Step | Hyp | Ref
| Expression |
1 | | hausdiag.x |
. . 3
⊢ 𝑋 = ∪
𝐽 |
2 | 1 | ishaus 21927 |
. 2
⊢ (𝐽 ∈ Haus ↔ (𝐽 ∈ Top ∧ ∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 (𝑎 ≠ 𝑏 → ∃𝑐 ∈ 𝐽 ∃𝑑 ∈ 𝐽 (𝑎 ∈ 𝑐 ∧ 𝑏 ∈ 𝑑 ∧ (𝑐 ∩ 𝑑) = ∅)))) |
3 | | txtop 22174 |
. . . . . 6
⊢ ((𝐽 ∈ Top ∧ 𝐽 ∈ Top) → (𝐽 ×t 𝐽) ∈ Top) |
4 | 3 | anidms 570 |
. . . . 5
⊢ (𝐽 ∈ Top → (𝐽 ×t 𝐽) ∈ Top) |
5 | | idssxp 5883 |
. . . . . 6
⊢ ( I
↾ 𝑋) ⊆ (𝑋 × 𝑋) |
6 | 1, 1 | txuni 22197 |
. . . . . . 7
⊢ ((𝐽 ∈ Top ∧ 𝐽 ∈ Top) → (𝑋 × 𝑋) = ∪ (𝐽 ×t 𝐽)) |
7 | 6 | anidms 570 |
. . . . . 6
⊢ (𝐽 ∈ Top → (𝑋 × 𝑋) = ∪ (𝐽 ×t 𝐽)) |
8 | 5, 7 | sseqtrid 3967 |
. . . . 5
⊢ (𝐽 ∈ Top → ( I ↾
𝑋) ⊆ ∪ (𝐽
×t 𝐽)) |
9 | | eqid 2798 |
. . . . . 6
⊢ ∪ (𝐽
×t 𝐽) =
∪ (𝐽 ×t 𝐽) |
10 | 9 | iscld2 21633 |
. . . . 5
⊢ (((𝐽 ×t 𝐽) ∈ Top ∧ ( I ↾
𝑋) ⊆ ∪ (𝐽
×t 𝐽))
→ (( I ↾ 𝑋)
∈ (Clsd‘(𝐽
×t 𝐽))
↔ (∪ (𝐽 ×t 𝐽) ∖ ( I ↾ 𝑋)) ∈ (𝐽 ×t 𝐽))) |
11 | 4, 8, 10 | syl2anc 587 |
. . . 4
⊢ (𝐽 ∈ Top → (( I ↾
𝑋) ∈
(Clsd‘(𝐽
×t 𝐽))
↔ (∪ (𝐽 ×t 𝐽) ∖ ( I ↾ 𝑋)) ∈ (𝐽 ×t 𝐽))) |
12 | | eltx 22173 |
. . . . 5
⊢ ((𝐽 ∈ Top ∧ 𝐽 ∈ Top) → ((∪ (𝐽
×t 𝐽)
∖ ( I ↾ 𝑋))
∈ (𝐽
×t 𝐽)
↔ ∀𝑒 ∈
(∪ (𝐽 ×t 𝐽) ∖ ( I ↾ 𝑋))∃𝑐 ∈ 𝐽 ∃𝑑 ∈ 𝐽 (𝑒 ∈ (𝑐 × 𝑑) ∧ (𝑐 × 𝑑) ⊆ (∪
(𝐽 ×t
𝐽) ∖ ( I ↾
𝑋))))) |
13 | 12 | anidms 570 |
. . . 4
⊢ (𝐽 ∈ Top → ((∪ (𝐽
×t 𝐽)
∖ ( I ↾ 𝑋))
∈ (𝐽
×t 𝐽)
↔ ∀𝑒 ∈
(∪ (𝐽 ×t 𝐽) ∖ ( I ↾ 𝑋))∃𝑐 ∈ 𝐽 ∃𝑑 ∈ 𝐽 (𝑒 ∈ (𝑐 × 𝑑) ∧ (𝑐 × 𝑑) ⊆ (∪
(𝐽 ×t
𝐽) ∖ ( I ↾
𝑋))))) |
14 | | eldif 3891 |
. . . . . . . . . 10
⊢ (𝑒 ∈ (∪ (𝐽
×t 𝐽)
∖ ( I ↾ 𝑋))
↔ (𝑒 ∈ ∪ (𝐽
×t 𝐽)
∧ ¬ 𝑒 ∈ ( I
↾ 𝑋))) |
15 | 7 | eqcomd 2804 |
. . . . . . . . . . . 12
⊢ (𝐽 ∈ Top → ∪ (𝐽
×t 𝐽) =
