| Step | Hyp | Ref
| Expression |
| 1 | | fcfval 24041 |
. . 3
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → ((𝐽 fClusf 𝐿)‘𝐹) = (𝐽 fClus ((𝑋 FilMap 𝐹)‘𝐿))) |
| 2 | 1 | eleq2d 2827 |
. 2
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → (𝐴 ∈ ((𝐽 fClusf 𝐿)‘𝐹) ↔ 𝐴 ∈ (𝐽 fClus ((𝑋 FilMap 𝐹)‘𝐿)))) |
| 3 | | simp1 1137 |
. . 3
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → 𝐽 ∈ (TopOn‘𝑋)) |
| 4 | | toponmax 22932 |
. . . 4
⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 ∈ 𝐽) |
| 5 | | filfbas 23856 |
. . . 4
⊢ (𝐿 ∈ (Fil‘𝑌) → 𝐿 ∈ (fBas‘𝑌)) |
| 6 | | id 22 |
. . . 4
⊢ (𝐹:𝑌⟶𝑋 → 𝐹:𝑌⟶𝑋) |
| 7 | | fmfil 23952 |
. . . 4
⊢ ((𝑋 ∈ 𝐽 ∧ 𝐿 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → ((𝑋 FilMap 𝐹)‘𝐿) ∈ (Fil‘𝑋)) |
| 8 | 4, 5, 6, 7 | syl3an 1161 |
. . 3
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → ((𝑋 FilMap 𝐹)‘𝐿) ∈ (Fil‘𝑋)) |
| 9 | | fclsopn 24022 |
. . 3
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ ((𝑋 FilMap 𝐹)‘𝐿) ∈ (Fil‘𝑋)) → (𝐴 ∈ (𝐽 fClus ((𝑋 FilMap 𝐹)‘𝐿)) ↔ (𝐴 ∈ 𝑋 ∧ ∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → ∀𝑥 ∈ ((𝑋 FilMap 𝐹)‘𝐿)(𝑜 ∩ 𝑥) ≠ ∅)))) |
| 10 | 3, 8, 9 | syl2anc 584 |
. 2
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → (𝐴 ∈ (𝐽 fClus ((𝑋 FilMap 𝐹)‘𝐿)) ↔ (𝐴 ∈ 𝑋 ∧ ∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → ∀𝑥 ∈ ((𝑋 FilMap 𝐹)‘𝐿)(𝑜 ∩ 𝑥) ≠ ∅)))) |
| 11 | | simpll1 1213 |
. . . . . . . . . 10
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑜 ∈ 𝐽) ∧ 𝑠 ∈ 𝐿) → 𝐽 ∈ (TopOn‘𝑋)) |
| 12 | 11, 4 | syl 17 |
. . . . . . . . 9
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑜 ∈ 𝐽) ∧ 𝑠 ∈ 𝐿) → 𝑋 ∈ 𝐽) |
| 13 | | simpll2 1214 |
. . . . . . . . . 10
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑜 ∈ 𝐽) ∧ 𝑠 ∈ 𝐿) → 𝐿 ∈ (Fil‘𝑌)) |
| 14 | 13, 5 | syl 17 |
. . . . . . . . 9
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑜 ∈ 𝐽) ∧ 𝑠 ∈ 𝐿) → 𝐿 ∈ (fBas‘𝑌)) |
| 15 | | simpll3 1215 |
. . . . . . . . 9
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑜 ∈ 𝐽) ∧ 𝑠 ∈ 𝐿) → 𝐹:𝑌⟶𝑋) |
| 16 | | simpl2 1193 |
. . . . . . . . . . . 12
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑜 ∈ 𝐽) → 𝐿 ∈ (Fil‘𝑌)) |
| 17 | | fgfil 23883 |
. . . . . . . . . . . 12
⊢ (𝐿 ∈ (Fil‘𝑌) → (𝑌filGen𝐿) = 𝐿) |
