𝑐 × 𝑑) ∩ ( I ∩ (𝑋 × V)))
59 inss1 3976 . . . . . . . . . . . . . . . . . 18 ((𝑐 × 𝑑) ∩ I ) ⊆ (𝑐 × 𝑑)
6059, 51syl5ss 3755 . . . . . . . . . . . . . . . . 17 (((𝐽 ∈ Top ∧ (𝑎𝑋𝑏𝑋)) ∧ (𝑐𝐽𝑑𝐽)) → ((𝑐 × 𝑑) ∩ I ) ⊆ (𝑋 × 𝑋))
61 ssv 3766 . . . . . . . . . . . . . . . . . 18 𝑋 ⊆ V
62 xpss2 5285 . . . . . . . . . . . . . . . . . 18 (𝑋 ⊆ V → (𝑋 × 𝑋) ⊆ (𝑋 × V))
6361, 62ax-mp 5 . . . . . . . . . . . . . . . . 17 (𝑋 × 𝑋) ⊆ (𝑋 × V)
6460, 63syl6ss 3756 . . . . . . . . . . . . . . . 16 (((𝐽 ∈ Top ∧ (𝑎𝑋𝑏𝑋)) ∧ (𝑐𝐽𝑑𝐽)) → ((𝑐 × 𝑑) ∩ I ) ⊆ (𝑋 × V))
65 df-ss 3729 . . . . . . . . . . . . . . . 16 (((𝑐 × 𝑑) ∩ I ) ⊆ (𝑋 × V) ↔ (((𝑐 × 𝑑) ∩ I ) ∩ (𝑋 × V)) = ((𝑐 × 𝑑) ∩ I ))
6664, 65sylib 208 . . . . . . . . . . . . . . 15 (((𝐽 ∈ Top ∧ (𝑎𝑋𝑏𝑋)) ∧ (𝑐𝐽𝑑𝐽)) → (((𝑐 × 𝑑) ∩ I ) ∩ (𝑋 × V)) = ((𝑐 × 𝑑) ∩ I ))
6758, 66syl5eqr 2808 . . . . . . . . . . . . . 14 (((𝐽 ∈ Top ∧ (𝑎𝑋𝑏𝑋)) ∧ (𝑐𝐽𝑑𝐽)) → ((𝑐 × 𝑑) ∩ ( I ∩ (𝑋 × V))) = ((𝑐 × 𝑑) ∩ I ))
6857, 67syl5eq 2806 . . . . . . . . . . . . 13 (((𝐽 ∈ Top ∧ (𝑎𝑋𝑏𝑋)) ∧ (𝑐𝐽𝑑𝐽)) → ((𝑐 × 𝑑) ∩ ( I ↾ 𝑋)) = ((𝑐 × 𝑑) ∩ I ))
6968eqeq1d 2762 . . . . . . . . . . . 12 (((𝐽 ∈ Top ∧ (𝑎𝑋𝑏𝑋)) ∧ (𝑐𝐽𝑑𝐽)) → (((𝑐 × 𝑑) ∩ ( I ↾ 𝑋)) = ∅ ↔ ((𝑐 × 𝑑) ∩ I ) = ∅))
70 opelxp 5303 . . . . . . . . . . . . . . . 16 (⟨𝑎, 𝑎⟩ ∈ (𝑐 × 𝑑) ↔ (𝑎𝑐𝑎𝑑))
71 df-br 4805 . . . . . . . . . . . . . . . 16 (𝑎(𝑐 × 𝑑)𝑎 ↔ ⟨𝑎, 𝑎⟩ ∈ (𝑐 × 𝑑))
72 elin 3939 . . . . . . . . . . . . . . . 16 (𝑎 ∈ (𝑐𝑑) ↔ (𝑎𝑐𝑎𝑑))
7370, 71, 723bitr4i 292 . . . . . . . . . . . . . . 15 (𝑎(𝑐 × 𝑑)𝑎𝑎 ∈ (𝑐𝑑))