(𝑋 × 𝑋)) |
16 | 15 | eleq2d 2875 |
. . . . . . . . . . 11
⊢ (𝐽 ∈ Top → (𝑒 ∈ ∪ (𝐽
×t 𝐽)
↔ 𝑒 ∈ (𝑋 × 𝑋))) |
17 | 16 | anbi1d 632 |
. . . . . . . . . 10
⊢ (𝐽 ∈ Top → ((𝑒 ∈ ∪ (𝐽
×t 𝐽)
∧ ¬ 𝑒 ∈ ( I
↾ 𝑋)) ↔ (𝑒 ∈ (𝑋 × 𝑋) ∧ ¬ 𝑒 ∈ ( I ↾ 𝑋)))) |
18 | 14, 17 | syl5bb 286 |
. . . . . . . . 9
⊢ (𝐽 ∈ Top → (𝑒 ∈ (∪ (𝐽
×t 𝐽)
∖ ( I ↾ 𝑋))
↔ (𝑒 ∈ (𝑋 × 𝑋) ∧ ¬ 𝑒 ∈ ( I ↾ 𝑋)))) |
19 | 18 | imbi1d 345 |
. . . . . . . 8
⊢ (𝐽 ∈ Top → ((𝑒 ∈ (∪ (𝐽
×t 𝐽)
∖ ( I ↾ 𝑋))
→ ∃𝑐 ∈
𝐽 ∃𝑑 ∈ 𝐽 (𝑒 ∈ (𝑐 × 𝑑) ∧ (𝑐 × 𝑑) ⊆ (∪
(𝐽 ×t
𝐽) ∖ ( I ↾
𝑋)))) ↔ ((𝑒 ∈ (𝑋 × 𝑋) ∧ ¬ 𝑒 ∈ ( I ↾ 𝑋)) → ∃𝑐 ∈ 𝐽 ∃𝑑 ∈ 𝐽 (𝑒 ∈ (𝑐 × 𝑑) ∧ (𝑐 × 𝑑) ⊆ (∪
(𝐽 ×t
𝐽) ∖ ( I ↾
𝑋)))))) |
20 | | impexp 454 |
. . . . . . . 8
⊢ (((𝑒 ∈ (𝑋 × 𝑋) ∧ ¬ 𝑒 ∈ ( I ↾ 𝑋)) → ∃𝑐 ∈ 𝐽 ∃𝑑 ∈ 𝐽 (𝑒 ∈ (𝑐 × 𝑑) ∧ (𝑐 × 𝑑) ⊆ (∪
(𝐽 ×t
𝐽) ∖ ( I ↾
𝑋)))) ↔ (𝑒 ∈ (𝑋 × 𝑋) → (¬ 𝑒 ∈ ( I ↾ 𝑋) → ∃𝑐 ∈ 𝐽 ∃𝑑 ∈ 𝐽 (𝑒 ∈ (𝑐 × 𝑑) ∧ (𝑐 × 𝑑) ⊆ (∪
(𝐽 ×t
𝐽) ∖ ( I ↾
𝑋)))))) |
21 | 19, 20 | syl6bb 290 |
. . . . . . 7
⊢ (𝐽 ∈ Top → ((𝑒 ∈ (∪ (𝐽
×t 𝐽)
∖ ( I ↾ 𝑋))
→ ∃𝑐 ∈
𝐽 ∃𝑑 ∈ 𝐽 (𝑒 ∈ (𝑐 × 𝑑) ∧ (𝑐 × 𝑑) ⊆ (∪
(𝐽 ×t
𝐽) ∖ ( I ↾
𝑋)))) ↔ (𝑒 ∈ (𝑋 × 𝑋) → (¬ 𝑒 ∈ ( I ↾ 𝑋) → ∃𝑐 ∈ 𝐽 ∃𝑑 ∈ 𝐽 (𝑒 ∈ (𝑐 × 𝑑) ∧ (𝑐 × 𝑑) ⊆ (∪
(𝐽 ×t
𝐽) ∖ ( I ↾
𝑋))))))) |
22 | 21 | ralbidv2 3160 |
. . . . . 6
⊢ (𝐽 ∈ Top →
(∀𝑒 ∈ (∪ (𝐽
×t 𝐽)
∖ ( I ↾ 𝑋))∃𝑐 ∈ 𝐽 ∃𝑑 ∈ 𝐽 (𝑒 ∈ (𝑐 × 𝑑) ∧ (𝑐 × 𝑑) ⊆ (∪
(𝐽 ×t
𝐽) ∖ ( I ↾
𝑋))) ↔ ∀𝑒 ∈ (𝑋 × 𝑋)(¬ 𝑒 ∈ ( I ↾ 𝑋) → ∃𝑐 ∈ 𝐽 ∃𝑑 ∈ 𝐽 (𝑒 ∈ (𝑐 × 𝑑) ∧ (𝑐 × 𝑑) ⊆ (∪
(𝐽 ×t
𝐽) ∖ ( I ↾
𝑋)))))) |
23 | | eleq1 2877 |
. . . . . . . . 9
⊢ (𝑒 = 〈𝑎, 𝑏〉 → (𝑒 ∈ ( I ↾ 𝑋) ↔ 〈𝑎, 𝑏〉 ∈ ( I ↾ 𝑋))) |
24 | 23 | notbid 321 |
. . . . . . . 8
⊢ (𝑒 = 〈𝑎, 𝑏〉 → (¬ 𝑒 ∈ ( I ↾ 𝑋) ↔ ¬ 〈𝑎, 𝑏〉 ∈ ( I ↾ 𝑋))) |