| 18 | 16, 17 | syl 17 |
. . . . . . . . . . 11
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑜 ∈ 𝐽) → (𝑌filGen𝐿) = 𝐿) |
| 19 | 18 | eleq2d 2827 |
. . . . . . . . . 10
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑜 ∈ 𝐽) → (𝑠 ∈ (𝑌filGen𝐿) ↔ 𝑠 ∈ 𝐿)) |
| 20 | 19 | biimpar 477 |
. . . . . . . . 9
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑜 ∈ 𝐽) ∧ 𝑠 ∈ 𝐿) → 𝑠 ∈ (𝑌filGen𝐿)) |
| 21 | | eqid 2737 |
. . . . . . . . . 10
⊢ (𝑌filGen𝐿) = (𝑌filGen𝐿) |
| 22 | 21 | imaelfm 23959 |
. . . . . . . . 9
⊢ (((𝑋 ∈ 𝐽 ∧ 𝐿 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑠 ∈ (𝑌filGen𝐿)) → (𝐹 “ 𝑠) ∈ ((𝑋 FilMap 𝐹)‘𝐿)) |
| 23 | 12, 14, 15, 20, 22 | syl31anc 1375 |
. . . . . . . 8
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑜 ∈ 𝐽) ∧ 𝑠 ∈ 𝐿) → (𝐹 “ 𝑠) ∈ ((𝑋 FilMap 𝐹)‘𝐿)) |
| 24 | | ineq2 4214 |
. . . . . . . . . 10
⊢ (𝑥 = (𝐹 “ 𝑠) → (𝑜 ∩ 𝑥) = (𝑜 ∩ (𝐹 “ 𝑠))) |
| 25 | 24 | neeq1d 3000 |
. . . . . . . . 9
⊢ (𝑥 = (𝐹 “ 𝑠) → ((𝑜 ∩ 𝑥) ≠ ∅ ↔ (𝑜 ∩ (𝐹 “ 𝑠)) ≠ ∅)) |
| 26 | 25 | rspcv 3618 |
. . . . . . . 8
⊢ ((𝐹 “ 𝑠) ∈ ((𝑋 FilMap 𝐹)‘𝐿) → (∀𝑥 ∈ ((𝑋 FilMap 𝐹)‘𝐿)(𝑜 ∩ 𝑥) ≠ ∅ → (𝑜 ∩ (𝐹 “ 𝑠)) ≠ ∅)) |
| 27 | 23, 26 | syl 17 |
. . . . . . 7
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑜 ∈ 𝐽) ∧ 𝑠 ∈ 𝐿) → (∀𝑥 ∈ ((𝑋 FilMap 𝐹)‘𝐿)(𝑜 ∩ 𝑥) ≠ ∅ → (𝑜 ∩ (𝐹 “ 𝑠)) ≠ ∅)) |
| 28 | 27 | ralrimdva 3154 |
. . . . . 6
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑜 ∈ 𝐽) → (∀𝑥 ∈ ((𝑋 FilMap 𝐹)‘𝐿)(𝑜 ∩ 𝑥) ≠ ∅ → ∀𝑠 ∈ 𝐿 (𝑜 ∩ (𝐹 “ 𝑠)) ≠ ∅)) |
| 29 | | elfm 23955 |
. . . . . . . . . . 11
⊢ ((𝑋 ∈ 𝐽 ∧ 𝐿 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → (𝑥 ∈ ((𝑋 FilMap 𝐹)‘𝐿) ↔ (𝑥 ⊆ 𝑋 ∧ ∃𝑠 ∈ 𝐿 (𝐹 “ 𝑠) ⊆ 𝑥))) |
| 30 | 4, 5, 6, 29 | syl3an 1161 |
. . . . . . . . . 10
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → (𝑥 ∈ ((𝑋 FilMap 𝐹)‘𝐿) ↔ (𝑥 ⊆ 𝑋 ∧ ∃𝑠 ∈ 𝐿 (𝐹 “ 𝑠) ⊆ 𝑥))) |
| 31 | 30 | adantr 480 |
. . . . . . . . 9
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑜 ∈ 𝐽) → (𝑥 ∈ ((𝑋 FilMap 𝐹)‘𝐿) ↔ (𝑥 ⊆ 𝑋 ∧ ∃𝑠 ∈ 𝐿 (𝐹 “ 𝑠) ⊆ 𝑥))) |
| 32 | 31 | simplbda 499 |
. . . . . . . 8
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑜 ∈ 𝐽) ∧ 𝑥 ∈ ((𝑋 FilMap 𝐹)‘𝐿)) → ∃𝑠 ∈ 𝐿 (𝐹 “ 𝑠) ⊆ 𝑥) |
| 33 | | r19.29r 3116 |
. . . . . . . . . 10
⊢
((∃𝑠 ∈
