Step |
Hyp |
Ref |
Expression |
1 |
|
elutop |
⊢ ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) → ( 𝐴 ∈ ( unifTop ‘ 𝑈 ) ↔ ( 𝐴 ⊆ 𝑋 ∧ ∀ 𝑥 ∈ 𝐴 ∃ 𝑡 ∈ 𝑈 ( 𝑡 “ { 𝑥 } ) ⊆ 𝐴 ) ) ) |
2 |
1
|
simprbda |
⊢ ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝐴 ∈ ( unifTop ‘ 𝑈 ) ) → 𝐴 ⊆ 𝑋 ) |
3 |
|
restutop |
⊢ ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝐴 ⊆ 𝑋 ) → ( ( unifTop ‘ 𝑈 ) ↾t 𝐴 ) ⊆ ( unifTop ‘ ( 𝑈 ↾t ( 𝐴 × 𝐴 ) ) ) ) |
4 |
2 3
|
syldan |
⊢ ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝐴 ∈ ( unifTop ‘ 𝑈 ) ) → ( ( unifTop ‘ 𝑈 ) ↾t 𝐴 ) ⊆ ( unifTop ‘ ( 𝑈 ↾t ( 𝐴 × 𝐴 ) ) ) ) |
5 |
|
trust |
⊢ ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝐴 ⊆ 𝑋 ) → ( 𝑈 ↾t ( 𝐴 × 𝐴 ) ) ∈ ( UnifOn ‘ 𝐴 ) ) |
6 |
2 5
|
syldan |
⊢ ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝐴 ∈ ( unifTop ‘ 𝑈 ) ) → ( 𝑈 ↾t ( 𝐴 × 𝐴 ) ) ∈ ( UnifOn ‘ 𝐴 ) ) |
7 |
|
elutop |
⊢ ( ( 𝑈 ↾t ( 𝐴 × 𝐴 ) ) ∈ ( UnifOn ‘ 𝐴 ) → ( 𝑏 ∈ ( unifTop ‘ ( 𝑈 ↾t ( 𝐴 × 𝐴 ) ) ) ↔ ( 𝑏 ⊆ 𝐴 ∧ ∀ 𝑥 ∈ 𝑏 ∃ 𝑢 ∈ ( 𝑈 ↾t ( 𝐴 × 𝐴 ) ) ( 𝑢 “ { 𝑥 } ) ⊆ 𝑏 ) ) ) |
8 |
6 7
|
syl |
⊢ ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝐴 ∈ ( unifTop ‘ 𝑈 ) ) → ( 𝑏 ∈ ( unifTop ‘ ( 𝑈 ↾t ( 𝐴 × 𝐴 ) ) ) ↔ ( 𝑏 ⊆ 𝐴 ∧ ∀ 𝑥 ∈ 𝑏 ∃ 𝑢 ∈ ( 𝑈 ↾t ( 𝐴 × 𝐴 ) ) ( 𝑢 “ { 𝑥 } ) ⊆ 𝑏 ) ) ) |
9 |
8
|
simprbda |
⊢ ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝐴 ∈ ( unifTop ‘ 𝑈 ) ) ∧ 𝑏 ∈ ( unifTop ‘ ( 𝑈 ↾t ( 𝐴 × 𝐴 ) ) ) ) → 𝑏 ⊆ 𝐴 ) |
10 |
2
|
adantr |
⊢ ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝐴 ∈ ( unifTop ‘ 𝑈 ) ) ∧ 𝑏 ∈ ( unifTop ‘ ( 𝑈 ↾t ( 𝐴 × 𝐴 ) ) ) ) → 𝐴 ⊆ 𝑋 ) |
11 |
9 10
|
sstrd |
⊢ ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝐴 ∈ ( unifTop ‘ 𝑈 ) ) ∧ 𝑏 ∈ ( unifTop ‘ ( 𝑈 ↾t ( 𝐴 × 𝐴 ) ) ) ) → 𝑏 ⊆ 𝑋 ) |
12 |
|
simp-9l |
⊢ ( ( ( ( ( ( ( ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝐴 ∈ ( unifTop ‘ 𝑈 ) ) ∧ 𝑏 ∈ ( unifTop ‘ ( 𝑈 ↾t ( 𝐴 × 𝐴 ) ) ) ) ∧ 𝑥 ∈ 𝑏 ) ∧ 𝑢 ∈ ( 𝑈 ↾t ( 𝐴 × 𝐴 ) ) ) ∧ ( 𝑢 “ { 𝑥 } ) ⊆ 𝑏 ) ∧ 𝑤 ∈ 𝑈 ) ∧ 𝑢 = ( 𝑤 ∩ ( 𝐴 × 𝐴 ) ) ) ∧ 𝑡 ∈ 𝑈 ) ∧ ( 𝑡 “ { 𝑥 } ) ⊆ 𝐴 ) → 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ) |
13 |
|
simplr |
⊢ ( ( ( ( ( ( ( ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝐴 ∈ ( unifTop ‘ 𝑈 ) ) ∧ 𝑏 ∈ ( unifTop ‘ ( 𝑈 ↾t ( 𝐴 × 𝐴 ) ) ) ) ∧ 𝑥 ∈ 𝑏 ) ∧ 𝑢 ∈ ( 𝑈 ↾t ( 𝐴 × 𝐴 ) ) ) ∧ ( 𝑢 “ { 𝑥 } ) ⊆ 𝑏 ) ∧ 𝑤 ∈ 𝑈 ) ∧ 𝑢 = ( 𝑤 ∩ ( 𝐴 × 𝐴 ) ) ) ∧ 𝑡 ∈ 𝑈 ) ∧ ( 𝑡 “ { 𝑥 } ) ⊆ 𝐴 ) → 𝑡 ∈ 𝑈 ) |
14 |
|
simp-4r |
⊢ ( ( ( ( ( ( ( ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝐴 ∈ ( unifTop ‘ 𝑈 ) ) ∧ 𝑏 ∈ ( unifTop ‘ ( 𝑈 ↾t ( 𝐴 × 𝐴 ) ) ) ) ∧ 𝑥 ∈ 𝑏 ) ∧ 𝑢 ∈ ( 𝑈 ↾t ( 𝐴 × 𝐴 ) ) ) ∧ ( 𝑢 “ { 𝑥 } ) ⊆ 𝑏 ) ∧ 𝑤 ∈ 𝑈 ) ∧ 𝑢 = ( 𝑤 ∩ ( 𝐴 × 𝐴 ) ) ) ∧ 𝑡 ∈ 𝑈 ) ∧ ( 𝑡 “ { 𝑥 } ) ⊆ 𝐴 ) → 𝑤 ∈ 𝑈 ) |
15 |
|
ustincl |
⊢ ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑡 ∈ 𝑈 ∧ 𝑤 ∈ 𝑈 ) → ( 𝑡 ∩ 𝑤 ) ∈ 𝑈 ) |
16 |
12 13 14 15
|
syl3anc |
⊢ ( ( ( ( ( ( ( ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝐴 ∈ ( unifTop ‘ 𝑈 ) ) ∧ 𝑏 ∈ ( unifTop ‘ ( 𝑈 ↾t ( 𝐴 × 𝐴 ) ) ) ) ∧ 𝑥 ∈ 𝑏 ) ∧ 𝑢 ∈ ( 𝑈 ↾t ( 𝐴 × 𝐴 ) ) ) ∧ ( 𝑢 “ { 𝑥 } ) ⊆ 𝑏 ) ∧ 𝑤 ∈ 𝑈 ) ∧ 𝑢 = ( 𝑤 ∩ ( 𝐴 × 𝐴 ) ) ) ∧ 𝑡 ∈ 𝑈 ) ∧ ( 𝑡 “ { 𝑥 } ) ⊆ 𝐴 ) → ( 𝑡 ∩ 𝑤 ) ∈ 𝑈 ) |
17 |
|
inimass |
⊢ ( ( 𝑡 ∩ 𝑤 ) “ { 𝑥 } ) ⊆ ( ( 𝑡 “ { 𝑥 } ) ∩ ( 𝑤 “ { 𝑥 } ) ) |
18 |
|
ssrin |
⊢ ( ( 𝑡 “ { 𝑥 } ) ⊆ 𝐴 → ( ( 𝑡 “ { 𝑥 } ) ∩ ( 𝑤 “ { 𝑥 } ) ) ⊆ ( 𝐴 ∩ ( 𝑤 “ { 𝑥 } ) ) ) |
19 |
18
|
adantl |
⊢ ( ( ( ( ( ( ( ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝐴 ∈ ( unifTop ‘ 𝑈 ) ) ∧ 𝑏 ∈ ( unifTop ‘ ( 𝑈 ↾t ( 𝐴 × 𝐴 ) ) ) ) ∧ 𝑥 ∈ 𝑏 ) ∧ 𝑢 ∈ ( 𝑈 ↾t ( 𝐴 × 𝐴 ) ) ) ∧ ( 𝑢 “ { 𝑥 } ) ⊆ 𝑏 ) ∧ 𝑤 ∈ 𝑈 ) ∧ 𝑢 = ( 𝑤 ∩ ( 𝐴 × 𝐴 ) ) ) ∧ 𝑡 ∈ 𝑈 ) ∧ ( 𝑡 “ { 𝑥 } ) ⊆ 𝐴 ) → ( ( 𝑡 “ { 𝑥 } ) ∩ ( 𝑤 “ { 𝑥 } ) ) ⊆ ( 𝐴 ∩ ( 𝑤 “ { 𝑥 } ) ) ) |
