Step | Hyp | Ref
| Expression |
1 | | f1ofo 6144 |
. . . . 5
⊢ (𝐹:𝑋–1-1-onto→𝑌 → 𝐹:𝑋–onto→𝑌) |
2 | | qtopcmp.1 |
. . . . . . 7
⊢ 𝑋 = ∪
𝐽 |
3 | 2 | elqtop2 21504 |
. . . . . 6
⊢ ((𝐽 ∈ TopBases ∧ 𝐹:𝑋–onto→𝑌) → (𝑥 ∈ (𝐽 qTop 𝐹) ↔ (𝑥 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑥) ∈ 𝐽))) |
4 | 2 | elqtop2 21504 |
. . . . . 6
⊢ ((𝐽 ∈ TopBases ∧ 𝐹:𝑋–onto→𝑌) → (𝑦 ∈ (𝐽 qTop 𝐹) ↔ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽))) |
5 | 3, 4 | anbi12d 747 |
. . . . 5
⊢ ((𝐽 ∈ TopBases ∧ 𝐹:𝑋–onto→𝑌) → ((𝑥 ∈ (𝐽 qTop 𝐹) ∧ 𝑦 ∈ (𝐽 qTop 𝐹)) ↔ ((𝑥 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑥) ∈ 𝐽) ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)))) |
6 | 1, 5 | sylan2 491 |
. . . 4
⊢ ((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) → ((𝑥 ∈ (𝐽 qTop 𝐹) ∧ 𝑦 ∈ (𝐽 qTop 𝐹)) ↔ ((𝑥 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑥) ∈ 𝐽) ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)))) |
7 | | simpl1l 1112 |
. . . . . . . . 9
⊢ ((((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑥 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑥) ∈ 𝐽) ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) → 𝐽 ∈ TopBases) |
8 | | simpl2r 1115 |
. . . . . . . . 9
⊢ ((((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑥 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑥) ∈ 𝐽) ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) → (◡𝐹 “ 𝑥) ∈ 𝐽) |
9 | | simpl3r 1117 |
. . . . . . . . 9
⊢ ((((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑥 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑥) ∈ 𝐽) ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) → (◡𝐹 “ 𝑦) ∈ 𝐽) |
10 | | simpl1r 1113 |
. . . . . . . . . . . 12
⊢ ((((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑥 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑥) ∈ 𝐽) ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) → 𝐹:𝑋–1-1-onto→𝑌) |
11 | | f1ocnv 6149 |
. . . . . . . . . . . 12
⊢ (𝐹:𝑋–1-1-onto→𝑌 → ◡𝐹:𝑌–1-1-onto→𝑋) |
12 | | f1ofn 6138 |
. . . . . . . . . . . 12
⊢ (◡𝐹:𝑌–1-1-onto→𝑋 → ◡𝐹 Fn 𝑌) |
13 | 10, 11, 12 | 3syl 18 |
. . . . . . . . . . 11
⊢ ((((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑥 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑥) ∈ 𝐽) ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) → ◡𝐹 Fn 𝑌) |
14 | | simpl2l 1114 |
. . . . . . . . . . 11
