| Step | Hyp | Ref
| Expression |
| 1 | | eqid 2737 |
. . . . . . 7
⊢ ∪ 𝐽 =
∪ 𝐽 |
| 2 | | eqid 2737 |
. . . . . . 7
⊢ ∪ 𝐾 =
∪ 𝐾 |
| 3 | 1, 2 | cnf 23254 |
. . . . . 6
⊢ (𝑓 ∈ (𝐽 Cn 𝐾) → 𝑓:∪ 𝐽⟶∪ 𝐾) |
| 4 | 3 | adantl 481 |
. . . . 5
⊢ (((𝐽 ∈ ran 𝑘Gen ∧
𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) → 𝑓:∪ 𝐽⟶∪ 𝐾) |
| 5 | | cnvimass 6100 |
. . . . . . . . 9
⊢ (◡𝑓 “ 𝑥) ⊆ dom 𝑓 |
| 6 | 4 | fdmd 6746 |
. . . . . . . . . 10
⊢ (((𝐽 ∈ ran 𝑘Gen ∧
𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) → dom 𝑓 = ∪ 𝐽) |
| 7 | 6 | adantr 480 |
. . . . . . . . 9
⊢ ((((𝐽 ∈ ran 𝑘Gen ∧
𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) → dom 𝑓 = ∪ 𝐽) |
| 8 | 5, 7 | sseqtrid 4026 |
. . . . . . . 8
⊢ ((((𝐽 ∈ ran 𝑘Gen ∧
𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) → (◡𝑓 “ 𝑥) ⊆ ∪ 𝐽) |
| 9 | | cnvresima 6250 |
. . . . . . . . . . . 12
⊢ (◡(𝑓 ↾ 𝑦) “ (𝑥 ∩ (𝑓 “ 𝑦))) = ((◡𝑓 “ (𝑥 ∩ (𝑓 “ 𝑦))) ∩ 𝑦) |
| 10 | 4 | ad2antrr 726 |
. . . . . . . . . . . . . . 15
⊢
(((((𝐽 ∈ ran
𝑘Gen ∧ 𝐾 ∈
Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) ∧ (𝑦 ∈ 𝒫 ∪ 𝐽
∧ (𝐽
↾t 𝑦)
∈ Comp)) → 𝑓:∪ 𝐽⟶∪ 𝐾) |
| 11 | | ffun 6739 |
. . . . . . . . . . . . . . 15
⊢ (𝑓:∪
𝐽⟶∪ 𝐾
→ Fun 𝑓) |
| 12 | | inpreima 7084 |
. . . . . . . . . . . . . . 15
⊢ (Fun
𝑓 → (◡𝑓 “ (𝑥 ∩ (𝑓 “ 𝑦))) = ((◡𝑓 “ 𝑥) ∩ (◡𝑓 “ (𝑓 “ 𝑦)))) |
| 13 | 10, 11, 12 | 3syl 18 |
. . . . . . . . . . . . . 14
⊢
(((((𝐽 ∈ ran
𝑘Gen ∧ 𝐾 ∈
Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) ∧ (𝑦 ∈ 𝒫 ∪ 𝐽
∧ (𝐽
↾t 𝑦)
∈ Comp)) → (◡𝑓 “ (𝑥 ∩ (𝑓 “ 𝑦))) = ((◡𝑓 “ 𝑥) ∩ (◡𝑓 “ (𝑓 “ 𝑦)))) |
| 14 | 13 | ineq1d 4219 |
. . . . . . . . . . . . 13
⊢
(((((𝐽 ∈ ran
𝑘Gen ∧ 𝐾 ∈
Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) ∧ (𝑦 ∈ 𝒫 ∪ 𝐽
∧ (𝐽
↾t 𝑦)
