| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | fveq2 6906 | . . 3
⊢ (𝑥 = 𝑋 → (𝐹‘𝑥) = (𝐹‘𝑋)) | 
| 2 | 1 | eleq1d 2826 | . 2
⊢ (𝑥 = 𝑋 → ((𝐹‘𝑥) ∈ ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹) ↔ (𝐹‘𝑋) ∈ ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹))) | 
| 3 |  | oveq2 7439 | . . . 4
⊢ (𝑎 = (((nei‘𝐽)‘{𝑋}) ↾t 𝐴) → ((𝐽 ↾t 𝐴) fLim 𝑎) = ((𝐽 ↾t 𝐴) fLim (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))) | 
| 4 |  | oveq2 7439 | . . . . . 6
⊢ (𝑎 = (((nei‘𝐽)‘{𝑋}) ↾t 𝐴) → (𝐾 fLimf 𝑎) = (𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))) | 
| 5 | 4 | fveq1d 6908 | . . . . 5
⊢ (𝑎 = (((nei‘𝐽)‘{𝑋}) ↾t 𝐴) → ((𝐾 fLimf 𝑎)‘𝐹) = ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹)) | 
| 6 | 5 | eleq2d 2827 | . . . 4
⊢ (𝑎 = (((nei‘𝐽)‘{𝑋}) ↾t 𝐴) → ((𝐹‘𝑥) ∈ ((𝐾 fLimf 𝑎)‘𝐹) ↔ (𝐹‘𝑥) ∈ ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹))) | 
| 7 | 3, 6 | raleqbidv 3346 | . . 3
⊢ (𝑎 = (((nei‘𝐽)‘{𝑋}) ↾t 𝐴) → (∀𝑥 ∈ ((𝐽 ↾t 𝐴) fLim 𝑎)(𝐹‘𝑥) ∈ ((𝐾 fLimf 𝑎)‘𝐹) ↔ ∀𝑥 ∈ ((𝐽 ↾t 𝐴) fLim (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))(𝐹‘𝑥) ∈ ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹))) | 
| 8 |  | flfcntr.1 | . . . . 5
⊢ (𝜑 → 𝐹 ∈ ((𝐽 ↾t 𝐴) Cn 𝐾)) | 
| 9 |  | flfcntr.j | . . . . . . . 8
⊢ (𝜑 → 𝐽 ∈ Top) | 
| 10 |  | flfcntr.c | . . . . . . . . 9
⊢ 𝐶 = ∪
𝐽 | 
| 11 | 10 | toptopon 22923 | . . . . . . . 8
⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝐶)) | 
| 12 | 9, 11 | sylib 218 | . . . . . . 7
⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝐶)) | 
| 13 |  | flfcntr.a | . . . . . . 7
⊢ (𝜑 → 𝐴 ⊆ 𝐶) | 
| 14 |  | resttopon 23169 | . . . . . . 7
⊢ ((𝐽 ∈ (TopOn‘𝐶) ∧ 𝐴 ⊆ 𝐶) → (𝐽 ↾t 𝐴) ∈ (TopOn‘𝐴)) | 
| 15 | 12, 13, 14 | syl2anc 584 | . . . . . 6
⊢ (𝜑 → (𝐽 ↾t 𝐴) ∈ (TopOn‘𝐴)) | 
| 16 |  | cntop2 23249 | . . . . . . . 8
⊢ (𝐹 ∈ ((𝐽 ↾t 𝐴) Cn 𝐾) → 𝐾 ∈ Top) | 
| 17 | 8, 16 | syl 17 | . . . . . . 7
⊢ (𝜑 → 𝐾 ∈ Top) | 
| 18 |  | flfcntr.b | . . . . . . . 8
⊢ 𝐵 = ∪
𝐾 | 
| 19 | 18 | toptopon 22923 | . . . . . . 7
⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘𝐵)) | 
| 20 | 17, 19 | sylib 218 | . . . . . 6
⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝐵)) | 
| 21 |  | cnflf 24010 | . . . . . 6