25 | | eleq1 2877 |
. . . . . . . . . 10
⊢ (𝑒 = 〈𝑎, 𝑏〉 → (𝑒 ∈ (𝑐 × 𝑑) ↔ 〈𝑎, 𝑏〉 ∈ (𝑐 × 𝑑))) |
26 | 25 | anbi1d 632 |
. . . . . . . . 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))))) |
27 | 26 | 2rexbidv 3259 |
. . . . . . . 8
⊢ (𝑒 = 〈𝑎, 𝑏〉 → (∃𝑐 ∈ 𝐽 ∃𝑑 ∈ 𝐽 (𝑒 ∈ (𝑐 × 𝑑) ∧ (𝑐 × 𝑑) ⊆ (∪
(𝐽 ×t
𝐽) ∖ ( I ↾
𝑋))) ↔ ∃𝑐 ∈ 𝐽 ∃𝑑 ∈ 𝐽 (〈𝑎, 𝑏〉 ∈ (𝑐 × 𝑑) ∧ (𝑐 × 𝑑) ⊆ (∪
(𝐽 ×t
𝐽) ∖ ( I ↾
𝑋))))) |
28 | 24, 27 | imbi12d 348 |
. . . . . . 7
⊢ (𝑒 = 〈𝑎, 𝑏〉 → ((¬ 𝑒 ∈ ( I ↾ 𝑋) → ∃𝑐 ∈ 𝐽 ∃𝑑 ∈ 𝐽 (𝑒 ∈ (𝑐 × 𝑑) ∧ (𝑐 × 𝑑) ⊆ (∪
(𝐽 ×t
𝐽) ∖ ( I ↾
𝑋)))) ↔ (¬
〈𝑎, 𝑏〉 ∈ ( I ↾ 𝑋) → ∃𝑐 ∈ 𝐽 ∃𝑑 ∈ 𝐽 (〈𝑎, 𝑏〉 ∈ (𝑐 × 𝑑) ∧ (𝑐 × 𝑑) ⊆ (∪
(𝐽 ×t
𝐽) ∖ ( I ↾
𝑋)))))) |
29 | 28 | ralxp 5676 |
. . . . . 6
⊢
(∀𝑒 ∈
(𝑋 × 𝑋)(¬ 𝑒 ∈ ( I ↾ 𝑋) → ∃𝑐 ∈ 𝐽 ∃𝑑 ∈ 𝐽 (𝑒 ∈ (𝑐 × 𝑑) ∧ (𝑐 × 𝑑) ⊆ (∪
(𝐽 ×t
𝐽) ∖ ( I ↾
𝑋)))) ↔ ∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 (¬ 〈𝑎, 𝑏〉 ∈ ( I ↾ 𝑋) → ∃𝑐 ∈ 𝐽 ∃𝑑 ∈ 𝐽 (〈𝑎, 𝑏〉 ∈ (𝑐 × 𝑑) ∧ (𝑐 × 𝑑) ⊆ (∪
(𝐽 ×t
𝐽) ∖ ( I ↾
𝑋))))) |
30 | 22, 29 | syl6bb 290 |
. . . . 5
⊢ (𝐽 ∈ Top →
(∀𝑒 ∈ (∪ (𝐽
×t 𝐽)
∖ ( I ↾ 𝑋))∃𝑐 ∈ 𝐽 ∃𝑑 ∈ 𝐽 (𝑒 ∈ (𝑐 × 𝑑) ∧ (𝑐 × 𝑑) ⊆ (∪
(𝐽 ×t
𝐽) ∖ ( I ↾
𝑋))) ↔ ∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 (¬ 〈𝑎, 𝑏〉 ∈ ( I ↾ 𝑋) → ∃𝑐 ∈ 𝐽 ∃𝑑 ∈ 𝐽 (〈𝑎, 𝑏〉 ∈ (𝑐 × 𝑑) ∧ (𝑐 × 𝑑) ⊆ (∪
(𝐽 ×t
𝐽) ∖ ( I ↾
𝑋)))))) |
31 | | vex 3444 |
. . . . . . . . . . 11
⊢ 𝑏 ∈ V |
32 | 31 | opelresi 5826 |
. . . . . . . . . 10
⊢
(〈𝑎, 𝑏〉 ∈ ( I ↾ 𝑋) ↔ (𝑎 ∈ 𝑋 ∧ 〈𝑎, 𝑏〉 ∈ I )) |
33 | | ibar 532 |
. . . . . . . . . . . 12
⊢ (𝑎 ∈ 𝑋 → (〈𝑎, 𝑏〉 ∈ I ↔ (𝑎 ∈ 𝑋 ∧ 〈𝑎, 𝑏〉 ∈ I ))) |
34 | 33 | adantr 484 |
. . . . . . . . . . 11