20 |
|
simpllr |
⊢ ( ( ( ( ( ( ( ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝐴 ∈ ( unifTop ‘ 𝑈 ) ) ∧ 𝑏 ∈ ( unifTop ‘ ( 𝑈 ↾t ( 𝐴 × 𝐴 ) ) ) ) ∧ 𝑥 ∈ 𝑏 ) ∧ 𝑢 ∈ ( 𝑈 ↾t ( 𝐴 × 𝐴 ) ) ) ∧ ( 𝑢 “ { 𝑥 } ) ⊆ 𝑏 ) ∧ 𝑤 ∈ 𝑈 ) ∧ 𝑢 = ( 𝑤 ∩ ( 𝐴 × 𝐴 ) ) ) ∧ 𝑡 ∈ 𝑈 ) ∧ ( 𝑡 “ { 𝑥 } ) ⊆ 𝐴 ) → 𝑢 = ( 𝑤 ∩ ( 𝐴 × 𝐴 ) ) ) |
21 |
20
|
imaeq1d |
⊢ ( ( ( ( ( ( ( ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝐴 ∈ ( unifTop ‘ 𝑈 ) ) ∧ 𝑏 ∈ ( unifTop ‘ ( 𝑈 ↾t ( 𝐴 × 𝐴 ) ) ) ) ∧ 𝑥 ∈ 𝑏 ) ∧ 𝑢 ∈ ( 𝑈 ↾t ( 𝐴 × 𝐴 ) ) ) ∧ ( 𝑢 “ { 𝑥 } ) ⊆ 𝑏 ) ∧ 𝑤 ∈ 𝑈 ) ∧ 𝑢 = ( 𝑤 ∩ ( 𝐴 × 𝐴 ) ) ) ∧ 𝑡 ∈ 𝑈 ) ∧ ( 𝑡 “ { 𝑥 } ) ⊆ 𝐴 ) → ( 𝑢 “ { 𝑥 } ) = ( ( 𝑤 ∩ ( 𝐴 × 𝐴 ) ) “ { 𝑥 } ) ) |
22 |
9
|
ad5antr |
⊢ ( ( ( ( ( ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝐴 ∈ ( unifTop ‘ 𝑈 ) ) ∧ 𝑏 ∈ ( unifTop ‘ ( 𝑈 ↾t ( 𝐴 × 𝐴 ) ) ) ) ∧ 𝑥 ∈ 𝑏 ) ∧ 𝑢 ∈ ( 𝑈 ↾t ( 𝐴 × 𝐴 ) ) ) ∧ ( 𝑢 “ { 𝑥 } ) ⊆ 𝑏 ) ∧ 𝑤 ∈ 𝑈 ) ∧ 𝑢 = ( 𝑤 ∩ ( 𝐴 × 𝐴 ) ) ) → 𝑏 ⊆ 𝐴 ) |
23 |
|
simp-5r |
⊢ ( ( ( ( ( ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝐴 ∈ ( unifTop ‘ 𝑈 ) ) ∧ 𝑏 ∈ ( unifTop ‘ ( 𝑈 ↾t ( 𝐴 × 𝐴 ) ) ) ) ∧ 𝑥 ∈ 𝑏 ) ∧ 𝑢 ∈ ( 𝑈 ↾t ( 𝐴 × 𝐴 ) ) ) ∧ ( 𝑢 “ { 𝑥 } ) ⊆ 𝑏 ) ∧ 𝑤 ∈ 𝑈 ) ∧ 𝑢 = ( 𝑤 ∩ ( 𝐴 × 𝐴 ) ) ) → 𝑥 ∈ 𝑏 ) |
24 |
22 23
|
sseldd |
⊢ ( ( ( ( ( ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝐴 ∈ ( unifTop ‘ 𝑈 ) ) ∧ 𝑏 ∈ ( unifTop ‘ ( 𝑈 ↾t ( 𝐴 × 𝐴 ) ) ) ) ∧ 𝑥 ∈ 𝑏 ) ∧ 𝑢 ∈ ( 𝑈 ↾t ( 𝐴 × 𝐴 ) ) ) ∧ ( 𝑢 “ { 𝑥 } ) ⊆ 𝑏 ) ∧ 𝑤 ∈ 𝑈 ) ∧ 𝑢 = ( 𝑤 ∩ ( 𝐴 × 𝐴 ) ) ) → 𝑥 ∈ 𝐴 ) |
25 |
24
|
ad2antrr |
⊢ ( ( ( ( ( ( ( ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝐴 ∈ ( unifTop ‘ 𝑈 ) ) ∧ 𝑏 ∈ ( unifTop ‘ ( 𝑈 ↾t ( 𝐴 × 𝐴 ) ) ) ) ∧ 𝑥 ∈ 𝑏 ) ∧ 𝑢 ∈ ( 𝑈 ↾t ( 𝐴 × 𝐴 ) ) ) ∧ ( 𝑢 “ { 𝑥 } ) ⊆ 𝑏 ) ∧ 𝑤 ∈ 𝑈 ) ∧ 𝑢 = ( 𝑤 ∩ ( 𝐴 × 𝐴 ) ) ) ∧ 𝑡 ∈ 𝑈 ) ∧ ( 𝑡 “ { 𝑥 } ) ⊆ 𝐴 ) → 𝑥 ∈ 𝐴 ) |
