Proof of Theorem cnfcf
| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | cncnp 23289 | . 2
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑥 ∈ 𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥)))) | 
| 2 |  | simplr 768 | . . . . . 6
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑥 ∈ 𝑋) → 𝐹:𝑋⟶𝑌) | 
| 3 |  | cnpfcf 24050 | . . . . . . 7
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝑥 ∈ 𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝑥 ∈ (𝐽 fClus 𝑓) → (𝐹‘𝑥) ∈ ((𝐾 fClusf 𝑓)‘𝐹))))) | 
| 4 | 3 | ad4ant124 1173 | . . . . . 6
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑥 ∈ 𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝑥 ∈ (𝐽 fClus 𝑓) → (𝐹‘𝑥) ∈ ((𝐾 fClusf 𝑓)‘𝐹))))) | 
| 5 | 2, 4 | mpbirand 707 | . . . . 5
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) ∧ 𝑥 ∈ 𝑋) → (𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥) ↔ ∀𝑓 ∈ (Fil‘𝑋)(𝑥 ∈ (𝐽 fClus 𝑓) → (𝐹‘𝑥) ∈ ((𝐾 fClusf 𝑓)‘𝐹)))) | 
| 6 | 5 | ralbidva 3175 | . . . 4
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) → (∀𝑥 ∈ 𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥) ↔ ∀𝑥 ∈ 𝑋 ∀𝑓 ∈ (Fil‘𝑋)(𝑥 ∈ (𝐽 fClus 𝑓) → (𝐹‘𝑥) ∈ ((𝐾 fClusf 𝑓)‘𝐹)))) | 
| 7 |  | ralcom 3288 | . . . . 5
⊢
(∀𝑥 ∈
𝑋 ∀𝑓 ∈ (Fil‘𝑋)(𝑥 ∈ (𝐽 fClus 𝑓) → (𝐹‘𝑥) ∈ ((𝐾 fClusf 𝑓)‘𝐹)) ↔ ∀𝑓 ∈ (Fil‘𝑋)∀𝑥 ∈ 𝑋 (𝑥 ∈ (𝐽 fClus 𝑓) → (𝐹‘𝑥) ∈ ((𝐾 fClusf 𝑓)‘𝐹))) | 
| 8 |  | eqid 2736 | . . . . . . . . . . . 12
⊢ ∪ 𝐽 =
∪ 𝐽 | 
| 9 | 8 | fclselbas 24025 | . . . . . . . . . . 11
⊢ (𝑥 ∈ (𝐽 fClus 𝑓) → 𝑥 ∈ ∪ 𝐽) | 
| 10 |  | toponuni 22921 | . . . . . . . . . . . . 13
⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽) | 
| 11 | 10 | ad2antrr 726 | . . . . . . . . . . . 12
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) → 𝑋 = ∪ 𝐽) | 
| 12 | 11 | eleq2d 2826 | . . . . . . . . . . 11
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) → (𝑥 ∈ 𝑋 ↔ 𝑥 ∈ ∪ 𝐽)) | 
| 13 | 9, 12 | imbitrrid 246 | . . . . . . . . . 10
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) → (𝑥 ∈ (𝐽 fClus 𝑓) → 𝑥 ∈ 𝑋)) | 
