| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | eqid 2736 | . . . . . . 7
⊢ ∪ 𝐾 =
∪ 𝐾 | 
| 2 | 1 | flimelbas 23977 | . . . . . 6
⊢ (𝑥 ∈ (𝐾 fLim 𝐹) → 𝑥 ∈ ∪ 𝐾) | 
| 3 | 2 | adantl 481 | . . . . 5
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑥 ∈ (𝐾 fLim 𝐹)) → 𝑥 ∈ ∪ 𝐾) | 
| 4 |  | simpl2 1192 | . . . . . . 7
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑥 ∈ (𝐾 fLim 𝐹)) → 𝐹 ∈ (Fil‘𝑋)) | 
| 5 |  | filunibas 23890 | . . . . . . 7
⊢ (𝐹 ∈ (Fil‘𝑋) → ∪ 𝐹 =
𝑋) | 
| 6 | 4, 5 | syl 17 | . . . . . 6
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑥 ∈ (𝐾 fLim 𝐹)) → ∪ 𝐹 = 𝑋) | 
| 7 | 1 | flimfil 23978 | . . . . . . . 8
⊢ (𝑥 ∈ (𝐾 fLim 𝐹) → 𝐹 ∈ (Fil‘∪ 𝐾)) | 
| 8 | 7 | adantl 481 | . . . . . . 7
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑥 ∈ (𝐾 fLim 𝐹)) → 𝐹 ∈ (Fil‘∪ 𝐾)) | 
| 9 |  | filunibas 23890 | . . . . . . 7
⊢ (𝐹 ∈ (Fil‘∪ 𝐾)
→ ∪ 𝐹 = ∪ 𝐾) | 
| 10 | 8, 9 | syl 17 | . . . . . 6
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑥 ∈ (𝐾 fLim 𝐹)) → ∪ 𝐹 = ∪
𝐾) | 
| 11 | 6, 10 | eqtr3d 2778 | . . . . 5
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑥 ∈ (𝐾 fLim 𝐹)) → 𝑋 = ∪ 𝐾) | 
| 12 | 3, 11 | eleqtrrd 2843 | . . . 4
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑥 ∈ (𝐾 fLim 𝐹)) → 𝑥 ∈ 𝑋) | 
| 13 |  | simpl1 1191 | . . . . . . 7
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑥 ∈ (𝐾 fLim 𝐹)) → 𝐽 ∈ (TopOn‘𝑋)) | 
| 14 |  | topontop 22920 | . . . . . . 7
⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top) | 
| 15 | 13, 14 | syl 17 | . . . . . 6
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑥 ∈ (𝐾 fLim 𝐹)) → 𝐽 ∈ Top) | 
| 16 |  | flimtop 23974 | . . . . . . 7
⊢ (𝑥 ∈ (𝐾 fLim 𝐹) → 𝐾 ∈ Top) | 
| 17 | 16 | adantl 481 | . . . . . 6
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑥 ∈ (𝐾 fLim 𝐹)) → 𝐾 ∈ Top) | 
| 18 |  | toponuni 22921 | . . . . . . . 8
⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽) | 
| 19 | 13, 18 | syl 17 | . . . . . . 7
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑥 ∈ (𝐾 fLim 𝐹)) → 𝑋 = ∪ 𝐽) | 
| 20 | 19, 11 | eqtr3d 2778 | . . . . . 6
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑥 ∈ (𝐾 fLim 𝐹)) → ∪ 𝐽 = ∪
𝐾) | 
| 21 |  | simpl3 1193 | . . . . . 6
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑥 ∈ (𝐾 fLim 𝐹)) → 𝐽 ⊆ 𝐾) | 
| 22 |  | eqid 2736 | . . . . . . 7
⊢ ∪ 𝐽 =
∪ 𝐽 | 
| 23 | 22, 1 | topssnei 23133 | . . . . . 6
⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ ∪ 𝐽 =
∪ 𝐾) ∧ 𝐽 ⊆ 𝐾) → ((nei‘𝐽)‘{𝑥}) ⊆ ((nei‘𝐾)‘{𝑥})) | 
| 24 | 15, 17, 20, 21, 23 | syl31anc 1374 | . . . . 5
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑥 ∈ (𝐾 fLim 𝐹)) → ((nei‘𝐽)‘{𝑥}) ⊆ ((nei‘𝐾)‘{𝑥})) | 
| 25 |  | flimneiss 23975 | . . . . . 6
⊢ (𝑥 ∈ (𝐾 fLim 𝐹) → ((nei‘𝐾)‘{𝑥}) ⊆ 𝐹) | 
| 26 | 25 | adantl 481 | . . . . 5
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑥 ∈ (𝐾 fLim 𝐹)) → ((nei‘𝐾)‘{𝑥}) ⊆ 𝐹) | 
| 27 | 24, 26 | sstrd 3993 | . . . 4
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑥 ∈ (𝐾 fLim 𝐹)) → ((nei‘𝐽)‘{𝑥}) ⊆ 𝐹) | 
| 28 |  | elflim 23980 | . . . . 5
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝑥 ∈ (𝐽 fLim 𝐹) ↔ (𝑥 ∈ 𝑋 ∧ ((nei‘𝐽)‘{𝑥}) ⊆ 𝐹))) | 
| 29 | 13, 4, 28 | syl2anc 584 | . . . 4
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑥 ∈ (𝐾 fLim 𝐹)) → (𝑥 ∈ (𝐽 fLim 𝐹) ↔ (𝑥 ∈ 𝑋 ∧ ((nei‘𝐽)‘{𝑥}) ⊆ 𝐹))) | 
| 30 | 12, 27, 29 | mpbir2and 713 | . . 3
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑥 ∈ (𝐾 fLim 𝐹)) → 𝑥 ∈ (𝐽 fLim 𝐹)) | 
| 31 | 30 | ex 412 | . 2
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐽 ⊆ 𝐾) → (𝑥 ∈ (𝐾 fLim 𝐹) → 𝑥 ∈ (𝐽 fLim 𝐹))) | 
| 32 | 31 | ssrdv 3988 | 1
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝐽 ⊆ 𝐾) → (𝐾 fLim 𝐹) ⊆ (𝐽 fLim 𝐹)) |