Step |
Hyp |
Ref |
Expression |
1 |
|
simpr |
⊢ ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑥 ⊆ ( unifTop ‘ 𝑈 ) ) → 𝑥 ⊆ ( unifTop ‘ 𝑈 ) ) |
2 |
|
utopval |
⊢ ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) → ( unifTop ‘ 𝑈 ) = { 𝑎 ∈ 𝒫 𝑋 ∣ ∀ 𝑝 ∈ 𝑎 ∃ 𝑣 ∈ 𝑈 ( 𝑣 “ { 𝑝 } ) ⊆ 𝑎 } ) |
3 |
|
ssrab2 |
⊢ { 𝑎 ∈ 𝒫 𝑋 ∣ ∀ 𝑝 ∈ 𝑎 ∃ 𝑣 ∈ 𝑈 ( 𝑣 “ { 𝑝 } ) ⊆ 𝑎 } ⊆ 𝒫 𝑋 |
4 |
2 3
|
eqsstrdi |
⊢ ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) → ( unifTop ‘ 𝑈 ) ⊆ 𝒫 𝑋 ) |
5 |
4
|
adantr |
⊢ ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑥 ⊆ ( unifTop ‘ 𝑈 ) ) → ( unifTop ‘ 𝑈 ) ⊆ 𝒫 𝑋 ) |
6 |
1 5
|
sstrd |
⊢ ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑥 ⊆ ( unifTop ‘ 𝑈 ) ) → 𝑥 ⊆ 𝒫 𝑋 ) |
7 |
|
sspwuni |
⊢ ( 𝑥 ⊆ 𝒫 𝑋 ↔ ∪ 𝑥 ⊆ 𝑋 ) |
8 |
6 7
|
sylib |
⊢ ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑥 ⊆ ( unifTop ‘ 𝑈 ) ) → ∪ 𝑥 ⊆ 𝑋 ) |
9 |
|
simp-4l |
⊢ ( ( ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑥 ⊆ ( unifTop ‘ 𝑈 ) ) ∧ 𝑝 ∈ ∪ 𝑥 ) ∧ 𝑦 ∈ 𝑥 ) ∧ 𝑝 ∈ 𝑦 ) → 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ) |
10 |
|
simp-4r |
⊢ ( ( ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑥 ⊆ ( unifTop ‘ 𝑈 ) ) ∧ 𝑝 ∈ ∪ 𝑥 ) ∧ 𝑦 ∈ 𝑥 ) ∧ 𝑝 ∈ 𝑦 ) → 𝑥 ⊆ ( unifTop ‘ 𝑈 ) ) |
11 |
|
simplr |
⊢ ( ( ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑥 ⊆ ( unifTop ‘ 𝑈 ) ) ∧ 𝑝 ∈ ∪ 𝑥 ) ∧ 𝑦 ∈ 𝑥 ) ∧ 𝑝 ∈ 𝑦 ) → 𝑦 ∈ 𝑥 ) |
12 |
10 11
|
sseldd |
⊢ ( ( ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑥 ⊆ ( unifTop ‘ 𝑈 ) ) ∧ 𝑝 ∈ ∪ 𝑥 ) ∧ 𝑦 ∈ 𝑥 ) ∧ 𝑝 ∈ 𝑦 ) → 𝑦 ∈ ( unifTop ‘ 𝑈 ) ) |
13 |
|
simpr |
⊢ ( ( ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑥 ⊆ ( unifTop ‘ 𝑈 ) ) ∧ 𝑝 ∈ ∪ 𝑥 ) ∧ 𝑦 ∈ 𝑥 ) ∧ 𝑝 ∈ 𝑦 ) → 𝑝 ∈ 𝑦 ) |
14 |
|
elutop |
⊢ ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) → ( 𝑦 ∈ ( unifTop ‘ 𝑈 ) ↔ ( 𝑦 ⊆ 𝑋 ∧ ∀ 𝑝 ∈ 𝑦 ∃ 𝑣 ∈ 𝑈 ( 𝑣 “ { 𝑝 } ) ⊆ 𝑦 ) ) ) |
15 |
14
|
biimpa |
⊢ ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑦 ∈ ( unifTop ‘ 𝑈 ) ) → ( 𝑦 ⊆ 𝑋 ∧ ∀ 𝑝 ∈ 𝑦 ∃ 𝑣 ∈ 𝑈 ( 𝑣 “ { 𝑝 } ) ⊆ 𝑦 ) ) |