⊢ ((𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋) → (〈𝑎, 𝑏〉 ∈ I ↔ (𝑎 ∈ 𝑋 ∧ 〈𝑎, 𝑏〉 ∈ I ))) |
35 | | df-br 5031 |
. . . . . . . . . . . 12
⊢ (𝑎 I 𝑏 ↔ 〈𝑎, 𝑏〉 ∈ I ) |
36 | 31 | ideq 5687 |
. . . . . . . . . . . 12
⊢ (𝑎 I 𝑏 ↔ 𝑎 = 𝑏) |
37 | 35, 36 | bitr3i 280 |
. . . . . . . . . . 11
⊢
(〈𝑎, 𝑏〉 ∈ I ↔ 𝑎 = 𝑏) |
38 | 34, 37 | bitr3di 289 |
. . . . . . . . . 10
⊢ ((𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋) → ((𝑎 ∈ 𝑋 ∧ 〈𝑎, 𝑏〉 ∈ I ) ↔ 𝑎 = 𝑏)) |
39 | 32, 38 | syl5bb 286 |
. . . . . . . . 9
⊢ ((𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋) → (〈𝑎, 𝑏〉 ∈ ( I ↾ 𝑋) ↔ 𝑎 = 𝑏)) |
40 | 39 | adantl 485 |
. . . . . . . 8
⊢ ((𝐽 ∈ Top ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → (〈𝑎, 𝑏〉 ∈ ( I ↾ 𝑋) ↔ 𝑎 = 𝑏)) |
41 | 40 | necon3bbid 3024 |
. . . . . . 7
⊢ ((𝐽 ∈ Top ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → (¬ 〈𝑎, 𝑏〉 ∈ ( I ↾ 𝑋) ↔ 𝑎 ≠ 𝑏)) |
42 | | elssuni 4830 |
. . . . . . . . . . . . . . . 16
⊢ (𝑐 ∈ 𝐽 → 𝑐 ⊆ ∪ 𝐽) |
43 | | elssuni 4830 |
. . . . . . . . . . . . . . . 16
⊢ (𝑑 ∈ 𝐽 → 𝑑 ⊆ ∪ 𝐽) |
44 | | xpss12 5534 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑐 ⊆ ∪ 𝐽
∧ 𝑑 ⊆ ∪ 𝐽)
→ (𝑐 × 𝑑) ⊆ (∪ 𝐽
× ∪ 𝐽)) |
45 | 42, 43, 44 | syl2an 598 |
. . . . . . . . . . . . . . 15
⊢ ((𝑐 ∈ 𝐽 ∧ 𝑑 ∈ 𝐽) → (𝑐 × 𝑑) ⊆ (∪ 𝐽 × ∪ 𝐽)) |
46 | 1, 1 | xpeq12i 5547 |
. . . . . . . . . . . . . . 15
⊢ (𝑋 × 𝑋) = (∪ 𝐽 × ∪ 𝐽) |
47 | 45, 46 | sseqtrrdi 3966 |
. . . . . . . . . . . . . 14
⊢ ((𝑐 ∈ 𝐽 ∧ 𝑑 ∈ 𝐽) → (𝑐 × 𝑑) ⊆ (𝑋 × 𝑋)) |
48 | 47 | adantl 485 |
. . . . . . . . . . . . 13
⊢ (((𝐽 ∈ Top ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) ∧ (𝑐 ∈ 𝐽 ∧ 𝑑 ∈ 𝐽)) → (𝑐 × 𝑑) ⊆ (𝑋 × 𝑋)) |