7473notbii 309 . . . . . . . . . . . . . 14 𝑎(𝑐 × 𝑑)𝑎 ↔ ¬ 𝑎 ∈ (𝑐𝑑))
7574albii 1896 . . . . . . . . . . . . 13 (∀𝑎 ¬ 𝑎(𝑐 × 𝑑)𝑎 ↔ ∀𝑎 ¬ 𝑎 ∈ (𝑐𝑑))
76 intirr 5672 . . . . . . . . . . . . 13 (((𝑐 × 𝑑) ∩ I ) = ∅ ↔ ∀𝑎 ¬ 𝑎(𝑐 × 𝑑)𝑎)
77 eq0 4072 . . . . . . . . . . . . 13 ((𝑐𝑑) = ∅ ↔ ∀𝑎 ¬ 𝑎 ∈ (𝑐𝑑))
7875, 76, 773bitr4i 292 . . . . . . . . . . . 12 (((𝑐 × 𝑑) ∩ I ) = ∅ ↔ (𝑐𝑑) = ∅)
7969, 78syl6bb 276 . . . . . . . . . . 11 (((𝐽 ∈ Top ∧ (𝑎𝑋𝑏𝑋)) ∧ (𝑐𝐽𝑑𝐽)) → (((𝑐 × 𝑑) ∩ ( I ↾ 𝑋)) = ∅ ↔ (𝑐𝑑) = ∅))
8055, 79bitr3d 270 . . . . . . . . . 10 (((𝐽 ∈ Top ∧ (𝑎𝑋𝑏𝑋)) ∧ (𝑐𝐽𝑑𝐽)) → ((𝑐 × 𝑑) ⊆ ( (𝐽 ×t 𝐽) ∖ ( I ↾ 𝑋)) ↔ (𝑐𝑑) = ∅))
8180anbi2d 742 . . . . . . . . 9 (((𝐽 ∈ Top ∧ (𝑎𝑋𝑏𝑋)) ∧ (𝑐𝐽𝑑𝐽)) → (((𝑎𝑐𝑏𝑑) ∧ (𝑐 × 𝑑) ⊆ ( (𝐽 ×t 𝐽) ∖ ( I ↾ 𝑋))) ↔ ((𝑎𝑐𝑏𝑑) ∧ (𝑐𝑑) = ∅)))
82 opelxp 5303 . . . . . . . . . 10 (⟨𝑎, 𝑏⟩ ∈ (𝑐 × 𝑑) ↔ (𝑎𝑐𝑏𝑑))
8382anbi1i 733 . . . . . . . . 9 ((⟨𝑎, 𝑏⟩ ∈ (𝑐 × 𝑑) ∧ (𝑐 × 𝑑) ⊆ ( (𝐽 ×t 𝐽) ∖ ( I ↾ 𝑋))) ↔ ((𝑎𝑐𝑏𝑑) ∧ (𝑐 × 𝑑) ⊆ ( (𝐽 ×t 𝐽) ∖ ( I ↾ 𝑋))))
84 df-3an 1074 . . . . . . . . 9 ((𝑎𝑐𝑏𝑑 ∧ (𝑐𝑑) = ∅) ↔ ((𝑎𝑐𝑏𝑑) ∧ (𝑐𝑑) = ∅))
8581, 83, 843bitr4g 303 . . . . . . . 8 (((𝐽 ∈ Top ∧ (𝑎𝑋𝑏𝑋)) ∧ (𝑐𝐽𝑑𝐽)) → ((⟨𝑎, 𝑏⟩ ∈ (𝑐 × 𝑑) ∧ (𝑐 × 𝑑) ⊆ ( (𝐽 ×t 𝐽) ∖ ( I ↾ 𝑋))) ↔ (𝑎𝑐𝑏𝑑 ∧ (𝑐𝑑) = ∅)))
86852rexbidva 3194 . . . . . . 7 ((𝐽 ∈ Top ∧ (𝑎𝑋𝑏𝑋)) → (∃𝑐𝐽𝑑𝐽 (⟨𝑎, 𝑏⟩ ∈ (𝑐 × 𝑑) ∧ (𝑐 × 𝑑) ⊆ ( (𝐽 ×t 𝐽) ∖ ( I ↾ 𝑋))) ↔ ∃𝑐𝐽𝑑𝐽 (𝑎𝑐𝑏𝑑 ∧ (𝑐𝑑) = ∅)))
8744, 86imbi12d 333 . . . . . 6 ((𝐽 ∈ Top ∧ (𝑎𝑋𝑏𝑋)) → ((¬ ⟨𝑎, 𝑏⟩ ∈ ( I ↾ 𝑋) → ∃𝑐𝐽𝑑𝐽 (⟨𝑎, 𝑏⟩ ∈ (𝑐 × 𝑑) ∧ (𝑐 × 𝑑) ⊆ ( (𝐽 ×t 𝐽) ∖ ( I ↾ 𝑋)))) ↔ (𝑎𝑏 → ∃𝑐𝐽𝑑𝐽 (𝑎𝑐𝑏𝑑 ∧ (𝑐𝑑) = ∅))))