𝐿 (𝐹 “ 𝑠) ⊆ 𝑥 ∧ ∀𝑠 ∈ 𝐿 (𝑜 ∩ (𝐹 “ 𝑠)) ≠ ∅) → ∃𝑠 ∈ 𝐿 ((𝐹 “ 𝑠) ⊆ 𝑥 ∧ (𝑜 ∩ (𝐹 “ 𝑠)) ≠ ∅)) |
| 34 | | sslin 4243 |
. . . . . . . . . . . 12
⊢ ((𝐹 “ 𝑠) ⊆ 𝑥 → (𝑜 ∩ (𝐹 “ 𝑠)) ⊆ (𝑜 ∩ 𝑥)) |
| 35 | | ssn0 4404 |
. . . . . . . . . . . 12
⊢ (((𝑜 ∩ (𝐹 “ 𝑠)) ⊆ (𝑜 ∩ 𝑥) ∧ (𝑜 ∩ (𝐹 “ 𝑠)) ≠ ∅) → (𝑜 ∩ 𝑥) ≠ ∅) |
| 36 | 34, 35 | sylan 580 |
. . . . . . . . . . 11
⊢ (((𝐹 “ 𝑠) ⊆ 𝑥 ∧ (𝑜 ∩ (𝐹 “ 𝑠)) ≠ ∅) → (𝑜 ∩ 𝑥) ≠ ∅) |
| 37 | 36 | rexlimivw 3151 |
. . . . . . . . . 10
⊢
(∃𝑠 ∈
𝐿 ((𝐹 “ 𝑠) ⊆ 𝑥 ∧ (𝑜 ∩ (𝐹 “ 𝑠)) ≠ ∅) → (𝑜 ∩ 𝑥) ≠ ∅) |
| 38 | 33, 37 | syl 17 |
. . . . . . . . 9
⊢
((∃𝑠 ∈
𝐿 (𝐹 “ 𝑠) ⊆ 𝑥 ∧ ∀𝑠 ∈ 𝐿 (𝑜 ∩ (𝐹 “ 𝑠)) ≠ ∅) → (𝑜 ∩ 𝑥) ≠ ∅) |
| 39 | 38 | ex 412 |
. . . . . . . 8
⊢
(∃𝑠 ∈
𝐿 (𝐹 “ 𝑠) ⊆ 𝑥 → (∀𝑠 ∈ 𝐿 (𝑜 ∩ (𝐹 “ 𝑠)) ≠ ∅ → (𝑜 ∩ 𝑥) ≠ ∅)) |
| 40 | 32, 39 | syl 17 |
. . . . . . 7
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑜 ∈ 𝐽) ∧ 𝑥 ∈ ((𝑋 FilMap 𝐹)‘𝐿)) → (∀𝑠 ∈ 𝐿 (𝑜 ∩ (𝐹 “ 𝑠)) ≠ ∅ → (𝑜 ∩ 𝑥) ≠ ∅)) |
| 41 | 40 | ralrimdva 3154 |
. . . . . 6
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑜 ∈ 𝐽) → (∀𝑠 ∈ 𝐿 (𝑜 ∩ (𝐹 “ 𝑠)) ≠ ∅ → ∀𝑥 ∈ ((𝑋 FilMap 𝐹)‘𝐿)(𝑜 ∩ 𝑥) ≠ ∅)) |
| 42 | 28, 41 | impbid 212 |
. . . . 5
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑜 ∈ 𝐽) → (∀𝑥 ∈ ((𝑋 FilMap 𝐹)‘𝐿)(𝑜 ∩ 𝑥) ≠ ∅ ↔ ∀𝑠 ∈ 𝐿 (𝑜 ∩ (𝐹 “ 𝑠)) ≠ ∅)) |
| 43 | 42 | imbi2d 340 |
. . . 4
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) ∧ 𝑜 ∈ 𝐽) → ((𝐴 ∈ 𝑜 → ∀𝑥 ∈ ((𝑋 FilMap 𝐹)‘𝐿)(𝑜 ∩ 𝑥) ≠ ∅) ↔ (𝐴 ∈ 𝑜 → ∀𝑠 ∈ 𝐿 (𝑜 ∩ (𝐹 “ 𝑠)) ≠ ∅))) |
| 44 | 43 | ralbidva 3176 |
. . 3
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → (∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → ∀𝑥 ∈ ((𝑋 FilMap 𝐹)‘𝐿)(𝑜 ∩ 𝑥) ≠ ∅) ↔ ∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → ∀𝑠 ∈ 𝐿 (𝑜 ∩ (𝐹 “ 𝑠)) ≠ ∅))) |
| 45 | 44 | anbi2d 630 |
. 2
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → ((𝐴 ∈ 𝑋 ∧ ∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → ∀𝑥 ∈ ((𝑋 FilMap 𝐹)‘𝐿)(𝑜 ∩ 𝑥) ≠ ∅)) ↔ (𝐴 ∈ 𝑋 ∧ ∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → ∀𝑠 ∈ 𝐿 (𝑜 ∩ (𝐹 “ 𝑠)) ≠ ∅)))) |
| 46 | 2, 10, 45 | 3bitrd 305 |
1
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐿 ∈ (Fil‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → (𝐴 ∈ ((𝐽 fClusf 𝐿)‘𝐹) ↔ (𝐴 ∈ 𝑋 ∧ ∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → ∀𝑠 ∈ 𝐿 (𝑜 ∩ (𝐹 “ 𝑠)) ≠ ∅)))) |