| Step | Hyp | Ref
| Expression |
| 1 | | hauseqlcld.f |
. . . . . . . . . 10
⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾)) |
| 2 | | eqid 2737 |
. . . . . . . . . . 11
⊢ ∪ 𝐽 =
∪ 𝐽 |
| 3 | | eqid 2737 |
. . . . . . . . . . 11
⊢ ∪ 𝐾 =
∪ 𝐾 |
| 4 | 2, 3 | cnf 23254 |
. . . . . . . . . 10
⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹:∪ 𝐽⟶∪ 𝐾) |
| 5 | 1, 4 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → 𝐹:∪ 𝐽⟶∪ 𝐾) |
| 6 | 5 | ffvelcdmda 7104 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑏 ∈ ∪ 𝐽) → (𝐹‘𝑏) ∈ ∪ 𝐾) |
| 7 | 6 | biantrurd 532 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑏 ∈ ∪ 𝐽) → (〈(𝐹‘𝑏), (𝐺‘𝑏)〉 ∈ I ↔ ((𝐹‘𝑏) ∈ ∪ 𝐾 ∧ 〈(𝐹‘𝑏), (𝐺‘𝑏)〉 ∈ I ))) |
| 8 | | fvex 6919 |
. . . . . . . . 9
⊢ (𝐺‘𝑏) ∈ V |
| 9 | 8 | ideq 5863 |
. . . . . . . 8
⊢ ((𝐹‘𝑏) I (𝐺‘𝑏) ↔ (𝐹‘𝑏) = (𝐺‘𝑏)) |
| 10 | | df-br 5144 |
. . . . . . . 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 6005 |
. . . . . . 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 6906 |
. . . . . . . . . 10
⊢ (𝑎 = 𝑏 → (𝐹‘𝑎) = (𝐹‘𝑏)) |
| 15 | | fveq2 6906 |
. . . . . . . . . 10
⊢ (𝑎 = 𝑏 → (𝐺‘𝑎) = (𝐺‘𝑏)) |
| 16 | 14, 15 | opeq12d 4881 |
. . . . . . . . 9
⊢ (𝑎 = 𝑏 → 〈(𝐹‘𝑎), (𝐺‘𝑎)〉 = 〈(𝐹‘𝑏), (𝐺‘𝑏)〉) |
| 17 | | eqid 2737 |
. . . . . . . . 9
⊢ (𝑎 ∈ ∪ 𝐽
↦ 〈(𝐹‘𝑎), (𝐺‘𝑎)〉) = (𝑎 ∈ ∪ 𝐽 ↦ 〈(𝐹‘𝑎), (𝐺‘𝑎)〉) |
| 18 | | opex 5469 |
. . . . . . . . 9
⊢
〈(𝐹‘𝑏), (𝐺‘𝑏)〉 ∈ V |
| 19 | 16, 17, 18 | fvmpt 7016 |
. . . . . . . 8
⊢ (𝑏 ∈ ∪ 𝐽
→ ((𝑎 ∈ ∪ 𝐽
↦ 〈(𝐹‘𝑎), (𝐺‘𝑎)〉)‘𝑏) = 〈(𝐹‘𝑏), (𝐺‘𝑏)〉) |
| 20 | 19 | adantl 481 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑏 ∈ ∪ 𝐽) → ((𝑎 ∈ ∪ 𝐽 ↦ 〈(𝐹‘𝑎), (𝐺‘𝑎)〉)‘𝑏) = 〈(𝐹‘𝑏), (𝐺‘𝑏)〉) |
| 21 | 20 | eleq1d 2826 |
. . . . . 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 579 |
. . . 4