26 |
|
inimasn |
⊢ ( 𝑥 ∈ V → ( ( 𝑤 ∩ ( 𝐴 × 𝐴 ) ) “ { 𝑥 } ) = ( ( 𝑤 “ { 𝑥 } ) ∩ ( ( 𝐴 × 𝐴 ) “ { 𝑥 } ) ) ) |
27 |
26
|
elv |
⊢ ( ( 𝑤 ∩ ( 𝐴 × 𝐴 ) ) “ { 𝑥 } ) = ( ( 𝑤 “ { 𝑥 } ) ∩ ( ( 𝐴 × 𝐴 ) “ { 𝑥 } ) ) |
28 |
|
xpimasn |
⊢ ( 𝑥 ∈ 𝐴 → ( ( 𝐴 × 𝐴 ) “ { 𝑥 } ) = 𝐴 ) |
29 |
28
|
ineq2d |
⊢ ( 𝑥 ∈ 𝐴 → ( ( 𝑤 “ { 𝑥 } ) ∩ ( ( 𝐴 × 𝐴 ) “ { 𝑥 } ) ) = ( ( 𝑤 “ { 𝑥 } ) ∩ 𝐴 ) ) |
30 |
27 29
|
eqtrid |
⊢ ( 𝑥 ∈ 𝐴 → ( ( 𝑤 ∩ ( 𝐴 × 𝐴 ) ) “ { 𝑥 } ) = ( ( 𝑤 “ { 𝑥 } ) ∩ 𝐴 ) ) |
31 |
|
incom |
⊢ ( ( 𝑤 “ { 𝑥 } ) ∩ 𝐴 ) = ( 𝐴 ∩ ( 𝑤 “ { 𝑥 } ) ) |
32 |
30 31
|
eqtrdi |
⊢ ( 𝑥 ∈ 𝐴 → ( ( 𝑤 ∩ ( 𝐴 × 𝐴 ) ) “ { 𝑥 } ) = ( 𝐴 ∩ ( 𝑤 “ { 𝑥 } ) ) ) |
33 |
25 32
|
syl |
⊢ ( ( ( ( ( ( ( ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝐴 ∈ ( unifTop ‘ 𝑈 ) ) ∧ 𝑏 ∈ ( unifTop ‘ ( 𝑈 ↾t ( 𝐴 × 𝐴 ) ) ) ) ∧ 𝑥 ∈ 𝑏 ) ∧ 𝑢 ∈ ( 𝑈 ↾t ( 𝐴 × 𝐴 ) ) ) ∧ ( 𝑢 “ { 𝑥 } ) ⊆ 𝑏 ) ∧ 𝑤 ∈ 𝑈 ) ∧ 𝑢 = ( 𝑤 ∩ ( 𝐴 × 𝐴 ) ) ) ∧ 𝑡 ∈ 𝑈 ) ∧ ( 𝑡 “ { 𝑥 } ) ⊆ 𝐴 ) → ( ( 𝑤 ∩ ( 𝐴 × 𝐴 ) ) “ { 𝑥 } ) = ( 𝐴 ∩ ( 𝑤 “ { 𝑥 } ) ) ) |
34 |
21 33
|
eqtrd |
⊢ ( ( ( ( ( ( ( ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝐴 ∈ ( unifTop ‘ 𝑈 ) ) ∧ 𝑏 ∈ ( unifTop ‘ ( 𝑈 ↾t ( 𝐴 × 𝐴 ) ) ) ) ∧ 𝑥 ∈ 𝑏 ) ∧ 𝑢 ∈ ( 𝑈 ↾t ( 𝐴 × 𝐴 ) ) ) ∧ ( 𝑢 “ { 𝑥 } ) ⊆ 𝑏 ) ∧ 𝑤 ∈ 𝑈 ) ∧ 𝑢 = ( 𝑤 ∩ ( 𝐴 × 𝐴 ) ) ) ∧ 𝑡 ∈ 𝑈 ) ∧ ( 𝑡 “ { 𝑥 } ) ⊆ 𝐴 ) → ( 𝑢 “ { 𝑥 } ) = ( 𝐴 ∩ ( 𝑤 “ { 𝑥 } ) ) ) |
35 |
19 34
|
sseqtrrd |
⊢ ( ( ( ( ( ( ( ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝐴 ∈ ( unifTop ‘ 𝑈 ) ) ∧ 𝑏 ∈ ( unifTop ‘ ( 𝑈 ↾t ( 𝐴 × 𝐴 ) ) ) ) ∧ 𝑥 ∈ 𝑏 ) ∧ 𝑢 ∈ ( 𝑈 ↾t ( 𝐴 × 𝐴 ) ) ) ∧ ( 𝑢 “ { 𝑥 } ) ⊆ 𝑏 ) ∧ 𝑤 ∈ 𝑈 ) ∧ 𝑢 = ( 𝑤 ∩ ( 𝐴 × 𝐴 ) ) ) ∧ 𝑡 ∈ 𝑈 ) ∧ ( 𝑡 “ { 𝑥 } ) ⊆ 𝐴 ) → ( ( 𝑡 “ { 𝑥 } ) ∩ ( 𝑤 “ { 𝑥 } ) ) ⊆ ( 𝑢 “ { 𝑥 } ) ) |
36 |
|
simp-5r |
⊢ ( ( ( ( ( ( ( ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝐴 ∈ ( unifTop ‘ 𝑈 ) ) ∧ 𝑏 ∈ ( unifTop ‘ ( 𝑈 ↾t ( 𝐴 × 𝐴 ) ) ) ) ∧ 𝑥 ∈ 𝑏 ) ∧ 𝑢 ∈ ( 𝑈 ↾t ( 𝐴 × 𝐴 ) ) ) ∧ ( 𝑢 “ { 𝑥 } ) ⊆ 𝑏 ) ∧ 𝑤 ∈ 𝑈 ) ∧ 𝑢 = ( 𝑤 ∩ ( 𝐴 × 𝐴 ) ) ) ∧ 𝑡 ∈ 𝑈 ) ∧ ( 𝑡 “ { 𝑥 } ) ⊆ 𝐴 ) → ( 𝑢 “ { 𝑥 } ) ⊆ 𝑏 ) |
37 |
35 36
|
sstrd |
⊢ ( ( ( ( ( ( ( ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝐴 ∈ ( unifTop ‘ 𝑈 ) ) ∧ 𝑏 ∈ ( unifTop ‘ ( 𝑈 ↾t ( 𝐴 × 𝐴 ) ) ) ) ∧ 𝑥 ∈ 𝑏 ) ∧ 𝑢 ∈ ( 𝑈 ↾t ( 𝐴 × 𝐴 ) ) ) ∧ ( 𝑢 “ { 𝑥 } ) ⊆ 𝑏 ) ∧ 𝑤 ∈ 𝑈 ) ∧ 𝑢 = ( 𝑤 ∩ ( 𝐴 × 𝐴 ) ) ) ∧ 𝑡 ∈ 𝑈 ) ∧ ( 𝑡 “ { 𝑥 } ) ⊆ 𝐴 ) → ( ( 𝑡 “ { 𝑥 } ) ∩ ( 𝑤 “ { 𝑥 } ) ) ⊆ 𝑏 ) |
38 |
17 37
|
sstrid |
⊢ ( ( ( ( ( ( ( ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝐴 ∈ ( unifTop ‘ 𝑈 ) ) ∧ 𝑏 ∈ ( unifTop ‘ ( 𝑈 ↾t ( 𝐴 × 𝐴 ) ) ) ) ∧ 𝑥 ∈ 𝑏 ) ∧ 𝑢 ∈ ( 𝑈 ↾t ( 𝐴 × 𝐴 ) ) ) ∧ ( 𝑢 “ { 𝑥 } ) ⊆ 𝑏 ) ∧ 𝑤 ∈ 𝑈 ) ∧ 𝑢 = ( 𝑤 ∩ ( 𝐴 × 𝐴 ) ) ) ∧ 𝑡 ∈ 𝑈 ) ∧ ( 𝑡 “ { 𝑥 } ) ⊆ 𝐴 ) → ( ( 𝑡 ∩ 𝑤 ) “ { 𝑥 } ) ⊆ 𝑏 ) |
39 |
|
imaeq1 |
⊢ ( 𝑣 = ( 𝑡 ∩ 𝑤 ) → ( 𝑣 “ { 𝑥 } ) = ( ( 𝑡 ∩ 𝑤 ) “ { 𝑥 } ) ) |
40 |
39
|
sseq1d |
⊢ ( 𝑣 = ( 𝑡 ∩ 𝑤 ) → ( ( 𝑣 “ { 𝑥 } ) ⊆ 𝑏 ↔ ( ( 𝑡 ∩ 𝑤 ) “ { 𝑥 } ) ⊆ 𝑏 ) ) |
41 |
40
|
rspcev |
⊢ ( ( ( 𝑡 ∩ 𝑤 ) ∈ 𝑈 ∧ ( ( 𝑡 ∩ 𝑤 ) “ { 𝑥 } ) ⊆ 𝑏 ) → ∃ 𝑣 ∈ 𝑈 ( 𝑣 “ { 𝑥 } ) ⊆ 𝑏 ) |
42 |
16 38 41
|
syl2anc |
⊢ ( ( ( ( ( ( ( ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝐴 ∈ ( unifTop ‘ 𝑈 ) ) ∧ 𝑏 ∈ ( unifTop ‘ ( 𝑈 ↾t ( 𝐴 × 𝐴 ) ) ) ) ∧ 𝑥 ∈ 𝑏 ) ∧ 𝑢 ∈ ( 𝑈 ↾t ( 𝐴 × 𝐴 ) ) ) ∧ ( 𝑢 “ { 𝑥 } ) ⊆ 𝑏 ) ∧ 𝑤 ∈ 𝑈 ) ∧ 𝑢 = ( 𝑤 ∩ ( 𝐴 × 𝐴 ) ) ) ∧ 𝑡 ∈ 𝑈 ) ∧ ( 𝑡 “ { 𝑥 } ) ⊆ 𝐴 ) → ∃ 𝑣 ∈ 𝑈 ( 𝑣 “ { 𝑥 } ) ⊆ 𝑏 ) |
43 |
|
simp-4l |
⊢ ( ( ( ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝐴 ∈ ( unifTop ‘ 𝑈 ) ) ∧ 𝑏 ∈ ( unifTop ‘ ( 𝑈 ↾t ( 𝐴 × 𝐴 ) ) ) ) ∧ 𝑥 ∈ 𝑏 ) ∧ 𝑢 ∈ ( 𝑈 ↾t ( 𝐴 × 𝐴 ) ) ) ∧ ( 𝑢 “ { 𝑥 } ) ⊆ 𝑏 ) → ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝐴 ∈ ( unifTop ‘ 𝑈 ) ) ) |
44 |
43
|
ad2antrr |
