Step |
Hyp |
Ref |
Expression |
1 |
|
tgcn.1 |
⊢ ( 𝜑 → 𝐽 ∈ ( TopOn ‘ 𝑋 ) ) |
2 |
|
tgcn.3 |
⊢ ( 𝜑 → 𝐾 = ( topGen ‘ 𝐵 ) ) |
3 |
|
tgcn.4 |
⊢ ( 𝜑 → 𝐾 ∈ ( TopOn ‘ 𝑌 ) ) |
4 |
|
iscn |
⊢ ( ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) ∧ 𝐾 ∈ ( TopOn ‘ 𝑌 ) ) → ( 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ↔ ( 𝐹 : 𝑋 ⟶ 𝑌 ∧ ∀ 𝑦 ∈ 𝐾 ( ◡ 𝐹 “ 𝑦 ) ∈ 𝐽 ) ) ) |
5 |
1 3 4
|
syl2anc |
⊢ ( 𝜑 → ( 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ↔ ( 𝐹 : 𝑋 ⟶ 𝑌 ∧ ∀ 𝑦 ∈ 𝐾 ( ◡ 𝐹 “ 𝑦 ) ∈ 𝐽 ) ) ) |
6 |
|
topontop |
⊢ ( 𝐾 ∈ ( TopOn ‘ 𝑌 ) → 𝐾 ∈ Top ) |
7 |
3 6
|
syl |
⊢ ( 𝜑 → 𝐾 ∈ Top ) |
8 |
2 7
|
eqeltrrd |
⊢ ( 𝜑 → ( topGen ‘ 𝐵 ) ∈ Top ) |
9 |
|
tgclb |
⊢ ( 𝐵 ∈ TopBases ↔ ( topGen ‘ 𝐵 ) ∈ Top ) |
10 |
8 9
|
sylibr |
⊢ ( 𝜑 → 𝐵 ∈ TopBases ) |
11 |
|
bastg |
⊢ ( 𝐵 ∈ TopBases → 𝐵 ⊆ ( topGen ‘ 𝐵 ) ) |
12 |
10 11
|
syl |
⊢ ( 𝜑 → 𝐵 ⊆ ( topGen ‘ 𝐵 ) ) |
13 |
12 2
|
sseqtrrd |
⊢ ( 𝜑 → 𝐵 ⊆ 𝐾 ) |
14 |
|
ssralv |
⊢ ( 𝐵 ⊆ 𝐾 → ( ∀ 𝑦 ∈ 𝐾 ( ◡ 𝐹 “ 𝑦 ) ∈ 𝐽 → ∀ 𝑦 ∈ 𝐵 ( ◡ 𝐹 “ 𝑦 ) ∈ 𝐽 ) ) |
15 |
13 14
|
syl |
⊢ ( 𝜑 → ( ∀ 𝑦 ∈ 𝐾 ( ◡ 𝐹 “ 𝑦 ) ∈ 𝐽 → ∀ 𝑦 ∈ 𝐵 ( ◡ 𝐹 “ 𝑦 ) ∈ 𝐽 ) ) |
16 |
2
|
eleq2d |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝐾 ↔ 𝑥 ∈ ( topGen ‘ 𝐵 ) ) ) |
17 |
|
eltg3 |
⊢ ( 𝐵 ∈ TopBases → ( 𝑥 ∈ ( topGen ‘ 𝐵 ) ↔ ∃ 𝑧 ( 𝑧 ⊆ 𝐵 ∧ 𝑥 = ∪ 𝑧 ) ) ) |