⊢ (((𝐽 ↾t 𝐴) ∈ (TopOn‘𝐴) ∧ 𝐾 ∈ (TopOn‘𝐵)) → (𝐹 ∈ ((𝐽 ↾t 𝐴) Cn 𝐾) ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑎 ∈ (Fil‘𝐴)∀𝑥 ∈ ((𝐽 ↾t 𝐴) fLim 𝑎)(𝐹‘𝑥) ∈ ((𝐾 fLimf 𝑎)‘𝐹)))) | 
| 22 | 15, 20, 21 | syl2anc 584 | . . . . 5
⊢ (𝜑 → (𝐹 ∈ ((𝐽 ↾t 𝐴) Cn 𝐾) ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑎 ∈ (Fil‘𝐴)∀𝑥 ∈ ((𝐽 ↾t 𝐴) fLim 𝑎)(𝐹‘𝑥) ∈ ((𝐾 fLimf 𝑎)‘𝐹)))) | 
| 23 | 8, 22 | mpbid 232 | . . . 4
⊢ (𝜑 → (𝐹:𝐴⟶𝐵 ∧ ∀𝑎 ∈ (Fil‘𝐴)∀𝑥 ∈ ((𝐽 ↾t 𝐴) fLim 𝑎)(𝐹‘𝑥) ∈ ((𝐾 fLimf 𝑎)‘𝐹))) | 
| 24 | 23 | simprd 495 | . . 3
⊢ (𝜑 → ∀𝑎 ∈ (Fil‘𝐴)∀𝑥 ∈ ((𝐽 ↾t 𝐴) fLim 𝑎)(𝐹‘𝑥) ∈ ((𝐾 fLimf 𝑎)‘𝐹)) | 
| 25 | 10 | sscls 23064 | . . . . . 6
⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝐶) → 𝐴 ⊆ ((cls‘𝐽)‘𝐴)) | 
| 26 | 9, 13, 25 | syl2anc 584 | . . . . 5
⊢ (𝜑 → 𝐴 ⊆ ((cls‘𝐽)‘𝐴)) | 
| 27 |  | flfcntr.y | . . . . 5
⊢ (𝜑 → 𝑋 ∈ 𝐴) | 
| 28 | 26, 27 | sseldd 3984 | . . . 4
⊢ (𝜑 → 𝑋 ∈ ((cls‘𝐽)‘𝐴)) | 
| 29 | 13, 27 | sseldd 3984 | . . . . 5
⊢ (𝜑 → 𝑋 ∈ 𝐶) | 
| 30 |  | trnei 23900 | . . . . 5
⊢ ((𝐽 ∈ (TopOn‘𝐶) ∧ 𝐴 ⊆ 𝐶 ∧ 𝑋 ∈ 𝐶) → (𝑋 ∈ ((cls‘𝐽)‘𝐴) ↔ (((nei‘𝐽)‘{𝑋}) ↾t 𝐴) ∈ (Fil‘𝐴))) | 
| 31 | 12, 13, 29, 30 | syl3anc 1373 | . . . 4
⊢ (𝜑 → (𝑋 ∈ ((cls‘𝐽)‘𝐴) ↔ (((nei‘𝐽)‘{𝑋}) ↾t 𝐴) ∈ (Fil‘𝐴))) | 
| 32 | 28, 31 | mpbid 232 | . . 3
⊢ (𝜑 → (((nei‘𝐽)‘{𝑋}) ↾t 𝐴) ∈ (Fil‘𝐴)) | 
| 33 | 7, 24, 32 | rspcdva 3623 | . 2
⊢ (𝜑 → ∀𝑥 ∈ ((𝐽 ↾t 𝐴) fLim (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))(𝐹‘𝑥) ∈ ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹)) | 
| 34 |  | neiflim 23982 | . . . 4
⊢ (((𝐽 ↾t 𝐴) ∈ (TopOn‘𝐴) ∧ 𝑋 ∈ 𝐴) → 𝑋 ∈ ((𝐽 ↾t 𝐴) fLim ((nei‘(𝐽 ↾t 𝐴))‘{𝑋}))) | 
| 35 | 15, 27, 34 | syl2anc 584 | . . 3
⊢ (𝜑 → 𝑋 ∈ ((𝐽 ↾t 𝐴) fLim ((nei‘(𝐽 ↾t 𝐴))‘{𝑋}))) | 
| 36 | 27 | snssd 4809 | . . . . 5
⊢ (𝜑 → {𝑋} ⊆ 𝐴) | 
| 37 | 10 | neitr 23188 | . . . . 5
⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝐶 ∧ {𝑋} ⊆ 𝐴) → ((nei‘(𝐽 ↾t 𝐴))‘{𝑋}) = (((nei‘𝐽)‘{𝑋}) ↾t 𝐴)) | 
| 38 | 9, 13, 36, 37 | syl3anc 1373 | . . . 4
⊢ (𝜑 → ((nei‘(𝐽 ↾t 𝐴))‘{𝑋}) = (((nei‘𝐽)‘{𝑋}) ↾t 𝐴)) | 
| 39 | 38 | oveq2d 7447 | . . 3
⊢ (𝜑 → ((𝐽 ↾t 𝐴) fLim ((nei‘(𝐽 ↾t 𝐴))‘{𝑋})) = ((𝐽 ↾t 𝐴) fLim (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))) | 
| 40 | 35, 39 | eleqtrd 2843 | . 2
⊢ (𝜑 → 𝑋 ∈ ((𝐽 ↾t 𝐴) fLim (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))) | 
| 41 | 2, 33, 40 | rspcdva 3623 | 1
⊢ (𝜑 → (𝐹‘𝑋) ∈ ((𝐾 fLimf (((nei‘𝐽)‘{𝑋}) ↾t 𝐴))‘𝐹)) |