⊢ ((((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑥 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑥) ∈ 𝐽) ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) → 𝑥 ⊆ 𝑌) |
15 | | inss1 3833 |
. . . . . . . . . . . 12
⊢ (𝑥 ∩ 𝑦) ⊆ 𝑥 |
16 | | simpr 477 |
. . . . . . . . . . . 12
⊢ ((((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑥 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑥) ∈ 𝐽) ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) → 𝑧 ∈ (𝑥 ∩ 𝑦)) |
17 | 15, 16 | sseldi 3601 |
. . . . . . . . . . 11
⊢ ((((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑥 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑥) ∈ 𝐽) ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) → 𝑧 ∈ 𝑥) |
18 | | fnfvima 6496 |
. . . . . . . . . . 11
⊢ ((◡𝐹 Fn 𝑌 ∧ 𝑥 ⊆ 𝑌 ∧ 𝑧 ∈ 𝑥) → (◡𝐹‘𝑧) ∈ (◡𝐹 “ 𝑥)) |
19 | 13, 14, 17, 18 | syl3anc 1326 |
. . . . . . . . . 10
⊢ ((((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑥 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑥) ∈ 𝐽) ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) → (◡𝐹‘𝑧) ∈ (◡𝐹 “ 𝑥)) |
20 | | simpl3l 1116 |
. . . . . . . . . . 11
⊢ ((((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑥 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑥) ∈ 𝐽) ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) → 𝑦 ⊆ 𝑌) |
21 | | inss2 3834 |
. . . . . . . . . . . 12
⊢ (𝑥 ∩ 𝑦) ⊆ 𝑦 |
22 | 21, 16 | sseldi 3601 |
. . . . . . . . . . 11
⊢ ((((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑥 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑥) ∈ 𝐽) ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) → 𝑧 ∈ 𝑦) |
23 | | fnfvima 6496 |
. . . . . . . . . . 11
⊢ ((◡𝐹 Fn 𝑌 ∧ 𝑦 ⊆ 𝑌 ∧ 𝑧 ∈ 𝑦) → (◡𝐹‘𝑧) ∈ (◡𝐹 “ 𝑦)) |
24 | 13, 20, 22, 23 | syl3anc 1326 |
. . . . . . . . . 10
⊢ ((((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑥 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑥) ∈ 𝐽) ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) → (◡𝐹‘𝑧) ∈ (◡𝐹 “ 𝑦)) |
25 | 19, 24 | elind 3798 |
. . . . . . . . 9
⊢ ((((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑥 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑥) ∈ 𝐽) ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) → (◡𝐹‘𝑧) ∈ ((◡𝐹 “ 𝑥) ∩ (◡𝐹 “ 𝑦))) |
26 | | basis2 20755 |
. . . . . . . . 9
⊢ (((𝐽 ∈ TopBases ∧ (◡𝐹 “ 𝑥) ∈ 𝐽) ∧ ((◡𝐹 “ 𝑦) ∈ 𝐽 ∧ (◡𝐹‘𝑧) ∈ ((◡𝐹 “ 𝑥) ∩ (◡𝐹 “ 𝑦)))) → ∃𝑤 ∈ 𝐽 ((◡𝐹‘𝑧) ∈ 𝑤 ∧ 𝑤 ⊆ ((◡𝐹 “ 𝑥) ∩ (◡𝐹 “ 𝑦)))) |