∈ Comp)) → ((◡𝑓 “ (𝑥 ∩ (𝑓 “ 𝑦))) ∩ 𝑦) = (((◡𝑓 “ 𝑥) ∩ (◡𝑓 “ (𝑓 “ 𝑦))) ∩ 𝑦)) |
| 15 | | in32 4230 |
. . . . . . . . . . . . . 14
⊢ (((◡𝑓 “ 𝑥) ∩ (◡𝑓 “ (𝑓 “ 𝑦))) ∩ 𝑦) = (((◡𝑓 “ 𝑥) ∩ 𝑦) ∩ (◡𝑓 “ (𝑓 “ 𝑦))) |
| 16 | | ssrin 4242 |
. . . . . . . . . . . . . . . . . 18
⊢ ((◡𝑓 “ 𝑥) ⊆ dom 𝑓 → ((◡𝑓 “ 𝑥) ∩ 𝑦) ⊆ (dom 𝑓 ∩ 𝑦)) |
| 17 | 5, 16 | ax-mp 5 |
. . . . . . . . . . . . . . . . 17
⊢ ((◡𝑓 “ 𝑥) ∩ 𝑦) ⊆ (dom 𝑓 ∩ 𝑦) |
| 18 | | dminss 6173 |
. . . . . . . . . . . . . . . . 17
⊢ (dom
𝑓 ∩ 𝑦) ⊆ (◡𝑓 “ (𝑓 “ 𝑦)) |
| 19 | 17, 18 | sstri 3993 |
. . . . . . . . . . . . . . . 16
⊢ ((◡𝑓 “ 𝑥) ∩ 𝑦) ⊆ (◡𝑓 “ (𝑓 “ 𝑦)) |
| 20 | 19 | a1i 11 |
. . . . . . . . . . . . . . 15
⊢
(((((𝐽 ∈ ran
𝑘Gen ∧ 𝐾 ∈
Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) ∧ (𝑦 ∈ 𝒫 ∪ 𝐽
∧ (𝐽
↾t 𝑦)
∈ Comp)) → ((◡𝑓 “ 𝑥) ∩ 𝑦) ⊆ (◡𝑓 “ (𝑓 “ 𝑦))) |
| 21 | | dfss2 3969 |
. . . . . . . . . . . . . . 15
⊢ (((◡𝑓 “ 𝑥) ∩ 𝑦) ⊆ (◡𝑓 “ (𝑓 “ 𝑦)) ↔ (((◡𝑓 “ 𝑥) ∩ 𝑦) ∩ (◡𝑓 “ (𝑓 “ 𝑦))) = ((◡𝑓 “ 𝑥) ∩ 𝑦)) |
| 22 | 20, 21 | sylib 218 |
. . . . . . . . . . . . . 14
⊢
(((((𝐽 ∈ ran
𝑘Gen ∧ 𝐾 ∈
Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) ∧ (𝑦 ∈ 𝒫 ∪ 𝐽
∧ (𝐽
↾t 𝑦)
∈ Comp)) → (((◡𝑓 “ 𝑥) ∩ 𝑦) ∩ (◡𝑓 “ (𝑓 “ 𝑦))) = ((◡𝑓 “ 𝑥) ∩ 𝑦)) |
| 23 | 15, 22 | eqtrid 2789 |
. . . . . . . . . . . . 13
⊢
(((((𝐽 ∈ ran
𝑘Gen ∧ 𝐾 ∈
Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) ∧ (𝑦 ∈ 𝒫 ∪ 𝐽
∧ (𝐽
↾t 𝑦)
∈ Comp)) → (((◡𝑓 “ 𝑥) ∩ (◡𝑓 “ (𝑓 “ 𝑦))) ∩ 𝑦) = ((◡𝑓 “ 𝑥) ∩ 𝑦)) |
| 24 | 14, 23 | eqtrd 2777 |
. . . . . . . . . . . 12
⊢
(((((𝐽 ∈ ran
𝑘Gen ∧ 𝐾 ∈
Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) ∧ (𝑦 ∈ 𝒫 ∪ 𝐽
∧ (𝐽
↾t 𝑦)
∈ Comp)) → ((◡𝑓 “ (𝑥 ∩ (𝑓 “ 𝑦))) ∩ 𝑦) = ((◡𝑓 “ 𝑥) ∩ 𝑦)) |
| 25 | 9, 24 | eqtrid 2789 |
. . . . . . . . . . 11
⊢
(((((𝐽 ∈ ran