18 |
10 17
|
syl |
⊢ ( 𝜑 → ( 𝑥 ∈ ( topGen ‘ 𝐵 ) ↔ ∃ 𝑧 ( 𝑧 ⊆ 𝐵 ∧ 𝑥 = ∪ 𝑧 ) ) ) |
19 |
16 18
|
bitrd |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝐾 ↔ ∃ 𝑧 ( 𝑧 ⊆ 𝐵 ∧ 𝑥 = ∪ 𝑧 ) ) ) |
20 |
|
ssralv |
⊢ ( 𝑧 ⊆ 𝐵 → ( ∀ 𝑦 ∈ 𝐵 ( ◡ 𝐹 “ 𝑦 ) ∈ 𝐽 → ∀ 𝑦 ∈ 𝑧 ( ◡ 𝐹 “ 𝑦 ) ∈ 𝐽 ) ) |
21 |
|
topontop |
⊢ ( 𝐽 ∈ ( TopOn ‘ 𝑋 ) → 𝐽 ∈ Top ) |
22 |
1 21
|
syl |
⊢ ( 𝜑 → 𝐽 ∈ Top ) |
23 |
|
iunopn |
⊢ ( ( 𝐽 ∈ Top ∧ ∀ 𝑦 ∈ 𝑧 ( ◡ 𝐹 “ 𝑦 ) ∈ 𝐽 ) → ∪ 𝑦 ∈ 𝑧 ( ◡ 𝐹 “ 𝑦 ) ∈ 𝐽 ) |
24 |
23
|
ex |
⊢ ( 𝐽 ∈ Top → ( ∀ 𝑦 ∈ 𝑧 ( ◡ 𝐹 “ 𝑦 ) ∈ 𝐽 → ∪ 𝑦 ∈ 𝑧 ( ◡ 𝐹 “ 𝑦 ) ∈ 𝐽 ) ) |
25 |
22 24
|
syl |
⊢ ( 𝜑 → ( ∀ 𝑦 ∈ 𝑧 ( ◡ 𝐹 “ 𝑦 ) ∈ 𝐽 → ∪ 𝑦 ∈ 𝑧 ( ◡ 𝐹 “ 𝑦 ) ∈ 𝐽 ) ) |
26 |
20 25
|
sylan9r |
⊢ ( ( 𝜑 ∧ 𝑧 ⊆ 𝐵 ) → ( ∀ 𝑦 ∈ 𝐵 ( ◡ 𝐹 “ 𝑦 ) ∈ 𝐽 → ∪ 𝑦 ∈ 𝑧 ( ◡ 𝐹 “ 𝑦 ) ∈ 𝐽 ) ) |
27 |
|
imaeq2 |
⊢ ( 𝑥 = ∪ 𝑧 → ( ◡ 𝐹 “ 𝑥 ) = ( ◡ 𝐹 “ ∪ 𝑧 ) ) |
28 |
|
imauni |
⊢ ( ◡ 𝐹 “ ∪ 𝑧 ) = ∪ 𝑦 ∈ 𝑧 ( ◡ 𝐹 “ 𝑦 ) |
29 |
27 28
|
eqtrdi |
⊢ ( 𝑥 = ∪ 𝑧 → ( ◡ 𝐹 “ 𝑥 ) = ∪ 𝑦 ∈ 𝑧 ( ◡ 𝐹 “ 𝑦 ) ) |
30 |
29
|
eleq1d |
⊢ ( 𝑥 = ∪ 𝑧 → ( ( ◡ 𝐹 “ 𝑥 ) ∈ 𝐽 ↔ ∪ 𝑦 ∈ 𝑧 ( ◡ 𝐹 “ 𝑦 ) ∈ 𝐽 ) ) |
31 |
30
|
imbi2d |
⊢ ( 𝑥 = ∪ 𝑧 → ( ( ∀ 𝑦 ∈ 𝐵 ( ◡ 𝐹 “ 𝑦 ) ∈ 𝐽 → ( ◡ 𝐹 “ 𝑥 ) ∈ 𝐽 ) ↔ ( ∀ 𝑦 ∈ 𝐵 ( ◡ 𝐹 “ 𝑦 ) ∈ 𝐽 → ∪ 𝑦 ∈ 𝑧 ( ◡ 𝐹 “ 𝑦 ) ∈ 𝐽 ) ) ) |
32 |
26 31
|
syl5ibrcom |