27 | 7, 8, 9, 25, 26 | syl22anc 1327 |
. . . . . . . 8
⊢ ((((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑥 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑥) ∈ 𝐽) ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) → ∃𝑤 ∈ 𝐽 ((◡𝐹‘𝑧) ∈ 𝑤 ∧ 𝑤 ⊆ ((◡𝐹 “ 𝑥) ∩ (◡𝐹 “ 𝑦)))) |
28 | 10 | adantr 481 |
. . . . . . . . . . 11
⊢
(((((𝐽 ∈
TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑥 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑥) ∈ 𝐽) ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) ∧ (𝑤 ∈ 𝐽 ∧ ((◡𝐹‘𝑧) ∈ 𝑤 ∧ 𝑤 ⊆ ((◡𝐹 “ 𝑥) ∩ (◡𝐹 “ 𝑦))))) → 𝐹:𝑋–1-1-onto→𝑌) |
29 | | simp2l 1087 |
. . . . . . . . . . . . . 14
⊢ (((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑥 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑥) ∈ 𝐽) ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) → 𝑥 ⊆ 𝑌) |
30 | 15, 29 | syl5ss 3614 |
. . . . . . . . . . . . 13
⊢ (((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑥 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑥) ∈ 𝐽) ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) → (𝑥 ∩ 𝑦) ⊆ 𝑌) |
31 | 30 | sselda 3603 |
. . . . . . . . . . . 12
⊢ ((((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑥 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑥) ∈ 𝐽) ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) → 𝑧 ∈ 𝑌) |
32 | 31 | adantr 481 |
. . . . . . . . . . 11
⊢
(((((𝐽 ∈
TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑥 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑥) ∈ 𝐽) ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) ∧ (𝑤 ∈ 𝐽 ∧ ((◡𝐹‘𝑧) ∈ 𝑤 ∧ 𝑤 ⊆ ((◡𝐹 “ 𝑥) ∩ (◡𝐹 “ 𝑦))))) → 𝑧 ∈ 𝑌) |
33 | | f1ocnvfv2 6533 |
. . . . . . . . . . 11
⊢ ((𝐹:𝑋–1-1-onto→𝑌 ∧ 𝑧 ∈ 𝑌) → (𝐹‘(◡𝐹‘𝑧)) = 𝑧) |
34 | 28, 32, 33 | syl2anc 693 |
. . . . . . . . . 10
⊢
(((((𝐽 ∈
TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑥 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑥) ∈ 𝐽) ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) ∧ (𝑤 ∈ 𝐽 ∧ ((◡𝐹‘𝑧) ∈ 𝑤 ∧ 𝑤 ⊆ ((◡𝐹 “ 𝑥) ∩ (◡𝐹 “ 𝑦))))) → (𝐹‘(◡𝐹‘𝑧)) = 𝑧) |
35 | | f1ofn 6138 |
. . . . . . . . . . . 12
⊢ (𝐹:𝑋–1-1-onto→𝑌 → 𝐹 Fn 𝑋) |
36 | 28, 35 | syl 17 |
. . . . . . . . . . 11
⊢
(((((𝐽 ∈
TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑥 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑥) ∈ 𝐽) ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) ∧ (𝑤 ∈ 𝐽 ∧ ((◡𝐹‘𝑧) ∈ 𝑤 ∧ 𝑤 ⊆ ((◡𝐹 “ 𝑥) ∩ (◡𝐹 “ 𝑦))))) → 𝐹 Fn 𝑋) |