⊢ ( ( ( ( ( ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝐴 ∈ ( unifTop ‘ 𝑈 ) ) ∧ 𝑏 ∈ ( unifTop ‘ ( 𝑈 ↾t ( 𝐴 × 𝐴 ) ) ) ) ∧ 𝑥 ∈ 𝑏 ) ∧ 𝑢 ∈ ( 𝑈 ↾t ( 𝐴 × 𝐴 ) ) ) ∧ ( 𝑢 “ { 𝑥 } ) ⊆ 𝑏 ) ∧ 𝑤 ∈ 𝑈 ) ∧ 𝑢 = ( 𝑤 ∩ ( 𝐴 × 𝐴 ) ) ) → ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝐴 ∈ ( unifTop ‘ 𝑈 ) ) ) |
45 |
1
|
simplbda |
⊢ ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝐴 ∈ ( unifTop ‘ 𝑈 ) ) → ∀ 𝑥 ∈ 𝐴 ∃ 𝑡 ∈ 𝑈 ( 𝑡 “ { 𝑥 } ) ⊆ 𝐴 ) |
46 |
45
|
r19.21bi |
⊢ ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝐴 ∈ ( unifTop ‘ 𝑈 ) ) ∧ 𝑥 ∈ 𝐴 ) → ∃ 𝑡 ∈ 𝑈 ( 𝑡 “ { 𝑥 } ) ⊆ 𝐴 ) |
47 |
44 24 46
|
syl2anc |
⊢ ( ( ( ( ( ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝐴 ∈ ( unifTop ‘ 𝑈 ) ) ∧ 𝑏 ∈ ( unifTop ‘ ( 𝑈 ↾t ( 𝐴 × 𝐴 ) ) ) ) ∧ 𝑥 ∈ 𝑏 ) ∧ 𝑢 ∈ ( 𝑈 ↾t ( 𝐴 × 𝐴 ) ) ) ∧ ( 𝑢 “ { 𝑥 } ) ⊆ 𝑏 ) ∧ 𝑤 ∈ 𝑈 ) ∧ 𝑢 = ( 𝑤 ∩ ( 𝐴 × 𝐴 ) ) ) → ∃ 𝑡 ∈ 𝑈 ( 𝑡 “ { 𝑥 } ) ⊆ 𝐴 ) |
48 |
42 47
|
r19.29a |
⊢ ( ( ( ( ( ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝐴 ∈ ( unifTop ‘ 𝑈 ) ) ∧ 𝑏 ∈ ( unifTop ‘ ( 𝑈 ↾t ( 𝐴 × 𝐴 ) ) ) ) ∧ 𝑥 ∈ 𝑏 ) ∧ 𝑢 ∈ ( 𝑈 ↾t ( 𝐴 × 𝐴 ) ) ) ∧ ( 𝑢 “ { 𝑥 } ) ⊆ 𝑏 ) ∧ 𝑤 ∈ 𝑈 ) ∧ 𝑢 = ( 𝑤 ∩ ( 𝐴 × 𝐴 ) ) ) → ∃ 𝑣 ∈ 𝑈 ( 𝑣 “ { 𝑥 } ) ⊆ 𝑏 ) |
49 |
|
simplr |
⊢ ( ( ( ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝐴 ∈ ( unifTop ‘ 𝑈 ) ) ∧ 𝑏 ∈ ( unifTop ‘ ( 𝑈 ↾t ( 𝐴 × 𝐴 ) ) ) ) ∧ 𝑥 ∈ 𝑏 ) ∧ 𝑢 ∈ ( 𝑈 ↾t ( 𝐴 × 𝐴 ) ) ) ∧ ( 𝑢 “ { 𝑥 } ) ⊆ 𝑏 ) → 𝑢 ∈ ( 𝑈 ↾t ( 𝐴 × 𝐴 ) ) ) |
50 |
|
sqxpexg |
⊢ ( 𝐴 ∈ ( unifTop ‘ 𝑈 ) → ( 𝐴 × 𝐴 ) ∈ V ) |
51 |
|
elrest |
⊢ ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ ( 𝐴 × 𝐴 ) ∈ V ) → ( 𝑢 ∈ ( 𝑈 ↾t ( 𝐴 × 𝐴 ) ) ↔ ∃ 𝑤 ∈ 𝑈 𝑢 = ( 𝑤 ∩ ( 𝐴 × 𝐴 ) ) ) ) |
52 |
50 51
|
sylan2 |
⊢ ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝐴 ∈ ( unifTop ‘ 𝑈 ) ) → ( 𝑢 ∈ ( 𝑈 ↾t ( 𝐴 × 𝐴 ) ) ↔ ∃ 𝑤 ∈ 𝑈 𝑢 = ( 𝑤 ∩ ( 𝐴 × 𝐴 ) ) ) ) |
53 |
52
|
biimpa |
