Step | Hyp | Ref
| Expression |
1 | | topontop 22384 |
. . 3
⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top) |
2 | | fvssunirn 6914 |
. . . 4
⊢
(Fil‘𝑌)
⊆ ∪ ran Fil |
3 | 2 | sseli 3976 |
. . 3
⊢ (𝐿 ∈ (Fil‘𝑌) → 𝐿 ∈ ∪ ran
Fil) |
4 | | unieq 4915 |
. . . . . 6
⊢ (𝑥 = 𝐽 → ∪ 𝑥 = ∪
𝐽) |
5 | | unieq 4915 |
. . . . . 6
⊢ (𝑦 = 𝐿 → ∪ 𝑦 = ∪
𝐿) |
6 | 4, 5 | oveqan12d 7415 |
. . . . 5
⊢ ((𝑥 = 𝐽 ∧ 𝑦 = 𝐿) → (∪ 𝑥 ↑m ∪ 𝑦) =
(∪ 𝐽 ↑m ∪ 𝐿)) |
7 | | simpl 484 |
. . . . . 6
⊢ ((𝑥 = 𝐽 ∧ 𝑦 = 𝐿) → 𝑥 = 𝐽) |
8 | 4 | adantr 482 |
. . . . . . . 8
⊢ ((𝑥 = 𝐽 ∧ 𝑦 = 𝐿) → ∪ 𝑥 = ∪
𝐽) |
9 | 8 | oveq1d 7411 |
. . . . . . 7
⊢ ((𝑥 = 𝐽 ∧ 𝑦 = 𝐿) → (∪ 𝑥 FilMap 𝑓) = (∪ 𝐽 FilMap 𝑓)) |
10 | | simpr 486 |
. . . . . . 7
⊢ ((𝑥 = 𝐽 ∧ 𝑦 = 𝐿) → 𝑦 = 𝐿) |
11 | 9, 10 | fveq12d 6888 |
. . . . . 6
⊢ ((𝑥 = 𝐽 ∧ 𝑦 = 𝐿) → ((∪
𝑥 FilMap 𝑓)‘𝑦) = ((∪ 𝐽 FilMap 𝑓)‘𝐿)) |
12 | 7, 11 | oveq12d 7414 |
. . . . 5
⊢ ((𝑥 = 𝐽 ∧ 𝑦 = 𝐿) → (𝑥 fLim ((∪ 𝑥 FilMap 𝑓)‘𝑦)) = (𝐽 fLim ((∪ 𝐽 FilMap 𝑓)‘𝐿))) |
13 | 6, 12 | mpteq12dv 5235 |
. . . 4
⊢ ((𝑥 = 𝐽 ∧ 𝑦 = 𝐿) → (𝑓 ∈ (∪ 𝑥 ↑m ∪ 𝑦)
↦ (𝑥 fLim ((∪ 𝑥
FilMap 𝑓)‘𝑦))) = (𝑓 ∈ (∪ 𝐽 ↑m ∪ 𝐿)
↦ (𝐽 fLim ((∪ 𝐽
FilMap 𝑓)‘𝐿)))) |
14 | | df-flf 23413 |
. . . 4
⊢ fLimf =
(𝑥 ∈ Top, 𝑦 ∈ ∪ ran Fil ↦ (𝑓 ∈ (∪ 𝑥 ↑m ∪ 𝑦)
↦ (𝑥 fLim ((∪ 𝑥
FilMap 𝑓)‘𝑦)))) |
15 | | ovex 7429 |
. . . . 5
⊢ (∪ 𝐽
↑m ∪ 𝐿) ∈ V |
16 | 15 | mptex 7212 |
. . . 4
⊢ (𝑓 ∈ (∪ 𝐽
↑m ∪ 𝐿) ↦ (𝐽 fLim ((∪ 𝐽 FilMap 𝑓)‘𝐿))) ∈ V |
17 | 13, 14, 16 | ovmpoa 7550 |
. . 3
⊢ ((𝐽 ∈ Top ∧ 𝐿 ∈ ∪ ran Fil) → (𝐽 fLimf 𝐿) = (𝑓 ∈ (∪ 𝐽 ↑m ∪ 𝐿)
↦ (𝐽 fLim ((∪ 𝐽
FilMap 𝑓)‘𝐿)))) |
18 | 1, 3, 17 | syl2an 597 |
. 2
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) → (𝐽 fLimf 𝐿) = (𝑓 ∈ (∪ 𝐽 ↑m ∪ 𝐿)
↦ (𝐽 fLim ((∪ 𝐽
FilMap 𝑓)‘𝐿)))) |
19 | | toponuni 22385 |
. . . . 5
⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽) |
20 | 19 | eqcomd 2739 |
. . . 4
⊢ (𝐽 ∈ (TopOn‘𝑋) → ∪ 𝐽 =
𝑋) |
21 | | filunibas 23354 |
. . . 4
⊢ (𝐿 ∈ (Fil‘𝑌) → ∪ 𝐿 =
𝑌) |
22 | 20, 21 | oveqan12d 7415 |
. . 3
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) → (∪
𝐽 ↑m ∪ 𝐿) =
(𝑋 ↑m 𝑌)) |
23 | 20 | adantr 482 |
. . . . . 6
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) → ∪ 𝐽 = 𝑋) |
24 | 23 | oveq1d 7411 |
. . . . 5
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) → (∪
𝐽 FilMap 𝑓) = (𝑋 FilMap 𝑓)) |
25 | 24 | fveq1d 6883 |
. . . 4
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) → ((∪
𝐽 FilMap 𝑓)‘𝐿) = ((𝑋 FilMap 𝑓)‘𝐿)) |
26 | 25 | oveq2d 7412 |
. . 3
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) → (𝐽 fLim ((∪ 𝐽 FilMap 𝑓)‘𝐿)) = (𝐽 fLim ((𝑋 FilMap 𝑓)‘𝐿))) |
27 | 22, 26 | mpteq12dv 5235 |
. 2
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) → (𝑓 ∈ (∪ 𝐽 ↑m ∪ 𝐿)
↦ (𝐽 fLim ((∪ 𝐽
FilMap 𝑓)‘𝐿))) = (𝑓 ∈ (𝑋 ↑m 𝑌) ↦ (𝐽 fLim ((𝑋 FilMap 𝑓)‘𝐿)))) |
28 | 18, 27 | eqtrd 2773 |
1
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) → (𝐽 fLimf 𝐿) = (𝑓 ∈ (𝑋 ↑m 𝑌) ↦ (𝐽 fLim ((𝑋 FilMap 𝑓)‘𝐿)))) |