37 | | simprrr 805 |
. . . . . . . . . . . . 13
⊢
(((((𝐽 ∈
TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑥 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑥) ∈ 𝐽) ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) ∧ (𝑤 ∈ 𝐽 ∧ ((◡𝐹‘𝑧) ∈ 𝑤 ∧ 𝑤 ⊆ ((◡𝐹 “ 𝑥) ∩ (◡𝐹 “ 𝑦))))) → 𝑤 ⊆ ((◡𝐹 “ 𝑥) ∩ (◡𝐹 “ 𝑦))) |
38 | | inss1 3833 |
. . . . . . . . . . . . 13
⊢ ((◡𝐹 “ 𝑥) ∩ (◡𝐹 “ 𝑦)) ⊆ (◡𝐹 “ 𝑥) |
39 | 37, 38 | syl6ss 3615 |
. . . . . . . . . . . 12
⊢
(((((𝐽 ∈
TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑥 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑥) ∈ 𝐽) ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) ∧ (𝑤 ∈ 𝐽 ∧ ((◡𝐹‘𝑧) ∈ 𝑤 ∧ 𝑤 ⊆ ((◡𝐹 “ 𝑥) ∩ (◡𝐹 “ 𝑦))))) → 𝑤 ⊆ (◡𝐹 “ 𝑥)) |
40 | | cnvimass 5485 |
. . . . . . . . . . . . 13
⊢ (◡𝐹 “ 𝑥) ⊆ dom 𝐹 |
41 | | f1odm 6141 |
. . . . . . . . . . . . . 14
⊢ (𝐹:𝑋–1-1-onto→𝑌 → dom 𝐹 = 𝑋) |
42 | 28, 41 | syl 17 |
. . . . . . . . . . . . 13
⊢
(((((𝐽 ∈
TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑥 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑥) ∈ 𝐽) ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) ∧ (𝑤 ∈ 𝐽 ∧ ((◡𝐹‘𝑧) ∈ 𝑤 ∧ 𝑤 ⊆ ((◡𝐹 “ 𝑥) ∩ (◡𝐹 “ 𝑦))))) → dom 𝐹 = 𝑋) |
43 | 40, 42 | syl5sseq 3653 |
. . . . . . . . . . . 12
⊢
(((((𝐽 ∈
TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑥 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑥) ∈ 𝐽) ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) ∧ (𝑤 ∈ 𝐽 ∧ ((◡𝐹‘𝑧) ∈ 𝑤 ∧ 𝑤 ⊆ ((◡𝐹 “ 𝑥) ∩ (◡𝐹 “ 𝑦))))) → (◡𝐹 “ 𝑥) ⊆ 𝑋) |
44 | 39, 43 | sstrd 3613 |
. . . . . . . . . . 11
⊢
(((((𝐽 ∈
TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑥 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑥) ∈ 𝐽) ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) ∧ (𝑤 ∈ 𝐽 ∧ ((◡𝐹‘𝑧) ∈ 𝑤 ∧ 𝑤 ⊆ ((◡𝐹 “ 𝑥) ∩ (◡𝐹 “ 𝑦))))) → 𝑤 ⊆ 𝑋) |
45 | | simprrl 804 |
. . . . . . . . . . 11
⊢
(((((𝐽 ∈
TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑥 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑥) ∈ 𝐽) ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) ∧ (𝑤 ∈ 𝐽 ∧ ((◡𝐹‘𝑧) ∈ 𝑤 ∧ 𝑤 ⊆ ((◡𝐹 “ 𝑥) ∩ (◡𝐹 “ 𝑦))))) → (◡𝐹‘𝑧) ∈ 𝑤) |
46 | | fnfvima 6496 |
. . . . . . . . . . 11
⊢ ((𝐹 Fn 𝑋 ∧ 𝑤 ⊆ 𝑋 ∧ (◡𝐹‘𝑧) ∈ 𝑤) → (𝐹‘(◡𝐹‘𝑧)) ∈ (𝐹 “ 𝑤)) |
47 | 36, 44, 45, 46 | syl3anc 1326 |
. . . . . . . . . 10
⊢
(((((𝐽 ∈
TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑥 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑥) ∈ 𝐽) ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) ∧ (𝑤 ∈ 𝐽 ∧ ((◡𝐹‘𝑧) ∈ 𝑤 ∧ 𝑤 ⊆ ((◡𝐹 “ 𝑥) ∩ (◡𝐹 “ 𝑦))))) → (𝐹‘(◡𝐹‘𝑧)) ∈ (𝐹 “ 𝑤)) |
48 | 34, 47 | eqeltrrd 2702 |
. . . . . . . . 9
⊢
(((((𝐽 ∈
TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑥 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑥) ∈ 𝐽) ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) ∧ (𝑤 ∈ 𝐽 ∧ ((◡𝐹‘𝑧) ∈ 𝑤 ∧ 𝑤 ⊆ ((◡𝐹 “ 𝑥) ∩ (◡𝐹 “ 𝑦))))) → 𝑧 ∈ (𝐹 “ 𝑤)) |
49 | | imassrn 5477 |
. . . . . . . . . . . 12
⊢ (𝐹 “ 𝑤) ⊆ ran 𝐹 |
50 | 28, 1 | syl 17 |
. . . . . . . . . . . . 13
⊢
(((((𝐽 ∈
TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑥 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑥) ∈ 𝐽) ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) ∧ (𝑤 ∈ 𝐽 ∧ ((◡𝐹‘𝑧) ∈ 𝑤 ∧ 𝑤 ⊆ ((◡𝐹 “ 𝑥) ∩ (◡𝐹 “ 𝑦))))) → 𝐹:𝑋–onto→𝑌) |
51 | | forn 6118 |
. . . . . . . . . . . . 13
⊢ (𝐹:𝑋–onto→𝑌 → ran 𝐹 = 𝑌) |
52 | 50, 51 | syl 17 |
. . . . . . . . . . . 12
⊢
(((((𝐽 ∈
TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑥 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑥) ∈ 𝐽) ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) ∧ (𝑤 ∈ 𝐽 ∧ ((◡𝐹‘𝑧) ∈ 𝑤 ∧ 𝑤 ⊆ ((◡𝐹 “ 𝑥) ∩ (◡𝐹 “ 𝑦))))) → ran 𝐹 = 𝑌) |
53 | 49, 52 | syl5sseq 3653 |
. . . . . . . . . . 11
⊢
(((((𝐽 ∈
TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑥 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑥) ∈ 𝐽) ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) ∧ (𝑤 ∈ 𝐽 ∧ ((◡𝐹‘𝑧) ∈ 𝑤 ∧ 𝑤 ⊆ ((◡𝐹 “ 𝑥) ∩ (◡𝐹 “ 𝑦))))) → (𝐹 “ 𝑤) ⊆ 𝑌) |
54 | | f1of1 6136 |
. . . . . . . . . . . . . 14
⊢ (𝐹:𝑋–1-1-onto→𝑌 → 𝐹:𝑋–1-1→𝑌) |
55 | 28, 54 | syl 17 |
. . . . . . . . . . . . 13
⊢
(((((𝐽 ∈
TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑥 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑥) ∈ 𝐽) ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) ∧ (𝑤 ∈ 𝐽 ∧ ((◡𝐹‘𝑧) ∈ 𝑤 ∧ 𝑤 ⊆ ((◡𝐹 “ 𝑥) ∩ (◡𝐹 “ 𝑦))))) → 𝐹:𝑋–1-1→𝑌) |
56 | | f1imacnv 6153 |
. . . . . . . . . . . . 13
⊢ ((𝐹:𝑋–1-1→𝑌 ∧ 𝑤 ⊆ 𝑋) → (◡𝐹 “ (𝐹 “ 𝑤)) = 𝑤) |
57 | 55, 44, 56 | syl2anc 693 |
. . . . . . . . . . . 12
⊢
(((((𝐽 ∈
TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑥 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑥) ∈ 𝐽) ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) ∧ (𝑤 ∈ 𝐽 ∧ ((◡𝐹‘𝑧) ∈ 𝑤 ∧ 𝑤 ⊆ ((◡𝐹 “ 𝑥) ∩ (◡𝐹 “ 𝑦))))) → (◡𝐹 “ (𝐹 “ 𝑤)) = 𝑤) |