𝑘Gen ∧ 𝐾 ∈
Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) ∧ (𝑦 ∈ 𝒫 ∪ 𝐽
∧ (𝐽
↾t 𝑦)
∈ Comp)) → (◡(𝑓 ↾ 𝑦) “ (𝑥 ∩ (𝑓 “ 𝑦))) = ((◡𝑓 “ 𝑥) ∩ 𝑦)) |
| 26 | | simpr 484 |
. . . . . . . . . . . . . . 15
⊢ (((𝐽 ∈ ran 𝑘Gen ∧
𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) → 𝑓 ∈ (𝐽 Cn 𝐾)) |
| 27 | 26 | ad2antrr 726 |
. . . . . . . . . . . . . 14
⊢
(((((𝐽 ∈ ran
𝑘Gen ∧ 𝐾 ∈
Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) ∧ (𝑦 ∈ 𝒫 ∪ 𝐽
∧ (𝐽
↾t 𝑦)
∈ Comp)) → 𝑓
∈ (𝐽 Cn 𝐾)) |
| 28 | | elpwi 4607 |
. . . . . . . . . . . . . . 15
⊢ (𝑦 ∈ 𝒫 ∪ 𝐽
→ 𝑦 ⊆ ∪ 𝐽) |
| 29 | 28 | ad2antrl 728 |
. . . . . . . . . . . . . 14
⊢
(((((𝐽 ∈ ran
𝑘Gen ∧ 𝐾 ∈
Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) ∧ (𝑦 ∈ 𝒫 ∪ 𝐽
∧ (𝐽
↾t 𝑦)
∈ Comp)) → 𝑦
⊆ ∪ 𝐽) |
| 30 | 1 | cnrest 23293 |
. . . . . . . . . . . . . 14
⊢ ((𝑓 ∈ (𝐽 Cn 𝐾) ∧ 𝑦 ⊆ ∪ 𝐽) → (𝑓 ↾ 𝑦) ∈ ((𝐽 ↾t 𝑦) Cn 𝐾)) |
| 31 | 27, 29, 30 | syl2anc 584 |
. . . . . . . . . . . . 13
⊢
(((((𝐽 ∈ ran
𝑘Gen ∧ 𝐾 ∈
Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) ∧ (𝑦 ∈ 𝒫 ∪ 𝐽
∧ (𝐽
↾t 𝑦)
∈ Comp)) → (𝑓
↾ 𝑦) ∈ ((𝐽 ↾t 𝑦) Cn 𝐾)) |
| 32 | | simpr 484 |
. . . . . . . . . . . . . . . 16
⊢ ((𝐽 ∈ ran 𝑘Gen ∧
𝐾 ∈ Top) → 𝐾 ∈ Top) |
| 33 | 32 | ad3antrrr 730 |
. . . . . . . . . . . . . . 15
⊢
(((((𝐽 ∈ ran
𝑘Gen ∧ 𝐾 ∈
Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) ∧ (𝑦 ∈ 𝒫 ∪ 𝐽
∧ (𝐽
↾t 𝑦)
∈ Comp)) → 𝐾
∈ Top) |
| 34 | | toptopon2 22924 |
. . . . . . . . . . . . . . 15
⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘∪ 𝐾)) |
| 35 | 33, 34 | sylib 218 |
. . . . . . . . . . . . . 14
⊢
(((((𝐽 ∈ ran
𝑘Gen ∧ 𝐾 ∈
Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) ∧ (𝑦 ∈ 𝒫 ∪ 𝐽
∧ (𝐽
↾t 𝑦)
∈ Comp)) → 𝐾
∈ (TopOn‘∪ 𝐾)) |
| 36 | | df-ima 5698 |