49 | 7 | ad2antrr 725 |
. . . . . . . . . . . . 13
⊢ (((𝐽 ∈ Top ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) ∧ (𝑐 ∈ 𝐽 ∧ 𝑑 ∈ 𝐽)) → (𝑋 × 𝑋) = ∪ (𝐽 ×t 𝐽)) |
50 | 48, 49 | sseqtrd 3955 |
. . . . . . . . . . . 12
⊢ (((𝐽 ∈ Top ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) ∧ (𝑐 ∈ 𝐽 ∧ 𝑑 ∈ 𝐽)) → (𝑐 × 𝑑) ⊆ ∪ (𝐽 ×t 𝐽)) |
51 | | reldisj 4359 |
. . . . . . . . . . . 12
⊢ ((𝑐 × 𝑑) ⊆ ∪ (𝐽 ×t 𝐽) → (((𝑐 × 𝑑) ∩ ( I ↾ 𝑋)) = ∅ ↔ (𝑐 × 𝑑) ⊆ (∪
(𝐽 ×t
𝐽) ∖ ( I ↾
𝑋)))) |
52 | 50, 51 | syl 17 |
. . . . . . . . . . 11
⊢ (((𝐽 ∈ Top ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) ∧ (𝑐 ∈ 𝐽 ∧ 𝑑 ∈ 𝐽)) → (((𝑐 × 𝑑) ∩ ( I ↾ 𝑋)) = ∅ ↔ (𝑐 × 𝑑) ⊆ (∪
(𝐽 ×t
𝐽) ∖ ( I ↾
𝑋)))) |
53 | | df-res 5531 |
. . . . . . . . . . . . . . 15
⊢ ( I
↾ 𝑋) = ( I ∩
(𝑋 ×
V)) |
54 | 53 | ineq2i 4136 |
. . . . . . . . . . . . . 14
⊢ ((𝑐 × 𝑑) ∩ ( I ↾ 𝑋)) = ((𝑐 × 𝑑) ∩ ( I ∩ (𝑋 × V))) |
55 | | inass 4146 |
. . . . . . . . . . . . . . 15
⊢ (((𝑐 × 𝑑) ∩ I ) ∩ (𝑋 × V)) = ((𝑐 × 𝑑) ∩ ( I ∩ (𝑋 × V))) |
56 | | inss1 4155 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑐 × 𝑑) ∩ I ) ⊆ (𝑐 × 𝑑) |
57 | 56, 48 | sstrid 3926 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝐽 ∈ Top ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) ∧ (𝑐 ∈ 𝐽 ∧ 𝑑 ∈ 𝐽)) → ((𝑐 × 𝑑) ∩ I ) ⊆ (𝑋 × 𝑋)) |
58 | | ssv 3939 |
. . . . . . . . . . . . . . . . . 18
⊢ 𝑋 ⊆ V |
59 | | xpss2 5539 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑋 ⊆ V → (𝑋 × 𝑋) ⊆ (𝑋 × V)) |
60 | 58, 59 | ax-mp 5 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑋 × 𝑋) ⊆ (𝑋 × V) |