⊢ ( ( 𝜑 ∧ 𝑧 ⊆ 𝐵 ) → ( 𝑥 = ∪ 𝑧 → ( ∀ 𝑦 ∈ 𝐵 ( ◡ 𝐹 “ 𝑦 ) ∈ 𝐽 → ( ◡ 𝐹 “ 𝑥 ) ∈ 𝐽 ) ) ) |
33 |
32
|
expimpd |
⊢ ( 𝜑 → ( ( 𝑧 ⊆ 𝐵 ∧ 𝑥 = ∪ 𝑧 ) → ( ∀ 𝑦 ∈ 𝐵 ( ◡ 𝐹 “ 𝑦 ) ∈ 𝐽 → ( ◡ 𝐹 “ 𝑥 ) ∈ 𝐽 ) ) ) |
34 |
33
|
exlimdv |
⊢ ( 𝜑 → ( ∃ 𝑧 ( 𝑧 ⊆ 𝐵 ∧ 𝑥 = ∪ 𝑧 ) → ( ∀ 𝑦 ∈ 𝐵 ( ◡ 𝐹 “ 𝑦 ) ∈ 𝐽 → ( ◡ 𝐹 “ 𝑥 ) ∈ 𝐽 ) ) ) |
35 |
19 34
|
sylbid |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝐾 → ( ∀ 𝑦 ∈ 𝐵 ( ◡ 𝐹 “ 𝑦 ) ∈ 𝐽 → ( ◡ 𝐹 “ 𝑥 ) ∈ 𝐽 ) ) ) |
36 |
35
|
imp |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐾 ) → ( ∀ 𝑦 ∈ 𝐵 ( ◡ 𝐹 “ 𝑦 ) ∈ 𝐽 → ( ◡ 𝐹 “ 𝑥 ) ∈ 𝐽 ) ) |
37 |
36
|
ralrimdva |
⊢ ( 𝜑 → ( ∀ 𝑦 ∈ 𝐵 ( ◡ 𝐹 “ 𝑦 ) ∈ 𝐽 → ∀ 𝑥 ∈ 𝐾 ( ◡ 𝐹 “ 𝑥 ) ∈ 𝐽 ) ) |
38 |
|
imaeq2 |
⊢ ( 𝑥 = 𝑦 → ( ◡ 𝐹 “ 𝑥 ) = ( ◡ 𝐹 “ 𝑦 ) ) |
39 |
38
|
eleq1d |
⊢ ( 𝑥 = 𝑦 → ( ( ◡ 𝐹 “ 𝑥 ) ∈ 𝐽 ↔ ( ◡ 𝐹 “ 𝑦 ) ∈ 𝐽 ) ) |
40 |
39
|
cbvralvw |
⊢ ( ∀ 𝑥 ∈ 𝐾 ( ◡ 𝐹 “ 𝑥 ) ∈ 𝐽 ↔ ∀ 𝑦 ∈ 𝐾 ( ◡ 𝐹 “ 𝑦 ) ∈ 𝐽 ) |
41 |
37 40
|
syl6ib |
⊢ ( 𝜑 → ( ∀ 𝑦 ∈ 𝐵 ( ◡ 𝐹 “ 𝑦 ) ∈ 𝐽 → ∀ 𝑦 ∈ 𝐾 ( ◡ 𝐹 “ 𝑦 ) ∈ 𝐽 ) ) |
42 |
15 41
|
impbid |
⊢ ( 𝜑 → ( ∀ 𝑦 ∈ 𝐾 ( ◡ 𝐹 “ 𝑦 ) ∈ 𝐽 ↔ ∀ 𝑦 ∈ 𝐵 ( ◡ 𝐹 “ 𝑦 ) ∈ 𝐽 ) ) |
43 |
42
|
anbi2d |
⊢ ( 𝜑 → ( ( 𝐹 : 𝑋 ⟶ 𝑌 ∧ ∀ 𝑦 ∈ 𝐾 ( ◡ 𝐹 “ 𝑦 ) ∈ 𝐽 ) ↔ ( 𝐹 : 𝑋 ⟶ 𝑌 ∧ ∀ 𝑦 ∈ 𝐵 ( ◡ 𝐹 “ 𝑦 ) ∈ 𝐽 ) ) ) |
44 |
5 43
|
bitrd |
⊢ ( 𝜑 → ( 𝐹 ∈ ( 𝐽 Cn 𝐾 ) ↔ ( 𝐹 : 𝑋 ⟶ 𝑌 ∧ ∀ 𝑦 ∈ 𝐵 ( ◡ 𝐹 “ 𝑦 ) ∈ 𝐽 ) ) ) |