58 | | simprl 794 |
. . . . . . . . . . . 12
⊢
(((((𝐽 ∈
TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑥 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑥) ∈ 𝐽) ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) ∧ (𝑤 ∈ 𝐽 ∧ ((◡𝐹‘𝑧) ∈ 𝑤 ∧ 𝑤 ⊆ ((◡𝐹 “ 𝑥) ∩ (◡𝐹 “ 𝑦))))) → 𝑤 ∈ 𝐽) |
59 | 57, 58 | eqeltrd 2701 |
. . . . . . . . . . 11
⊢
(((((𝐽 ∈
TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑥 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑥) ∈ 𝐽) ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) ∧ (𝑤 ∈ 𝐽 ∧ ((◡𝐹‘𝑧) ∈ 𝑤 ∧ 𝑤 ⊆ ((◡𝐹 “ 𝑥) ∩ (◡𝐹 “ 𝑦))))) → (◡𝐹 “ (𝐹 “ 𝑤)) ∈ 𝐽) |
60 | 7 | adantr 481 |
. . . . . . . . . . . 12
⊢
(((((𝐽 ∈
TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑥 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑥) ∈ 𝐽) ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) ∧ (𝑤 ∈ 𝐽 ∧ ((◡𝐹‘𝑧) ∈ 𝑤 ∧ 𝑤 ⊆ ((◡𝐹 “ 𝑥) ∩ (◡𝐹 “ 𝑦))))) → 𝐽 ∈ TopBases) |
61 | 2 | elqtop2 21504 |
. . . . . . . . . . . 12
⊢ ((𝐽 ∈ TopBases ∧ 𝐹:𝑋–onto→𝑌) → ((𝐹 “ 𝑤) ∈ (𝐽 qTop 𝐹) ↔ ((𝐹 “ 𝑤) ⊆ 𝑌 ∧ (◡𝐹 “ (𝐹 “ 𝑤)) ∈ 𝐽))) |
62 | 60, 50, 61 | syl2anc 693 |
. . . . . . . . . . 11
⊢
(((((𝐽 ∈
TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑥 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑥) ∈ 𝐽) ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) ∧ (𝑤 ∈ 𝐽 ∧ ((◡𝐹‘𝑧) ∈ 𝑤 ∧ 𝑤 ⊆ ((◡𝐹 “ 𝑥) ∩ (◡𝐹 “ 𝑦))))) → ((𝐹 “ 𝑤) ∈ (𝐽 qTop 𝐹) ↔ ((𝐹 “ 𝑤) ⊆ 𝑌 ∧ (◡𝐹 “ (𝐹 “ 𝑤)) ∈ 𝐽))) |
63 | 53, 59, 62 | mpbir2and 957 |
. . . . . . . . . 10
⊢
(((((𝐽 ∈
TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑥 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑥) ∈ 𝐽) ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) ∧ (𝑤 ∈ 𝐽 ∧ ((◡𝐹‘𝑧) ∈ 𝑤 ∧ 𝑤 ⊆ ((◡𝐹 “ 𝑥) ∩ (◡𝐹 “ 𝑦))))) → (𝐹 “ 𝑤) ∈ (𝐽 qTop 𝐹)) |
64 | | fnfun 5988 |
. . . . . . . . . . . . . 14
⊢ (𝐹 Fn 𝑋 → Fun 𝐹) |
65 | | inpreima 6342 |
. . . . . . . . . . . . . 14
⊢ (Fun
𝐹 → (◡𝐹 “ (𝑥 ∩ 𝑦)) = ((◡𝐹 “ 𝑥) ∩ (◡𝐹 “ 𝑦))) |
66 | 36, 64, 65 | 3syl 18 |
. . . . . . . . . . . . 13
⊢
(((((𝐽 ∈
TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑥 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑥) ∈ 𝐽) ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) ∧ (𝑤 ∈ 𝐽 ∧ ((◡𝐹‘𝑧) ∈ 𝑤 ∧ 𝑤 ⊆ ((◡𝐹 “ 𝑥) ∩ (◡𝐹 “ 𝑦))))) → (◡𝐹 “ (𝑥 ∩ 𝑦)) = ((◡𝐹 “ 𝑥) ∩ (◡𝐹 “ 𝑦))) |
67 | 37, 66 | sseqtr4d 3642 |
. . . . . . . . . . . 12