61 | 57, 60 | sstrdi 3927 |
. . . . . . . . . . . . . . . 16
⊢ (((𝐽 ∈ Top ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) ∧ (𝑐 ∈ 𝐽 ∧ 𝑑 ∈ 𝐽)) → ((𝑐 × 𝑑) ∩ I ) ⊆ (𝑋 × V)) |
62 | | df-ss 3898 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑐 × 𝑑) ∩ I ) ⊆ (𝑋 × V) ↔ (((𝑐 × 𝑑) ∩ I ) ∩ (𝑋 × V)) = ((𝑐 × 𝑑) ∩ I )) |
63 | 61, 62 | sylib 221 |
. . . . . . . . . . . . . . 15
⊢ (((𝐽 ∈ Top ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) ∧ (𝑐 ∈ 𝐽 ∧ 𝑑 ∈ 𝐽)) → (((𝑐 × 𝑑) ∩ I ) ∩ (𝑋 × V)) = ((𝑐 × 𝑑) ∩ I )) |
64 | 55, 63 | syl5eqr 2847 |
. . . . . . . . . . . . . 14
⊢ (((𝐽 ∈ Top ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) ∧ (𝑐 ∈ 𝐽 ∧ 𝑑 ∈ 𝐽)) → ((𝑐 × 𝑑) ∩ ( I ∩ (𝑋 × V))) = ((𝑐 × 𝑑) ∩ I )) |
65 | 54, 64 | syl5eq 2845 |
. . . . . . . . . . . . 13
⊢ (((𝐽 ∈ Top ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) ∧ (𝑐 ∈ 𝐽 ∧ 𝑑 ∈ 𝐽)) → ((𝑐 × 𝑑) ∩ ( I ↾ 𝑋)) = ((𝑐 × 𝑑) ∩ I )) |
66 | 65 | eqeq1d 2800 |
. . . . . . . . . . . 12
⊢ (((𝐽 ∈ Top ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) ∧ (𝑐 ∈ 𝐽 ∧ 𝑑 ∈ 𝐽)) → (((𝑐 × 𝑑) ∩ ( I ↾ 𝑋)) = ∅ ↔ ((𝑐 × 𝑑) ∩ I ) = ∅)) |
67 | | opelxp 5555 |
. . . . . . . . . . . . . . . 16
⊢
(〈𝑎, 𝑎〉 ∈ (𝑐 × 𝑑) ↔ (𝑎 ∈ 𝑐 ∧ 𝑎 ∈ 𝑑)) |
68 | | df-br 5031 |
. . . . . . . . . . . . . . . 16
⊢ (𝑎(𝑐 × 𝑑)𝑎 ↔ 〈𝑎, 𝑎〉 ∈ (𝑐 × 𝑑)) |
69 | | elin 3897 |
. . . . . . . . . . . . . . . 16
⊢ (𝑎 ∈ (𝑐 ∩ 𝑑) ↔ (𝑎 ∈ 𝑐 ∧ 𝑎 ∈ 𝑑)) |
70 | 67, 68, 69 | 3bitr4i 306 |
. . . . . . . . . . . . . . 15
⊢ (𝑎(𝑐 × 𝑑)𝑎 ↔ 𝑎 ∈ (𝑐 ∩ 𝑑)) |
71 | 70 | notbii 323 |
. . . . . . . . . . . . . 14