88872ralbidva 3126 . . . . 5 (𝐽 ∈ Top → (∀𝑎𝑋𝑏𝑋 (¬ ⟨𝑎, 𝑏⟩ ∈ ( I ↾ 𝑋) → ∃𝑐𝐽𝑑𝐽 (⟨𝑎, 𝑏⟩ ∈ (𝑐 × 𝑑) ∧ (𝑐 × 𝑑) ⊆ ( (𝐽 ×t 𝐽) ∖ ( I ↾ 𝑋)))) ↔ ∀𝑎𝑋𝑏𝑋 (𝑎𝑏 → ∃𝑐𝐽𝑑𝐽 (𝑎𝑐𝑏𝑑 ∧ (𝑐𝑑) = ∅))))
8933, 88bitrd 268 . . . 4 (𝐽 ∈ Top → (∀𝑒 ∈ ( (𝐽 ×t 𝐽) ∖ ( I ↾ 𝑋))∃𝑐𝐽𝑑𝐽 (𝑒 ∈ (𝑐 × 𝑑) ∧ (𝑐 × 𝑑) ⊆ ( (𝐽 ×t 𝐽) ∖ ( I ↾ 𝑋))) ↔ ∀𝑎𝑋𝑏𝑋 (𝑎𝑏 → ∃𝑐𝐽𝑑𝐽 (𝑎𝑐𝑏𝑑 ∧ (𝑐𝑑) = ∅))))
9014, 16, 893bitrrd 295 . . 3 (𝐽 ∈ Top → (∀𝑎𝑋𝑏𝑋 (𝑎𝑏 → ∃𝑐𝐽𝑑𝐽 (𝑎𝑐𝑏𝑑 ∧ (𝑐𝑑) = ∅)) ↔ ( I ↾ 𝑋) ∈ (Clsd‘(𝐽 ×t 𝐽))))
9190pm5.32i 672 . 2 ((𝐽 ∈ Top ∧ ∀𝑎𝑋𝑏𝑋 (𝑎𝑏 → ∃𝑐𝐽𝑑𝐽 (𝑎𝑐𝑏𝑑 ∧ (𝑐𝑑) = ∅))) ↔ (𝐽 ∈ Top ∧ ( I ↾ 𝑋) ∈ (Clsd‘(𝐽 ×t 𝐽))))
922, 91bitri 264 1 (𝐽 ∈ Haus ↔ (𝐽 ∈ Top ∧ ( I ↾ 𝑋) ∈ (Clsd‘(𝐽 ×t 𝐽))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383  w3a 1072  wal 1630   = wceq 1632  wcel 2139  wne 2932  wral 3050  wrex 3051  Vcvv 3340  cdif 3712  cin 3714  wss 3715  c0 4058  cop 4327   cuni 4588   class class class wbr 4804   I cid 5173   × cxp 5264  cres 5268  wf 6045  1-1-ontowf1o 6048  cfv 6049  (class class class)co 6813  Topctop 20900  Clsdccld 21022  Hauscha 21314   ×t ctx 21565
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-1st 7333  df-2nd 7334  df-topgen 16306  df-top 20901  df-topon 20918  df-bases 20952  df-cld 21025  df-haus 21321  df-tx 21567
This theorem is referenced by:  hauseqlcld  21651  tgphaus  22121  qtophaus  30212
  Copyright terms: Public domain W3C validator