. . . . . . . . . . . . . . . 16
⊢ (𝑓 “ 𝑦) = ran (𝑓 ↾ 𝑦) |
| 37 | 36 | eqimss2i 4045 |
. . . . . . . . . . . . . . 15
⊢ ran
(𝑓 ↾ 𝑦) ⊆ (𝑓 “ 𝑦) |
| 38 | 37 | a1i 11 |
. . . . . . . . . . . . . 14
⊢
(((((𝐽 ∈ ran
𝑘Gen ∧ 𝐾 ∈
Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) ∧ (𝑦 ∈ 𝒫 ∪ 𝐽
∧ (𝐽
↾t 𝑦)
∈ Comp)) → ran (𝑓
↾ 𝑦) ⊆ (𝑓 “ 𝑦)) |
| 39 | | imassrn 6089 |
. . . . . . . . . . . . . . 15
⊢ (𝑓 “ 𝑦) ⊆ ran 𝑓 |
| 40 | 10 | frnd 6744 |
. . . . . . . . . . . . . . 15
⊢
(((((𝐽 ∈ ran
𝑘Gen ∧ 𝐾 ∈
Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) ∧ (𝑦 ∈ 𝒫 ∪ 𝐽
∧ (𝐽
↾t 𝑦)
∈ Comp)) → ran 𝑓
⊆ ∪ 𝐾) |
| 41 | 39, 40 | sstrid 3995 |
. . . . . . . . . . . . . 14
⊢
(((((𝐽 ∈ ran
𝑘Gen ∧ 𝐾 ∈
Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) ∧ (𝑦 ∈ 𝒫 ∪ 𝐽
∧ (𝐽
↾t 𝑦)
∈ Comp)) → (𝑓
“ 𝑦) ⊆ ∪ 𝐾) |
| 42 | | cnrest2 23294 |
. . . . . . . . . . . . . 14
⊢ ((𝐾 ∈ (TopOn‘∪ 𝐾)
∧ ran (𝑓 ↾ 𝑦) ⊆ (𝑓 “ 𝑦) ∧ (𝑓 “ 𝑦) ⊆ ∪ 𝐾) → ((𝑓 ↾ 𝑦) ∈ ((𝐽 ↾t 𝑦) Cn 𝐾) ↔ (𝑓 ↾ 𝑦) ∈ ((𝐽 ↾t 𝑦) Cn (𝐾 ↾t (𝑓 “ 𝑦))))) |
| 43 | 35, 38, 41, 42 | syl3anc 1373 |
. . . . . . . . . . . . 13
⊢
(((((𝐽 ∈ ran
𝑘Gen ∧ 𝐾 ∈
Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) ∧ (𝑦 ∈ 𝒫 ∪ 𝐽
∧ (𝐽
↾t 𝑦)
∈ Comp)) → ((𝑓
↾ 𝑦) ∈ ((𝐽 ↾t 𝑦) Cn 𝐾) ↔ (𝑓 ↾ 𝑦) ∈ ((𝐽 ↾t 𝑦) Cn (𝐾 ↾t (𝑓 “ 𝑦))))) |
| 44 | 31, 43 | mpbid 232 |
. . . . . . . . . . . 12
⊢
(((((𝐽 ∈ ran
𝑘Gen ∧ 𝐾 ∈
Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) ∧ (𝑦 ∈ 𝒫 ∪ 𝐽
∧ (𝐽
↾t 𝑦)
∈ Comp)) → (𝑓
↾ 𝑦) ∈ ((𝐽 ↾t 𝑦) Cn (𝐾 ↾t (𝑓 “ 𝑦)))) |
| 45 | | simplr 769 |
. . . . . . . . . . . . 13
⊢
(((((𝐽 ∈ ran
𝑘Gen ∧ 𝐾 ∈
Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) ∧ (𝑦 ∈ 𝒫 ∪ 𝐽
∧ (𝐽
↾t 𝑦)
∈ Comp)) → 𝑥
∈ (𝑘Gen‘𝐾)) |
| 46 | | simprr 773 |
. . . . . . . . . . . . . 14
⊢
(((((𝐽 ∈ ran