⊢ (¬
𝑎(𝑐 × 𝑑)𝑎 ↔ ¬ 𝑎 ∈ (𝑐 ∩ 𝑑)) |
72 | 71 | albii 1821 |
. . . . . . . . . . . . 13
⊢
(∀𝑎 ¬
𝑎(𝑐 × 𝑑)𝑎 ↔ ∀𝑎 ¬ 𝑎 ∈ (𝑐 ∩ 𝑑)) |
73 | | intirr 5945 |
. . . . . . . . . . . . 13
⊢ (((𝑐 × 𝑑) ∩ I ) = ∅ ↔ ∀𝑎 ¬ 𝑎(𝑐 × 𝑑)𝑎) |
74 | | eq0 4258 |
. . . . . . . . . . . . 13
⊢ ((𝑐 ∩ 𝑑) = ∅ ↔ ∀𝑎 ¬ 𝑎 ∈ (𝑐 ∩ 𝑑)) |
75 | 72, 73, 74 | 3bitr4i 306 |
. . . . . . . . . . . 12
⊢ (((𝑐 × 𝑑) ∩ I ) = ∅ ↔ (𝑐 ∩ 𝑑) = ∅) |
76 | 66, 75 | syl6bb 290 |
. . . . . . . . . . 11
⊢ (((𝐽 ∈ Top ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) ∧ (𝑐 ∈ 𝐽 ∧ 𝑑 ∈ 𝐽)) → (((𝑐 × 𝑑) ∩ ( I ↾ 𝑋)) = ∅ ↔ (𝑐 ∩ 𝑑) = ∅)) |
77 | 52, 76 | bitr3d 284 |
. . . . . . . . . 10
⊢ (((𝐽 ∈ Top ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) ∧ (𝑐 ∈ 𝐽 ∧ 𝑑 ∈ 𝐽)) → ((𝑐 × 𝑑) ⊆ (∪
(𝐽 ×t
𝐽) ∖ ( I ↾
𝑋)) ↔ (𝑐 ∩ 𝑑) = ∅)) |
78 | 77 | anbi2d 631 |
. . . . . . . . 9
⊢ (((𝐽 ∈ Top ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) ∧ (𝑐 ∈ 𝐽 ∧ 𝑑 ∈ 𝐽)) → (((𝑎 ∈ 𝑐 ∧ 𝑏 ∈ 𝑑) ∧ (𝑐 × 𝑑) ⊆ (∪
(𝐽 ×t
𝐽) ∖ ( I ↾
𝑋))) ↔ ((𝑎 ∈ 𝑐 ∧ 𝑏 ∈ 𝑑) ∧ (𝑐 ∩ 𝑑) = ∅))) |
79 | | opelxp 5555 |
. . . . . . . . . 10
⊢
(〈𝑎, 𝑏〉 ∈ (𝑐 × 𝑑) ↔ (𝑎 ∈ 𝑐 ∧ 𝑏 ∈ 𝑑)) |
80 | 79 | anbi1i 626 |
. . . . . . . . 9
⊢
((〈𝑎, 𝑏〉 ∈ (𝑐 × 𝑑) ∧ (𝑐 × 𝑑) ⊆ (∪
(𝐽 ×t
𝐽) ∖ ( I ↾
𝑋))) ↔ ((𝑎 ∈ 𝑐 ∧ 𝑏 ∈ 𝑑) ∧ (𝑐 × 𝑑) ⊆ (∪
(𝐽 ×t
𝐽) ∖ ( I ↾
𝑋)))) |
81 | | df-3an 1086 |
. . . . . . . . 9
⊢ ((𝑎 ∈ 𝑐 ∧ 𝑏 ∈ 𝑑 ∧ (𝑐 ∩ 𝑑) = ∅) ↔ ((𝑎 ∈ 𝑐 ∧ 𝑏 ∈ 𝑑) ∧ (𝑐 ∩ 𝑑) = ∅)) |
82 | 78, 80, 81 | 3bitr4g 317 |
. . . . . . . 8
⊢ (((𝐽 ∈ Top ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) ∧ (𝑐 ∈ 𝐽 ∧ 𝑑 ∈ 𝐽)) → ((〈𝑎, 𝑏〉 ∈ (𝑐 × 𝑑) ∧ (𝑐 × 𝑑) ⊆ (∪
(𝐽 ×t
𝐽) ∖ ( I ↾