⊢ (𝜑 → ((𝑏 ∈ ∪ 𝐽 ∧ (𝐹‘𝑏) = (𝐺‘𝑏)) ↔ (𝑏 ∈ ∪ 𝐽 ∧ ((𝑎 ∈ ∪ 𝐽 ↦ 〈(𝐹‘𝑎), (𝐺‘𝑎)〉)‘𝑏) ∈ ( I ↾ ∪ 𝐾)))) |
| 24 | 5 | ffnd 6737 |
. . . . . . 7
⊢ (𝜑 → 𝐹 Fn ∪ 𝐽) |
| 25 | | hauseqlcld.g |
. . . . . . . . 9
⊢ (𝜑 → 𝐺 ∈ (𝐽 Cn 𝐾)) |
| 26 | 2, 3 | cnf 23254 |
. . . . . . . . 9
⊢ (𝐺 ∈ (𝐽 Cn 𝐾) → 𝐺:∪ 𝐽⟶∪ 𝐾) |
| 27 | 25, 26 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → 𝐺:∪ 𝐽⟶∪ 𝐾) |
| 28 | 27 | ffnd 6737 |
. . . . . . 7
⊢ (𝜑 → 𝐺 Fn ∪ 𝐽) |
| 29 | | fndmin 7065 |
. . . . . . 7
⊢ ((𝐹 Fn ∪
𝐽 ∧ 𝐺 Fn ∪ 𝐽) → dom (𝐹 ∩ 𝐺) = {𝑏 ∈ ∪ 𝐽 ∣ (𝐹‘𝑏) = (𝐺‘𝑏)}) |
| 30 | 24, 28, 29 | syl2anc 584 |
. . . . . 6
⊢ (𝜑 → dom (𝐹 ∩ 𝐺) = {𝑏 ∈ ∪ 𝐽 ∣ (𝐹‘𝑏) = (𝐺‘𝑏)}) |
| 31 | 30 | eleq2d 2827 |
. . . . 5
⊢ (𝜑 → (𝑏 ∈ dom (𝐹 ∩ 𝐺) ↔ 𝑏 ∈ {𝑏 ∈ ∪ 𝐽 ∣ (𝐹‘𝑏) = (𝐺‘𝑏)})) |
| 32 | | rabid 3458 |
. . . . 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 5469 |
. . . . . 6
⊢
〈(𝐹‘𝑎), (𝐺‘𝑎)〉 ∈ V |
| 35 | 34, 17 | fnmpti 6711 |
. . . . 5
⊢ (𝑎 ∈ ∪ 𝐽
↦ 〈(𝐹‘𝑎), (𝐺‘𝑎)〉) Fn ∪
𝐽 |
| 36 | | elpreima 7078 |
. . . . 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 2735 |
. 2
⊢ (𝜑 → dom (𝐹 ∩ 𝐺) = (◡(𝑎 ∈ ∪ 𝐽 ↦ 〈(𝐹‘𝑎), (𝐺‘𝑎)〉) “ ( I ↾ ∪ 𝐾))) |
| 40 | 2, 17 | txcnmpt 23632 |
. . . 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 584 |
. . 3
⊢ (𝜑 → (𝑎 ∈ ∪ 𝐽 ↦ 〈(𝐹‘𝑎), (𝐺‘𝑎)〉) ∈ (𝐽 Cn (𝐾 ×t 𝐾))) |
| 42 | | hauseqlcld.k |
. . . 4
⊢ (𝜑 → 𝐾 ∈ Haus) |
| 43 | 3 | hausdiag 23653 |
. . . . 5
⊢ (𝐾 ∈ Haus ↔ (𝐾 ∈ Top ∧ ( I ↾
∪ 𝐾) ∈ (Clsd‘(𝐾 ×t 𝐾)))) |
| 44 | 43 | simprbi 496 |
. . . 4
⊢ (𝐾 ∈ Haus → ( I ↾
∪ 𝐾) ∈ (Clsd‘(𝐾 ×t 𝐾))) |
| 45 | 42, 44 | syl 17 |
. . 3
⊢ (𝜑 → ( I ↾ ∪ 𝐾)
∈ (Clsd‘(𝐾
×t 𝐾))) |
| 46 | | cnclima 23276 |
. . 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 584 |
. 2
⊢ (𝜑 → (◡(𝑎 ∈ ∪ 𝐽 ↦ 〈(𝐹‘𝑎), (𝐺‘𝑎)〉) “ ( I ↾ ∪ 𝐾))
∈ (Clsd‘𝐽)) |
| 48 | 39, 47 | eqeltrd 2841 |
1
⊢ (𝜑 → dom (𝐹 ∩ 𝐺) ∈ (Clsd‘𝐽)) |