⊢ ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝐴 ∈ ( unifTop ‘ 𝑈 ) ) ∧ 𝑢 ∈ ( 𝑈 ↾t ( 𝐴 × 𝐴 ) ) ) → ∃ 𝑤 ∈ 𝑈 𝑢 = ( 𝑤 ∩ ( 𝐴 × 𝐴 ) ) ) |
54 |
43 49 53
|
syl2anc |
⊢ ( ( ( ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝐴 ∈ ( unifTop ‘ 𝑈 ) ) ∧ 𝑏 ∈ ( unifTop ‘ ( 𝑈 ↾t ( 𝐴 × 𝐴 ) ) ) ) ∧ 𝑥 ∈ 𝑏 ) ∧ 𝑢 ∈ ( 𝑈 ↾t ( 𝐴 × 𝐴 ) ) ) ∧ ( 𝑢 “ { 𝑥 } ) ⊆ 𝑏 ) → ∃ 𝑤 ∈ 𝑈 𝑢 = ( 𝑤 ∩ ( 𝐴 × 𝐴 ) ) ) |
55 |
48 54
|
r19.29a |
⊢ ( ( ( ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝐴 ∈ ( unifTop ‘ 𝑈 ) ) ∧ 𝑏 ∈ ( unifTop ‘ ( 𝑈 ↾t ( 𝐴 × 𝐴 ) ) ) ) ∧ 𝑥 ∈ 𝑏 ) ∧ 𝑢 ∈ ( 𝑈 ↾t ( 𝐴 × 𝐴 ) ) ) ∧ ( 𝑢 “ { 𝑥 } ) ⊆ 𝑏 ) → ∃ 𝑣 ∈ 𝑈 ( 𝑣 “ { 𝑥 } ) ⊆ 𝑏 ) |
56 |
8
|
simplbda |
⊢ ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝐴 ∈ ( unifTop ‘ 𝑈 ) ) ∧ 𝑏 ∈ ( unifTop ‘ ( 𝑈 ↾t ( 𝐴 × 𝐴 ) ) ) ) → ∀ 𝑥 ∈ 𝑏 ∃ 𝑢 ∈ ( 𝑈 ↾t ( 𝐴 × 𝐴 ) ) ( 𝑢 “ { 𝑥 } ) ⊆ 𝑏 ) |
57 |
56
|
r19.21bi |
⊢ ( ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝐴 ∈ ( unifTop ‘ 𝑈 ) ) ∧ 𝑏 ∈ ( unifTop ‘ ( 𝑈 ↾t ( 𝐴 × 𝐴 ) ) ) ) ∧ 𝑥 ∈ 𝑏 ) → ∃ 𝑢 ∈ ( 𝑈 ↾t ( 𝐴 × 𝐴 ) ) ( 𝑢 “ { 𝑥 } ) ⊆ 𝑏 ) |
58 |
55 57
|
r19.29a |
⊢ ( ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝐴 ∈ ( unifTop ‘ 𝑈 ) ) ∧ 𝑏 ∈ ( unifTop ‘ ( 𝑈 ↾t ( 𝐴 × 𝐴 ) ) ) ) ∧ 𝑥 ∈ 𝑏 ) → ∃ 𝑣 ∈ 𝑈 ( 𝑣 “ { 𝑥 } ) ⊆ 𝑏 ) |
59 |
58
|
ralrimiva |
⊢ ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝐴 ∈ ( unifTop ‘ 𝑈 ) ) ∧ 𝑏 ∈ ( unifTop ‘ ( 𝑈 ↾t ( 𝐴 × 𝐴 ) ) ) ) → ∀ 𝑥 ∈ 𝑏 ∃ 𝑣 ∈ 𝑈 ( 𝑣 “ { 𝑥 } ) ⊆ 𝑏 ) |
60 |
|
elutop |
⊢ ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) → ( 𝑏 ∈ ( unifTop ‘ 𝑈 ) ↔ ( 𝑏 ⊆ 𝑋 ∧ ∀ 𝑥 ∈ 𝑏 ∃ 𝑣 ∈ 𝑈 ( 𝑣 “ { 𝑥 } ) ⊆ 𝑏 ) ) ) |
61 |
60
|
ad2antrr |
⊢ ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝐴 ∈ ( unifTop ‘ 𝑈 ) ) ∧ 𝑏 ∈ ( unifTop ‘ ( 𝑈 ↾t ( 𝐴 × 𝐴 ) ) ) ) → ( 𝑏 ∈ ( unifTop ‘ 𝑈 ) ↔ ( 𝑏 ⊆ 𝑋 ∧ ∀ 𝑥 ∈ 𝑏 ∃ 𝑣 ∈ 𝑈 ( 𝑣 “ { 𝑥 } ) ⊆ 𝑏 ) ) ) |
