Step | Hyp | Ref
| Expression |
1 | | hauseqlcld.f |
. . . . . . . . . 10
⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾)) |
2 | | eqid 2733 |
. . . . . . . . . . 11
⊢ ∪ 𝐽 =
∪ 𝐽 |
3 | | eqid 2733 |
. . . . . . . . . . 11
⊢ ∪ 𝐾 =
∪ 𝐾 |
4 | 2, 3 | cnf 22620 |
. . . . . . . . . 10
⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:∪ 𝐽⟶∪ 𝐾) |
5 | 1, 4 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → 𝐹:∪ 𝐽⟶∪ 𝐾) |
6 | 5 | ffvelcdmda 7039 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑏 ∈ ∪ 𝐽) → (𝐹‘𝑏) ∈ ∪ 𝐾) |
7 | 6 | biantrurd 534 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑏 ∈ ∪ 𝐽) → (⟨(𝐹‘𝑏), (𝐺‘𝑏)⟩ ∈ I ↔ ((𝐹‘𝑏) ∈ ∪ 𝐾 ∧ ⟨(𝐹‘𝑏), (𝐺‘𝑏)⟩ ∈ I ))) |
8 | | fvex 6859 |
. . . . . . . . 9
⊢ (𝐺‘𝑏) ∈ V |
9 | 8 | ideq 5812 |
. . . . . . . 8
⊢ ((𝐹‘𝑏) I (𝐺‘𝑏) ↔ (𝐹‘𝑏) = (𝐺‘𝑏)) |
10 | | df-br 5110 |
. . . . . . . 8
⊢ ((𝐹‘𝑏) I (𝐺‘𝑏) ↔ ⟨(𝐹‘𝑏), (𝐺‘𝑏)⟩ ∈ I ) |
11 | 9, 10 | bitr3i 277 |
. . . . . . 7
⊢ ((𝐹‘𝑏) = (𝐺‘𝑏) ↔ ⟨(𝐹‘𝑏), (𝐺‘𝑏)⟩ ∈ I ) |
12 | 8 | opelresi 5949 |
. . . . . . 7
⊢
(⟨(𝐹‘𝑏), (𝐺‘𝑏)⟩ ∈ ( I ↾ ∪ 𝐾)
↔ ((𝐹‘𝑏) ∈ ∪ 𝐾
∧ ⟨(𝐹‘𝑏), (𝐺‘𝑏)⟩ ∈ I )) |
13 | 7, 11, 12 | 3bitr4g 314 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑏 ∈ ∪ 𝐽) → ((𝐹‘𝑏) = (𝐺‘𝑏) ↔ ⟨(𝐹‘𝑏), (𝐺‘𝑏)⟩ ∈ ( I ↾ ∪ 𝐾))) |
14 | | fveq2 6846 |
. . . . . . . . . 10
⊢ (𝑎 = 𝑏 → (𝐹‘𝑎) = (𝐹‘𝑏)) |
15 | | fveq2 6846 |
. . . . . . . . . 10
⊢ (𝑎 = 𝑏 → (𝐺‘𝑎) = (𝐺‘𝑏)) |
16 | 14, 15 | opeq12d 4842 |
. . . . . . . . 9
⊢ (𝑎 = 𝑏 → ⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩ = ⟨(𝐹‘𝑏), (𝐺‘𝑏)⟩) |
17 | | eqid 2733 |
. . . . . . . . 9
⊢ (𝑎 ∈ ∪ 𝐽
↦ ⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩) = (𝑎 ∈ ∪ 𝐽 ↦ ⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩) |
18 | | opex 5425 |
. . . . . . . . 9
⊢
⟨(𝐹‘𝑏), (𝐺‘𝑏)⟩ ∈ V |
19 | 16, 17, 18 | fvmpt 6952 |
. . . . . . . 8
⊢ (𝑏 ∈ ∪ 𝐽
→ ((𝑎 ∈ ∪ 𝐽