⊢
(((((𝐽 ∈
TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑥 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑥) ∈ 𝐽) ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) ∧ (𝑤 ∈ 𝐽 ∧ ((◡𝐹‘𝑧) ∈ 𝑤 ∧ 𝑤 ⊆ ((◡𝐹 “ 𝑥) ∩ (◡𝐹 “ 𝑦))))) → 𝑤 ⊆ (◡𝐹 “ (𝑥 ∩ 𝑦))) |
68 | 36, 64 | syl 17 |
. . . . . . . . . . . . 13
⊢
(((((𝐽 ∈
TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑥 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑥) ∈ 𝐽) ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) ∧ (𝑤 ∈ 𝐽 ∧ ((◡𝐹‘𝑧) ∈ 𝑤 ∧ 𝑤 ⊆ ((◡𝐹 “ 𝑥) ∩ (◡𝐹 “ 𝑦))))) → Fun 𝐹) |
69 | 39, 40 | syl6ss 3615 |
. . . . . . . . . . . . 13
⊢
(((((𝐽 ∈
TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑥 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑥) ∈ 𝐽) ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) ∧ (𝑤 ∈ 𝐽 ∧ ((◡𝐹‘𝑧) ∈ 𝑤 ∧ 𝑤 ⊆ ((◡𝐹 “ 𝑥) ∩ (◡𝐹 “ 𝑦))))) → 𝑤 ⊆ dom 𝐹) |
70 | | funimass3 6333 |
. . . . . . . . . . . . 13
⊢ ((Fun
𝐹 ∧ 𝑤 ⊆ dom 𝐹) → ((𝐹 “ 𝑤) ⊆ (𝑥 ∩ 𝑦) ↔ 𝑤 ⊆ (◡𝐹 “ (𝑥 ∩ 𝑦)))) |
71 | 68, 69, 70 | syl2anc 693 |
. . . . . . . . . . . 12
⊢
(((((𝐽 ∈
TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑥 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑥) ∈ 𝐽) ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) ∧ (𝑤 ∈ 𝐽 ∧ ((◡𝐹‘𝑧) ∈ 𝑤 ∧ 𝑤 ⊆ ((◡𝐹 “ 𝑥) ∩ (◡𝐹 “ 𝑦))))) → ((𝐹 “ 𝑤) ⊆ (𝑥 ∩ 𝑦) ↔ 𝑤 ⊆ (◡𝐹 “ (𝑥 ∩ 𝑦)))) |
72 | 67, 71 | mpbird 247 |
. . . . . . . . . . 11
⊢
(((((𝐽 ∈
TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑥 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑥) ∈ 𝐽) ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) ∧ (𝑤 ∈ 𝐽 ∧ ((◡𝐹‘𝑧) ∈ 𝑤 ∧ 𝑤 ⊆ ((◡𝐹 “ 𝑥) ∩ (◡𝐹 “ 𝑦))))) → (𝐹 “ 𝑤) ⊆ (𝑥 ∩ 𝑦)) |
73 | | vex 3203 |
. . . . . . . . . . . . 13
⊢ 𝑥 ∈ V |
74 | 73 | inex1 4799 |
. . . . . . . . . . . 12
⊢ (𝑥 ∩ 𝑦) ∈ V |
75 | 74 | elpw2 4828 |
. . . . . . . . . . 11
⊢ ((𝐹 “ 𝑤) ∈ 𝒫 (𝑥 ∩ 𝑦) ↔ (𝐹 “ 𝑤) ⊆ (𝑥 ∩ 𝑦)) |
76 | 72, 75 | sylibr 224 |
. . . . . . . . . 10
⊢
(((((𝐽 ∈
TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑥 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑥) ∈ 𝐽) ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) ∧ (𝑤 ∈ 𝐽 ∧ ((◡𝐹‘𝑧) ∈ 𝑤 ∧ 𝑤 ⊆ ((◡𝐹 “ 𝑥) ∩ (◡𝐹 “ 𝑦))))) → (𝐹 “ 𝑤) ∈ 𝒫 (𝑥 ∩ 𝑦)) |
77 | 63, 76 | elind 3798 |
. . . . . . . . 9
⊢
(((((𝐽 ∈
TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑥 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑥) ∈ 𝐽) ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) ∧ (𝑤 ∈ 𝐽 ∧ ((◡𝐹‘𝑧) ∈ 𝑤 ∧ 𝑤 ⊆ ((◡𝐹 “ 𝑥) ∩ (◡𝐹 “ 𝑦))))) → (𝐹 “ 𝑤) ∈ ((𝐽 qTop 𝐹) ∩ 𝒫 (𝑥 ∩ 𝑦))) |
78 | | elunii 4441 |
. . . . . . . . 9
⊢ ((𝑧 ∈ (𝐹 “ 𝑤) ∧ (𝐹 “ 𝑤) ∈ ((𝐽 qTop 𝐹) ∩ 𝒫 (𝑥 ∩ 𝑦))) → 𝑧 ∈ ∪ ((𝐽 qTop 𝐹) ∩ 𝒫 (𝑥 ∩ 𝑦))) |
79 | 48, 77, 78 | syl2anc 693 |
. . . . . . . 8
⊢
(((((𝐽 ∈
TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑥 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑥) ∈ 𝐽) ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) ∧ (𝑤 ∈ 𝐽 ∧ ((◡𝐹‘𝑧) ∈ 𝑤 ∧ 𝑤 ⊆ ((◡𝐹 “ 𝑥) ∩ (◡𝐹 “ 𝑦))))) → 𝑧 ∈ ∪ ((𝐽 qTop 𝐹) ∩ 𝒫 (𝑥 ∩ 𝑦))) |
80 | 27, 79 | rexlimddv 3035 |
. . . . . . 7
⊢ ((((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑥 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑥) ∈ 𝐽) ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) ∧ 𝑧 ∈ (𝑥 ∩ 𝑦)) → 𝑧 ∈ ∪ ((𝐽 qTop 𝐹) ∩ 𝒫 (𝑥 ∩ 𝑦))) |
81 | 80 | ex 450 |
. . . . . 6
⊢ (((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑥 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑥) ∈ 𝐽) ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) → (𝑧 ∈ (𝑥 ∩ 𝑦) → 𝑧 ∈ ∪ ((𝐽 qTop 𝐹) ∩ 𝒫 (𝑥 ∩ 𝑦)))) |
82 | 81 | ssrdv 3609 |
. . . . 5
⊢ (((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) ∧ (𝑥 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑥) ∈ 𝐽) ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) → (𝑥 ∩ 𝑦) ⊆ ∪
((𝐽 qTop 𝐹) ∩ 𝒫 (𝑥 ∩ 𝑦))) |
83 | 82 | 3expib 1268 |
. . . 4
⊢ ((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) → (((𝑥 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑥) ∈ 𝐽) ∧ (𝑦 ⊆ 𝑌 ∧ (◡𝐹 “ 𝑦) ∈ 𝐽)) → (𝑥 ∩ 𝑦) ⊆ ∪
((𝐽 qTop 𝐹) ∩ 𝒫 (𝑥 ∩ 𝑦)))) |
84 | 6, 83 | sylbid 230 |
. . 3
⊢ ((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) → ((𝑥 ∈ (𝐽 qTop 𝐹) ∧ 𝑦 ∈ (𝐽 qTop 𝐹)) → (𝑥 ∩ 𝑦) ⊆ ∪
((𝐽 qTop 𝐹) ∩ 𝒫 (𝑥 ∩ 𝑦)))) |
85 | 84 | ralrimivv 2970 |
. 2
⊢ ((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) → ∀𝑥 ∈ (𝐽 qTop 𝐹)∀𝑦 ∈ (𝐽 qTop 𝐹)(𝑥 ∩ 𝑦) ⊆ ∪
((𝐽 qTop 𝐹) ∩ 𝒫 (𝑥 ∩ 𝑦))) |
86 | | ovex 6678 |
. . 3
⊢ (𝐽 qTop 𝐹) ∈ V |
87 | | isbasisg 20751 |
. . 3
⊢ ((𝐽 qTop 𝐹) ∈ V → ((𝐽 qTop 𝐹) ∈ TopBases ↔ ∀𝑥 ∈ (𝐽 qTop 𝐹)∀𝑦 ∈ (𝐽 qTop 𝐹)(𝑥 ∩ 𝑦) ⊆ ∪
((𝐽 qTop 𝐹) ∩ 𝒫 (𝑥 ∩ 𝑦)))) |
88 | 86, 87 | ax-mp 5 |
. 2
⊢ ((𝐽 qTop 𝐹) ∈ TopBases ↔ ∀𝑥 ∈ (𝐽 qTop 𝐹)∀𝑦 ∈ (𝐽 qTop 𝐹)(𝑥 ∩ 𝑦) ⊆ ∪
((𝐽 qTop 𝐹) ∩ 𝒫 (𝑥 ∩ 𝑦))) |
89 | 85, 88 | sylibr 224 |
1
⊢ ((𝐽 ∈ TopBases ∧ 𝐹:𝑋–1-1-onto→𝑌) → (𝐽 qTop 𝐹) ∈ TopBases) |