𝑘Gen ∧ 𝐾 ∈
Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) ∧ (𝑦 ∈ 𝒫 ∪ 𝐽
∧ (𝐽
↾t 𝑦)
∈ Comp)) → (𝐽
↾t 𝑦)
∈ Comp) |
| 47 | | imacmp 23405 |
. . . . . . . . . . . . . 14
⊢ ((𝑓 ∈ (𝐽 Cn 𝐾) ∧ (𝐽 ↾t 𝑦) ∈ Comp) → (𝐾 ↾t (𝑓 “ 𝑦)) ∈ Comp) |
| 48 | 27, 46, 47 | syl2anc 584 |
. . . . . . . . . . . . 13
⊢
(((((𝐽 ∈ ran
𝑘Gen ∧ 𝐾 ∈
Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) ∧ (𝑦 ∈ 𝒫 ∪ 𝐽
∧ (𝐽
↾t 𝑦)
∈ Comp)) → (𝐾
↾t (𝑓
“ 𝑦)) ∈
Comp) |
| 49 | | kgeni 23545 |
. . . . . . . . . . . . 13
⊢ ((𝑥 ∈
(𝑘Gen‘𝐾)
∧ (𝐾
↾t (𝑓
“ 𝑦)) ∈ Comp)
→ (𝑥 ∩ (𝑓 “ 𝑦)) ∈ (𝐾 ↾t (𝑓 “ 𝑦))) |
| 50 | 45, 48, 49 | syl2anc 584 |
. . . . . . . . . . . 12
⊢
(((((𝐽 ∈ ran
𝑘Gen ∧ 𝐾 ∈
Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) ∧ (𝑦 ∈ 𝒫 ∪ 𝐽
∧ (𝐽
↾t 𝑦)
∈ Comp)) → (𝑥
∩ (𝑓 “ 𝑦)) ∈ (𝐾 ↾t (𝑓 “ 𝑦))) |
| 51 | | cnima 23273 |
. . . . . . . . . . . 12
⊢ (((𝑓 ↾ 𝑦) ∈ ((𝐽 ↾t 𝑦) Cn (𝐾 ↾t (𝑓 “ 𝑦))) ∧ (𝑥 ∩ (𝑓 “ 𝑦)) ∈ (𝐾 ↾t (𝑓 “ 𝑦))) → (◡(𝑓 ↾ 𝑦) “ (𝑥 ∩ (𝑓 “ 𝑦))) ∈ (𝐽 ↾t 𝑦)) |
| 52 | 44, 50, 51 | syl2anc 584 |
. . . . . . . . . . 11
⊢
(((((𝐽 ∈ ran
𝑘Gen ∧ 𝐾 ∈
Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) ∧ (𝑦 ∈ 𝒫 ∪ 𝐽
∧ (𝐽
↾t 𝑦)
∈ Comp)) → (◡(𝑓 ↾ 𝑦) “ (𝑥 ∩ (𝑓 “ 𝑦))) ∈ (𝐽 ↾t 𝑦)) |
| 53 | 25, 52 | eqeltrrd 2842 |
. . . . . . . . . 10
⊢
(((((𝐽 ∈ ran
𝑘Gen ∧ 𝐾 ∈
Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) ∧ (𝑦 ∈ 𝒫 ∪ 𝐽
∧ (𝐽
↾t 𝑦)
∈ Comp)) → ((◡𝑓 “ 𝑥) ∩ 𝑦) ∈ (𝐽 ↾t 𝑦)) |
| 54 | 53 | expr 456 |
. . . . . . . . 9
⊢
(((((𝐽 ∈ ran
𝑘Gen ∧ 𝐾 ∈
Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) ∧ 𝑦 ∈ 𝒫 ∪ 𝐽)
→ ((𝐽
↾t 𝑦)
∈ Comp → ((◡𝑓 “ 𝑥) ∩ 𝑦) ∈ (𝐽 ↾t 𝑦))) |
| 55 | 54 | ralrimiva 3146 |
. . . . . . . 8
⊢ ((((𝐽 ∈ ran 𝑘Gen ∧
𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) → ∀𝑦 ∈ 𝒫 ∪ 𝐽((𝐽 ↾t 𝑦) ∈ Comp → ((◡𝑓 “ 𝑥) ∩ 𝑦) ∈ (𝐽 ↾t 𝑦))) |