↦ ⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩)‘𝑏) = ⟨(𝐹‘𝑏), (𝐺‘𝑏)⟩) |
20 | 19 | adantl 483 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑏 ∈ ∪ 𝐽) → ((𝑎 ∈ ∪ 𝐽 ↦ ⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩)‘𝑏) = ⟨(𝐹‘𝑏), (𝐺‘𝑏)⟩) |
21 | 20 | eleq1d 2819 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑏 ∈ ∪ 𝐽) → (((𝑎 ∈ ∪ 𝐽 ↦ ⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩)‘𝑏) ∈ ( I ↾ ∪ 𝐾)
↔ ⟨(𝐹‘𝑏), (𝐺‘𝑏)⟩ ∈ ( I ↾ ∪ 𝐾))) |
22 | 13, 21 | bitr4d 282 |
. . . . 5
⊢ ((𝜑 ∧ 𝑏 ∈ ∪ 𝐽) → ((𝐹‘𝑏) = (𝐺‘𝑏) ↔ ((𝑎 ∈ ∪ 𝐽 ↦ ⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩)‘𝑏) ∈ ( I ↾ ∪ 𝐾))) |
23 | 22 | pm5.32da 580 |
. . . 4
⊢ (𝜑 → ((𝑏 ∈ ∪ 𝐽 ∧ (𝐹‘𝑏) = (𝐺‘𝑏)) ↔ (𝑏 ∈ ∪ 𝐽 ∧ ((𝑎 ∈ ∪ 𝐽 ↦ ⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩)‘𝑏) ∈ ( I ↾ ∪ 𝐾)))) |
24 | 5 | ffnd 6673 |
. . . . . . 7
⊢ (𝜑 → 𝐹 Fn ∪ 𝐽) |
25 | | hauseqlcld.g |
. . . . . . . . 9
⊢ (𝜑 → 𝐺 ∈ (𝐽 Cn 𝐾)) |
26 | 2, 3 | cnf 22620 |
. . . . . . . . 9
⊢ (𝐺 ∈ (𝐽 Cn 𝐾) → 𝐺:∪ 𝐽⟶∪ 𝐾) |
27 | 25, 26 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → 𝐺:∪ 𝐽⟶∪ 𝐾) |
28 | 27 | ffnd 6673 |
. . . . . . 7
⊢ (𝜑 → 𝐺 Fn ∪ 𝐽) |
29 | | fndmin 6999 |
. . . . . . 7
⊢ ((𝐹 Fn ∪
𝐽 ∧ 𝐺 Fn ∪ 𝐽) → dom (𝐹 ∩ 𝐺) = {𝑏 ∈ ∪ 𝐽 ∣ (𝐹‘𝑏) = (𝐺‘𝑏)}) |
30 | 24, 28, 29 | syl2anc 585 |
. . . . . 6
⊢ (𝜑 → dom (𝐹 ∩ 𝐺) = {𝑏 ∈ ∪ 𝐽 ∣ (𝐹‘𝑏) = (𝐺‘𝑏)}) |
31 | 30 | eleq2d 2820 |
. . . . 5
⊢ (𝜑 → (𝑏 ∈ dom (𝐹 ∩ 𝐺) ↔ 𝑏 ∈ {𝑏 ∈ ∪ 𝐽 ∣ (𝐹‘𝑏) = (𝐺‘𝑏)})) |
32 | | rabid 3426 |
. . . . 5
⊢ (𝑏 ∈ {𝑏 ∈ ∪ 𝐽 ∣ (𝐹‘𝑏) = (𝐺‘𝑏)} ↔ (𝑏 ∈ ∪ 𝐽 ∧ (𝐹‘𝑏) = (𝐺‘𝑏))) |
33 | 31, 32 | bitrdi 287 |
. . . 4
⊢ (𝜑 → (𝑏 ∈ dom (𝐹 ∩ 𝐺) ↔ (𝑏 ∈ ∪ 𝐽 ∧ (𝐹‘𝑏) = (𝐺‘𝑏)))) |
34 | | opex 5425 |
. . . . . 6
⊢
⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩ ∈ V |
35 | 34, 17 | fnmpti 6648 |
. . . . 5
⊢ (𝑎 ∈ ∪ 𝐽
↦ ⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩) Fn ∪
𝐽 |
36 | | elpreima 7012 |
. . . . 5
