| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | topontop 22920 | . . 3
⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top) | 
| 2 |  | fvssunirn 6938 | . . . 4
⊢
(Fil‘𝑌)
⊆ ∪ ran Fil | 
| 3 | 2 | sseli 3978 | . . 3
⊢ (𝐿 ∈ (Fil‘𝑌) → 𝐿 ∈ ∪ ran
Fil) | 
| 4 |  | unieq 4917 | . . . . . 6
⊢ (𝑥 = 𝐽 → ∪ 𝑥 = ∪
𝐽) | 
| 5 |  | unieq 4917 | . . . . . 6
⊢ (𝑦 = 𝐿 → ∪ 𝑦 = ∪
𝐿) | 
| 6 | 4, 5 | oveqan12d 7451 | . . . . 5
⊢ ((𝑥 = 𝐽 ∧ 𝑦 = 𝐿) → (∪ 𝑥 ↑m ∪ 𝑦) =
(∪ 𝐽 ↑m ∪ 𝐿)) | 
| 7 |  | simpl 482 | . . . . . 6
⊢ ((𝑥 = 𝐽 ∧ 𝑦 = 𝐿) → 𝑥 = 𝐽) | 
| 8 | 4 | adantr 480 | . . . . . . . 8
⊢ ((𝑥 = 𝐽 ∧ 𝑦 = 𝐿) → ∪ 𝑥 = ∪
𝐽) | 
| 9 | 8 | oveq1d 7447 | . . . . . . 7
⊢ ((𝑥 = 𝐽 ∧ 𝑦 = 𝐿) → (∪ 𝑥 FilMap 𝑓) = (∪ 𝐽 FilMap 𝑓)) | 
| 10 |  | simpr 484 | . . . . . . 7
⊢ ((𝑥 = 𝐽 ∧ 𝑦 = 𝐿) → 𝑦 = 𝐿) | 
| 11 | 9, 10 | fveq12d 6912 | . . . . . 6
⊢ ((𝑥 = 𝐽 ∧ 𝑦 = 𝐿) → ((∪
𝑥 FilMap 𝑓)‘𝑦) = ((∪ 𝐽 FilMap 𝑓)‘𝐿)) | 
| 12 | 7, 11 | oveq12d 7450 | . . . . 5
⊢ ((𝑥 = 𝐽 ∧ 𝑦 = 𝐿) → (𝑥 fLim ((∪ 𝑥 FilMap 𝑓)‘𝑦)) = (𝐽 fLim ((∪ 𝐽 FilMap 𝑓)‘𝐿))) | 
| 13 | 6, 12 | mpteq12dv 5232 | . . . 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 23949 | . . . 4
⊢  fLimf =
(𝑥 ∈ Top, 𝑦 ∈ ∪ ran Fil ↦ (𝑓 ∈ (∪ 𝑥 ↑m ∪ 𝑦)
↦ (𝑥 fLim ((∪ 𝑥
FilMap 𝑓)‘𝑦)))) | 
| 15 |  | ovex 7465 | . . . . 5
⊢ (∪ 𝐽
↑m ∪ 𝐿) ∈ V | 
| 16 | 15 | mptex 7244 | . . . 4
⊢ (𝑓 ∈ (∪ 𝐽
↑m ∪ 𝐿) ↦ (𝐽 fLim ((∪ 𝐽 FilMap 𝑓)‘𝐿))) ∈ V | 
| 17 | 13, 14, 16 | ovmpoa 7589 | . . 3
⊢ ((𝐽 ∈ Top ∧ 𝐿 ∈ ∪ ran Fil) → (𝐽 fLimf 𝐿) = (𝑓 ∈ (∪ 𝐽 ↑m ∪ 𝐿)
↦ (𝐽 fLim ((∪ 𝐽
FilMap 𝑓)‘𝐿)))) | 
| 18 | 1, 3, 17 | syl2an 596 | . 2
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) → (𝐽 fLimf 𝐿) = (𝑓 ∈ (∪ 𝐽 ↑m ∪ 𝐿)
↦ (𝐽 fLim ((∪ 𝐽
FilMap 𝑓)‘𝐿)))) | 
| 19 |  | toponuni 22921 | . . . . 5
⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽) | 
| 20 | 19 | eqcomd 2742 | . . . 4
⊢ (𝐽 ∈ (TopOn‘𝑋) → ∪ 𝐽 =
𝑋) | 
| 21 |  | filunibas 23890 | . . . 4
⊢ (𝐿 ∈ (Fil‘𝑌) → ∪ 𝐿 =
𝑌) | 
| 22 | 20, 21 | oveqan12d 7451 | . . 3
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) → (∪
𝐽 ↑m ∪ 𝐿) =
(𝑋 ↑m 𝑌)) | 
| 23 | 20 | adantr 480 | . . . . . 6
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) → ∪ 𝐽 = 𝑋) | 
| 24 | 23 | oveq1d 7447 | . . . . 5
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) → (∪
𝐽 FilMap 𝑓) = (𝑋 FilMap 𝑓)) | 
| 25 | 24 | fveq1d 6907 | . . . 4
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) → ((∪
𝐽 FilMap 𝑓)‘𝐿) = ((𝑋 FilMap 𝑓)‘𝐿)) | 
| 26 | 25 | oveq2d 7448 | . . 3
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) → (𝐽 fLim ((∪ 𝐽 FilMap 𝑓)‘𝐿)) = (𝐽 fLim ((𝑋 FilMap 𝑓)‘𝐿))) | 
| 27 | 22, 26 | mpteq12dv 5232 | . 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 2776 | 1
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌)) → (𝐽 fLimf 𝐿) = (𝑓 ∈ (𝑋 ↑m 𝑌) ↦ (𝐽 fLim ((𝑋 FilMap 𝑓)‘𝐿)))) |