| 14 | 13 | pm4.71rd 562 | . . . . . . . . 9
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) → (𝑥 ∈ (𝐽 fClus 𝑓) ↔ (𝑥 ∈ 𝑋 ∧ 𝑥 ∈ (𝐽 fClus 𝑓)))) | 
| 15 | 14 | imbi1d 341 | . . . . . . . 8
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) → ((𝑥 ∈ (𝐽 fClus 𝑓) → (𝐹‘𝑥) ∈ ((𝐾 fClusf 𝑓)‘𝐹)) ↔ ((𝑥 ∈ 𝑋 ∧ 𝑥 ∈ (𝐽 fClus 𝑓)) → (𝐹‘𝑥) ∈ ((𝐾 fClusf 𝑓)‘𝐹)))) | 
| 16 |  | impexp 450 | . . . . . . . 8
⊢ (((𝑥 ∈ 𝑋 ∧ 𝑥 ∈ (𝐽 fClus 𝑓)) → (𝐹‘𝑥) ∈ ((𝐾 fClusf 𝑓)‘𝐹)) ↔ (𝑥 ∈ 𝑋 → (𝑥 ∈ (𝐽 fClus 𝑓) → (𝐹‘𝑥) ∈ ((𝐾 fClusf 𝑓)‘𝐹)))) | 
| 17 | 15, 16 | bitrdi 287 | . . . . . . 7
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) → ((𝑥 ∈ (𝐽 fClus 𝑓) → (𝐹‘𝑥) ∈ ((𝐾 fClusf 𝑓)‘𝐹)) ↔ (𝑥 ∈ 𝑋 → (𝑥 ∈ (𝐽 fClus 𝑓) → (𝐹‘𝑥) ∈ ((𝐾 fClusf 𝑓)‘𝐹))))) | 
| 18 | 17 | ralbidv2 3173 | . . . . . 6
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) → (∀𝑥 ∈ (𝐽 fClus 𝑓)(𝐹‘𝑥) ∈ ((𝐾 fClusf 𝑓)‘𝐹) ↔ ∀𝑥 ∈ 𝑋 (𝑥 ∈ (𝐽 fClus 𝑓) → (𝐹‘𝑥) ∈ ((𝐾 fClusf 𝑓)‘𝐹)))) | 
| 19 | 18 | ralbidv 3177 | . . . . 5
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) → (∀𝑓 ∈ (Fil‘𝑋)∀𝑥 ∈ (𝐽 fClus 𝑓)(𝐹‘𝑥) ∈ ((𝐾 fClusf 𝑓)‘𝐹) ↔ ∀𝑓 ∈ (Fil‘𝑋)∀𝑥 ∈ 𝑋 (𝑥 ∈ (𝐽 fClus 𝑓) → (𝐹‘𝑥) ∈ ((𝐾 fClusf 𝑓)‘𝐹)))) | 
| 20 | 7, 19 | bitr4id 290 | . . . 4
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) → (∀𝑥 ∈ 𝑋 ∀𝑓 ∈ (Fil‘𝑋)(𝑥 ∈ (𝐽 fClus 𝑓) → (𝐹‘𝑥) ∈ ((𝐾 fClusf 𝑓)‘𝐹)) ↔ ∀𝑓 ∈ (Fil‘𝑋)∀𝑥 ∈ (𝐽 fClus 𝑓)(𝐹‘𝑥) ∈ ((𝐾 fClusf 𝑓)‘𝐹))) | 
| 21 | 6, 20 | bitrd 279 | . . 3
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) ∧ 𝐹:𝑋⟶𝑌) → (∀𝑥 ∈ 𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥) ↔ ∀𝑓 ∈ (Fil‘𝑋)∀𝑥 ∈ (𝐽 fClus 𝑓)(𝐹‘𝑥) ∈ ((𝐾 fClusf 𝑓)‘𝐹))) | 
| 22 | 21 | pm5.32da 579 | . 2
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → ((𝐹:𝑋⟶𝑌 ∧ ∀𝑥 ∈ 𝑋 𝐹 ∈ ((𝐽 CnP 𝐾)‘𝑥)) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑓 ∈ (Fil‘𝑋)∀𝑥 ∈ (𝐽 fClus 𝑓)(𝐹‘𝑥) ∈ ((𝐾 fClusf 𝑓)‘𝐹)))) | 
| 23 | 1, 22 | bitrd 279 | 1
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽 Cn 𝐾) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑓 ∈ (Fil‘𝑋)∀𝑥 ∈ (𝐽 fClus 𝑓)(𝐹‘𝑥) ∈ ((𝐾 fClusf 𝑓)‘𝐹)))) |