⊢ ((𝑎 ∈ ∪ 𝐽
↦ ⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩) Fn ∪
𝐽 → (𝑏 ∈ (◡(𝑎 ∈ ∪ 𝐽 ↦ ⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩) “ ( I ↾ ∪ 𝐾))
↔ (𝑏 ∈ ∪ 𝐽
∧ ((𝑎 ∈ ∪ 𝐽
↦ ⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩)‘𝑏) ∈ ( I ↾ ∪ 𝐾)))) |
37 | 35, 36 | mp1i 13 |
. . . 4
⊢ (𝜑 → (𝑏 ∈ (◡(𝑎 ∈ ∪ 𝐽 ↦ ⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩) “ ( I ↾ ∪ 𝐾))
↔ (𝑏 ∈ ∪ 𝐽
∧ ((𝑎 ∈ ∪ 𝐽
↦ ⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩)‘𝑏) ∈ ( I ↾ ∪ 𝐾)))) |
38 | 23, 33, 37 | 3bitr4d 311 |
. . 3
⊢ (𝜑 → (𝑏 ∈ dom (𝐹 ∩ 𝐺) ↔ 𝑏 ∈ (◡(𝑎 ∈ ∪ 𝐽 ↦ ⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩) “ ( I ↾ ∪ 𝐾)))) |
39 | 38 | eqrdv 2731 |
. 2
⊢ (𝜑 → dom (𝐹 ∩ 𝐺) = (◡(𝑎 ∈ ∪ 𝐽 ↦ ⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩) “ ( I ↾ ∪ 𝐾))) |
40 | 2, 17 | txcnmpt 22998 |
. . . 4
⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝐺 ∈ (𝐽 Cn 𝐾)) → (𝑎 ∈ ∪ 𝐽 ↦ ⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩) ∈ (𝐽 Cn (𝐾 ×t 𝐾))) |
41 | 1, 25, 40 | syl2anc 585 |
. . 3
⊢ (𝜑 → (𝑎 ∈ ∪ 𝐽 ↦ ⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩) ∈ (𝐽 Cn (𝐾 ×t 𝐾))) |
42 | | hauseqlcld.k |
. . . 4
⊢ (𝜑 → 𝐾 ∈ Haus) |
43 | 3 | hausdiag 23019 |
. . . . 5
⊢ (𝐾 ∈ Haus ↔ (𝐾 ∈ Top ∧ ( I ↾
∪ 𝐾) ∈ (Clsd‘(𝐾 ×t 𝐾)))) |
44 | 43 | simprbi 498 |
. . . 4
⊢ (𝐾 ∈ Haus → ( I ↾
∪ 𝐾) ∈ (Clsd‘(𝐾 ×t 𝐾))) |
45 | 42, 44 | syl 17 |
. . 3
⊢ (𝜑 → ( I ↾ ∪ 𝐾)
∈ (Clsd‘(𝐾
×t 𝐾))) |
46 | | cnclima 22642 |
. . 3
⊢ (((𝑎 ∈ ∪ 𝐽
↦ ⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩) ∈ (𝐽 Cn (𝐾 ×t 𝐾)) ∧ ( I ↾ ∪ 𝐾)
∈ (Clsd‘(𝐾
×t 𝐾)))
→ (◡(𝑎 ∈ ∪ 𝐽 ↦ ⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩) “ ( I ↾ ∪ 𝐾))
∈ (Clsd‘𝐽)) |
47 | 41, 45, 46 | syl2anc 585 |
. 2
⊢ (𝜑 → (◡(𝑎 ∈ ∪ 𝐽 ↦ ⟨(𝐹‘𝑎), (𝐺‘𝑎)⟩) “ ( I ↾ ∪ 𝐾))
∈ (Clsd‘𝐽)) |
48 | 39, 47 | eqeltrd 2834 |
1
⊢ (𝜑 → dom (𝐹 ∩ 𝐺) ∈ (Clsd‘𝐽)) |