| 56 | | kgentop 23550 |
. . . . . . . . . . 11
⊢ (𝐽 ∈ ran 𝑘Gen →
𝐽 ∈
Top) |
| 57 | 56 | ad3antrrr 730 |
. . . . . . . . . 10
⊢ ((((𝐽 ∈ ran 𝑘Gen ∧
𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) → 𝐽 ∈ Top) |
| 58 | | toptopon2 22924 |
. . . . . . . . . 10
⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽)) |
| 59 | 57, 58 | sylib 218 |
. . . . . . . . 9
⊢ ((((𝐽 ∈ ran 𝑘Gen ∧
𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) → 𝐽 ∈ (TopOn‘∪ 𝐽)) |
| 60 | | elkgen 23544 |
. . . . . . . . 9
⊢ (𝐽 ∈ (TopOn‘∪ 𝐽)
→ ((◡𝑓 “ 𝑥) ∈ (𝑘Gen‘𝐽) ↔ ((◡𝑓 “ 𝑥) ⊆ ∪ 𝐽 ∧ ∀𝑦 ∈ 𝒫 ∪ 𝐽((𝐽 ↾t 𝑦) ∈ Comp → ((◡𝑓 “ 𝑥) ∩ 𝑦) ∈ (𝐽 ↾t 𝑦))))) |
| 61 | 59, 60 | syl 17 |
. . . . . . . 8
⊢ ((((𝐽 ∈ ran 𝑘Gen ∧
𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) → ((◡𝑓 “ 𝑥) ∈ (𝑘Gen‘𝐽) ↔ ((◡𝑓 “ 𝑥) ⊆ ∪ 𝐽 ∧ ∀𝑦 ∈ 𝒫 ∪ 𝐽((𝐽 ↾t 𝑦) ∈ Comp → ((◡𝑓 “ 𝑥) ∩ 𝑦) ∈ (𝐽 ↾t 𝑦))))) |
| 62 | 8, 55, 61 | mpbir2and 713 |
. . . . . . 7
⊢ ((((𝐽 ∈ ran 𝑘Gen ∧
𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) → (◡𝑓 “ 𝑥) ∈ (𝑘Gen‘𝐽)) |
| 63 | | kgenidm 23555 |
. . . . . . . 8
⊢ (𝐽 ∈ ran 𝑘Gen →
(𝑘Gen‘𝐽) =
𝐽) |
| 64 | 63 | ad3antrrr 730 |
. . . . . . 7
⊢ ((((𝐽 ∈ ran 𝑘Gen ∧
𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) → (𝑘Gen‘𝐽) = 𝐽) |
| 65 | 62, 64 | eleqtrd 2843 |
. . . . . 6
⊢ ((((𝐽 ∈ ran 𝑘Gen ∧
𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) ∧ 𝑥 ∈ (𝑘Gen‘𝐾)) → (◡𝑓 “ 𝑥) ∈ 𝐽) |
| 66 | 65 | ralrimiva 3146 |
. . . . 5
⊢ (((𝐽 ∈ ran 𝑘Gen ∧
𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) → ∀𝑥 ∈ (𝑘Gen‘𝐾)(◡𝑓 “ 𝑥) ∈ 𝐽) |
| 67 | 56, 58 | sylib 218 |
. . . . . . 7
⊢ (𝐽 ∈ ran 𝑘Gen →
𝐽 ∈ (TopOn‘∪ 𝐽)) |
| 68 | | kgentopon 23546 |
. . . . . . . 8