𝑋))) ↔ (𝑎 ∈ 𝑐 ∧ 𝑏 ∈ 𝑑 ∧ (𝑐 ∩ 𝑑) = ∅))) |
83 | 82 | 2rexbidva 3258 |
. . . . . . 7
⊢ ((𝐽 ∈ Top ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → (∃𝑐 ∈ 𝐽 ∃𝑑 ∈ 𝐽 (〈𝑎, 𝑏〉 ∈ (𝑐 × 𝑑) ∧ (𝑐 × 𝑑) ⊆ (∪
(𝐽 ×t
𝐽) ∖ ( I ↾
𝑋))) ↔ ∃𝑐 ∈ 𝐽 ∃𝑑 ∈ 𝐽 (𝑎 ∈ 𝑐 ∧ 𝑏 ∈ 𝑑 ∧ (𝑐 ∩ 𝑑) = ∅))) |
84 | 41, 83 | imbi12d 348 |
. . . . . 6
⊢ ((𝐽 ∈ Top ∧ (𝑎 ∈ 𝑋 ∧ 𝑏 ∈ 𝑋)) → ((¬ 〈𝑎, 𝑏〉 ∈ ( I ↾ 𝑋) → ∃𝑐 ∈ 𝐽 ∃𝑑 ∈ 𝐽 (〈𝑎, 𝑏〉 ∈ (𝑐 × 𝑑) ∧ (𝑐 × 𝑑) ⊆ (∪
(𝐽 ×t
𝐽) ∖ ( I ↾
𝑋)))) ↔ (𝑎 ≠ 𝑏 → ∃𝑐 ∈ 𝐽 ∃𝑑 ∈ 𝐽 (𝑎 ∈ 𝑐 ∧ 𝑏 ∈ 𝑑 ∧ (𝑐 ∩ 𝑑) = ∅)))) |
85 | 84 | 2ralbidva 3163 |
. . . . 5
⊢ (𝐽 ∈ Top →
(∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 (¬ 〈𝑎, 𝑏〉 ∈ ( I ↾ 𝑋) → ∃𝑐 ∈ 𝐽 ∃𝑑 ∈ 𝐽 (〈𝑎, 𝑏〉 ∈ (𝑐 × 𝑑) ∧ (𝑐 × 𝑑) ⊆ (∪
(𝐽 ×t
𝐽) ∖ ( I ↾
𝑋)))) ↔ ∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 (𝑎 ≠ 𝑏 → ∃𝑐 ∈ 𝐽 ∃𝑑 ∈ 𝐽 (𝑎 ∈ 𝑐 ∧ 𝑏 ∈ 𝑑 ∧ (𝑐 ∩ 𝑑) = ∅)))) |
86 | 30, 85 | bitrd 282 |
. . . 4
⊢ (𝐽 ∈ Top →
(∀𝑒 ∈ (∪ (𝐽
×t 𝐽)
∖ ( I ↾ 𝑋))∃𝑐 ∈ 𝐽 ∃𝑑 ∈ 𝐽 (𝑒 ∈ (𝑐 × 𝑑) ∧ (𝑐 × 𝑑) ⊆ (∪
(𝐽 ×t
𝐽) ∖ ( I ↾
𝑋))) ↔ ∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 (𝑎 ≠ 𝑏 → ∃𝑐 ∈ 𝐽 ∃𝑑 ∈ 𝐽 (𝑎 ∈ 𝑐 ∧ 𝑏 ∈ 𝑑 ∧ (𝑐 ∩ 𝑑) = ∅)))) |
87 | 11, 13, 86 | 3bitrrd 309 |
. . 3
⊢ (𝐽 ∈ Top →
(∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 (𝑎 ≠ 𝑏 → ∃𝑐 ∈ 𝐽 ∃𝑑 ∈ 𝐽 (𝑎 ∈ 𝑐 ∧ 𝑏 ∈ 𝑑 ∧ (𝑐 ∩ 𝑑) = ∅)) ↔ ( I ↾ 𝑋) ∈ (Clsd‘(𝐽 ×t 𝐽)))) |
88 | 87 | pm5.32i 578 |
. 2
⊢ ((𝐽 ∈ Top ∧ ∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 (𝑎 ≠ 𝑏 → ∃𝑐 ∈ 𝐽 ∃𝑑 ∈ 𝐽 (𝑎 ∈ 𝑐 ∧ 𝑏 ∈ 𝑑 ∧ (𝑐 ∩ 𝑑) = ∅))) ↔ (𝐽 ∈ Top ∧ ( I ↾ 𝑋) ∈ (Clsd‘(𝐽 ×t 𝐽)))) |
89 | 2, 88 | bitri 278 |
1
⊢ (𝐽 ∈ Haus ↔ (𝐽 ∈ Top ∧ ( I ↾
𝑋) ∈
(Clsd‘(𝐽
×t 𝐽)))) |