62 |
11 59 61
|
mpbir2and |
⊢ ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝐴 ∈ ( unifTop ‘ 𝑈 ) ) ∧ 𝑏 ∈ ( unifTop ‘ ( 𝑈 ↾t ( 𝐴 × 𝐴 ) ) ) ) → 𝑏 ∈ ( unifTop ‘ 𝑈 ) ) |
63 |
|
df-ss |
⊢ ( 𝑏 ⊆ 𝐴 ↔ ( 𝑏 ∩ 𝐴 ) = 𝑏 ) |
64 |
9 63
|
sylib |
⊢ ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝐴 ∈ ( unifTop ‘ 𝑈 ) ) ∧ 𝑏 ∈ ( unifTop ‘ ( 𝑈 ↾t ( 𝐴 × 𝐴 ) ) ) ) → ( 𝑏 ∩ 𝐴 ) = 𝑏 ) |
65 |
64
|
eqcomd |
⊢ ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝐴 ∈ ( unifTop ‘ 𝑈 ) ) ∧ 𝑏 ∈ ( unifTop ‘ ( 𝑈 ↾t ( 𝐴 × 𝐴 ) ) ) ) → 𝑏 = ( 𝑏 ∩ 𝐴 ) ) |
66 |
|
ineq1 |
⊢ ( 𝑎 = 𝑏 → ( 𝑎 ∩ 𝐴 ) = ( 𝑏 ∩ 𝐴 ) ) |
67 |
66
|
rspceeqv |
⊢ ( ( 𝑏 ∈ ( unifTop ‘ 𝑈 ) ∧ 𝑏 = ( 𝑏 ∩ 𝐴 ) ) → ∃ 𝑎 ∈ ( unifTop ‘ 𝑈 ) 𝑏 = ( 𝑎 ∩ 𝐴 ) ) |
68 |
62 65 67
|
syl2anc |
⊢ ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝐴 ∈ ( unifTop ‘ 𝑈 ) ) ∧ 𝑏 ∈ ( unifTop ‘ ( 𝑈 ↾t ( 𝐴 × 𝐴 ) ) ) ) → ∃ 𝑎 ∈ ( unifTop ‘ 𝑈 ) 𝑏 = ( 𝑎 ∩ 𝐴 ) ) |
69 |
|
fvex |
⊢ ( unifTop ‘ 𝑈 ) ∈ V |
70 |
|
elrest |
⊢ ( ( ( unifTop ‘ 𝑈 ) ∈ V ∧ 𝐴 ∈ ( unifTop ‘ 𝑈 ) ) → ( 𝑏 ∈ ( ( unifTop ‘ 𝑈 ) ↾t 𝐴 ) ↔ ∃ 𝑎 ∈ ( unifTop ‘ 𝑈 ) 𝑏 = ( 𝑎 ∩ 𝐴 ) ) ) |
71 |
69 70
|
mpan |
⊢ ( 𝐴 ∈ ( unifTop ‘ 𝑈 ) → ( 𝑏 ∈ ( ( unifTop ‘ 𝑈 ) ↾t 𝐴 ) ↔ ∃ 𝑎 ∈ ( unifTop ‘ 𝑈 ) 𝑏 = ( 𝑎 ∩ 𝐴 ) ) ) |
72 |
71
|
ad2antlr |
⊢ ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝐴 ∈ ( unifTop ‘ 𝑈 ) ) ∧ 𝑏 ∈ ( unifTop ‘ ( 𝑈 ↾t ( 𝐴 × 𝐴 ) ) ) ) → ( 𝑏 ∈ ( ( unifTop ‘ 𝑈 ) ↾t 𝐴 ) ↔ ∃ 𝑎 ∈ ( unifTop ‘ 𝑈 ) 𝑏 = ( 𝑎 ∩ 𝐴 ) ) ) |
73 |
68 72
|
mpbird |
⊢ ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝐴 ∈ ( unifTop ‘ 𝑈 ) ) ∧ 𝑏 ∈ ( unifTop ‘ ( 𝑈 ↾t ( 𝐴 × 𝐴 ) ) ) ) → 𝑏 ∈ ( ( unifTop ‘ 𝑈 ) ↾t 𝐴 ) ) |
74 |
4 73
|
eqelssd |
⊢ ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝐴 ∈ ( unifTop ‘ 𝑈 ) ) → ( ( unifTop ‘ 𝑈 ) ↾t 𝐴 ) = ( unifTop ‘ ( 𝑈 ↾t ( 𝐴 × 𝐴 ) ) ) ) |