⊢ (𝐾 ∈ (TopOn‘∪ 𝐾)
→ (𝑘Gen‘𝐾) ∈ (TopOn‘∪ 𝐾)) |
| 69 | 34, 68 | sylbi 217 |
. . . . . . 7
⊢ (𝐾 ∈ Top →
(𝑘Gen‘𝐾)
∈ (TopOn‘∪ 𝐾)) |
| 70 | | iscn 23243 |
. . . . . . 7
⊢ ((𝐽 ∈ (TopOn‘∪ 𝐽)
∧ (𝑘Gen‘𝐾) ∈ (TopOn‘∪ 𝐾))
→ (𝑓 ∈ (𝐽 Cn (𝑘Gen‘𝐾)) ↔ (𝑓:∪ 𝐽⟶∪ 𝐾
∧ ∀𝑥 ∈
(𝑘Gen‘𝐾)(◡𝑓 “ 𝑥) ∈ 𝐽))) |
| 71 | 67, 69, 70 | syl2an 596 |
. . . . . 6
⊢ ((𝐽 ∈ ran 𝑘Gen ∧
𝐾 ∈ Top) → (𝑓 ∈ (𝐽 Cn (𝑘Gen‘𝐾)) ↔ (𝑓:∪ 𝐽⟶∪ 𝐾
∧ ∀𝑥 ∈
(𝑘Gen‘𝐾)(◡𝑓 “ 𝑥) ∈ 𝐽))) |
| 72 | 71 | adantr 480 |
. . . . 5
⊢ (((𝐽 ∈ ran 𝑘Gen ∧
𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) → (𝑓 ∈ (𝐽 Cn (𝑘Gen‘𝐾)) ↔ (𝑓:∪ 𝐽⟶∪ 𝐾
∧ ∀𝑥 ∈
(𝑘Gen‘𝐾)(◡𝑓 “ 𝑥) ∈ 𝐽))) |
| 73 | 4, 66, 72 | mpbir2and 713 |
. . . 4
⊢ (((𝐽 ∈ ran 𝑘Gen ∧
𝐾 ∈ Top) ∧ 𝑓 ∈ (𝐽 Cn 𝐾)) → 𝑓 ∈ (𝐽 Cn (𝑘Gen‘𝐾))) |
| 74 | 73 | ex 412 |
. . 3
⊢ ((𝐽 ∈ ran 𝑘Gen ∧
𝐾 ∈ Top) → (𝑓 ∈ (𝐽 Cn 𝐾) → 𝑓 ∈ (𝐽 Cn (𝑘Gen‘𝐾)))) |
| 75 | 74 | ssrdv 3989 |
. 2
⊢ ((𝐽 ∈ ran 𝑘Gen ∧
𝐾 ∈ Top) → (𝐽 Cn 𝐾) ⊆ (𝐽 Cn (𝑘Gen‘𝐾))) |
| 76 | 69 | adantl 481 |
. . . 4
⊢ ((𝐽 ∈ ran 𝑘Gen ∧
𝐾 ∈ Top) →
(𝑘Gen‘𝐾)
∈ (TopOn‘∪ 𝐾)) |
| 77 | | toponcom 22934 |
. . . 4
⊢ ((𝐾 ∈ Top ∧
(𝑘Gen‘𝐾)
∈ (TopOn‘∪ 𝐾)) → 𝐾 ∈ (TopOn‘∪ (𝑘Gen‘𝐾))) |
| 78 | 32, 76, 77 | syl2anc 584 |
. . 3
⊢ ((𝐽 ∈ ran 𝑘Gen ∧
𝐾 ∈ Top) → 𝐾 ∈ (TopOn‘∪ (𝑘Gen‘𝐾))) |
| 79 | | kgenss 23551 |
. . . 4
⊢ (𝐾 ∈ Top → 𝐾 ⊆
(𝑘Gen‘𝐾)) |
| 80 | 79 | adantl 481 |
. . 3
⊢ ((𝐽 ∈ ran 𝑘Gen ∧
𝐾 ∈ Top) → 𝐾 ⊆
(𝑘Gen‘𝐾)) |
| 81 | | eqid 2737 |
. . . 4
⊢ ∪ (𝑘Gen‘𝐾) = ∪
(𝑘Gen‘𝐾) |
| 82 | 81 | cnss2 23285 |
. . 3
⊢ ((𝐾 ∈ (TopOn‘∪ (𝑘Gen‘𝐾)) ∧ 𝐾 ⊆ (𝑘Gen‘𝐾)) → (𝐽 Cn (𝑘Gen‘𝐾)) ⊆ (𝐽 Cn 𝐾)) |
| 83 | 78, 80, 82 | syl2anc 584 |
. 2
⊢ ((𝐽 ∈ ran 𝑘Gen ∧
𝐾 ∈ Top) → (𝐽 Cn (𝑘Gen‘𝐾)) ⊆ (𝐽 Cn 𝐾)) |
| 84 | 75, 83 | eqssd 4001 |
1
⊢ ((𝐽 ∈ ran 𝑘Gen ∧
𝐾 ∈ Top) → (𝐽 Cn 𝐾) = (𝐽 Cn (𝑘Gen‘𝐾))) |