16 |
15
|
simprd |
⊢ ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑦 ∈ ( unifTop ‘ 𝑈 ) ) → ∀ 𝑝 ∈ 𝑦 ∃ 𝑣 ∈ 𝑈 ( 𝑣 “ { 𝑝 } ) ⊆ 𝑦 ) |
17 |
16
|
r19.21bi |
⊢ ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑦 ∈ ( unifTop ‘ 𝑈 ) ) ∧ 𝑝 ∈ 𝑦 ) → ∃ 𝑣 ∈ 𝑈 ( 𝑣 “ { 𝑝 } ) ⊆ 𝑦 ) |
18 |
9 12 13 17
|
syl21anc |
⊢ ( ( ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑥 ⊆ ( unifTop ‘ 𝑈 ) ) ∧ 𝑝 ∈ ∪ 𝑥 ) ∧ 𝑦 ∈ 𝑥 ) ∧ 𝑝 ∈ 𝑦 ) → ∃ 𝑣 ∈ 𝑈 ( 𝑣 “ { 𝑝 } ) ⊆ 𝑦 ) |
19 |
|
r19.41v |
⊢ ( ∃ 𝑣 ∈ 𝑈 ( ( 𝑣 “ { 𝑝 } ) ⊆ 𝑦 ∧ 𝑦 ∈ 𝑥 ) ↔ ( ∃ 𝑣 ∈ 𝑈 ( 𝑣 “ { 𝑝 } ) ⊆ 𝑦 ∧ 𝑦 ∈ 𝑥 ) ) |
20 |
|
ssuni |
⊢ ( ( ( 𝑣 “ { 𝑝 } ) ⊆ 𝑦 ∧ 𝑦 ∈ 𝑥 ) → ( 𝑣 “ { 𝑝 } ) ⊆ ∪ 𝑥 ) |
21 |
20
|
reximi |
⊢ ( ∃ 𝑣 ∈ 𝑈 ( ( 𝑣 “ { 𝑝 } ) ⊆ 𝑦 ∧ 𝑦 ∈ 𝑥 ) → ∃ 𝑣 ∈ 𝑈 ( 𝑣 “ { 𝑝 } ) ⊆ ∪ 𝑥 ) |
22 |
19 21
|
sylbir |
⊢ ( ( ∃ 𝑣 ∈ 𝑈 ( 𝑣 “ { 𝑝 } ) ⊆ 𝑦 ∧ 𝑦 ∈ 𝑥 ) → ∃ 𝑣 ∈ 𝑈 ( 𝑣 “ { 𝑝 } ) ⊆ ∪ 𝑥 ) |
23 |
18 11 22
|
syl2anc |
⊢ ( ( ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑥 ⊆ ( unifTop ‘ 𝑈 ) ) ∧ 𝑝 ∈ ∪ 𝑥 ) ∧ 𝑦 ∈ 𝑥 ) ∧ 𝑝 ∈ 𝑦 ) → ∃ 𝑣 ∈ 𝑈 ( 𝑣 “ { 𝑝 } ) ⊆ ∪ 𝑥 ) |
24 |
|
eluni2 |
⊢ ( 𝑝 ∈ ∪ 𝑥 ↔ ∃ 𝑦 ∈ 𝑥 𝑝 ∈ 𝑦 ) |
25 |
24
|
biimpi |
⊢ ( 𝑝 ∈ ∪ 𝑥 → ∃ 𝑦 ∈ 𝑥 𝑝 ∈ 𝑦 ) |
26 |
25
|
adantl |
⊢ ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑥 ⊆ ( unifTop ‘ 𝑈 ) ) ∧ 𝑝 ∈ ∪ 𝑥 ) → ∃ 𝑦 ∈ 𝑥 𝑝 ∈ 𝑦 ) |
27 |
23 26
|
r19.29a |
⊢ ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑥 ⊆ ( unifTop ‘ 𝑈 ) ) ∧ 𝑝 ∈ ∪ 𝑥 ) → ∃ 𝑣 ∈ 𝑈 ( 𝑣 “ { 𝑝 } ) ⊆ ∪ 𝑥 ) |
28 |
27
|
ralrimiva |
⊢ ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑥 ⊆ ( unifTop ‘ 𝑈 ) ) → ∀ 𝑝 ∈ ∪ 𝑥 ∃ 𝑣 ∈ 𝑈 ( 𝑣 “ { 𝑝 } ) ⊆ ∪ 𝑥 ) |
29 |
|
elutop |
⊢ ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) → ( ∪ 𝑥 ∈ ( unifTop ‘ 𝑈 ) ↔ ( ∪ 𝑥 ⊆ 𝑋 ∧ ∀ 𝑝 ∈ ∪ 𝑥 ∃ 𝑣 ∈ 𝑈 ( 𝑣 “ { 𝑝 } ) ⊆ ∪ 𝑥 ) ) ) |
30 |
29
|
adantr |
⊢ ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑥 ⊆ ( unifTop ‘ 𝑈 ) ) → ( ∪ 𝑥 ∈ ( unifTop ‘ 𝑈 ) ↔ ( ∪ 𝑥 ⊆ 𝑋 ∧ ∀ 𝑝 ∈ ∪ 𝑥 ∃ 𝑣 ∈ 𝑈 ( 𝑣 “ { 𝑝 } ) ⊆ ∪ 𝑥 ) ) ) |
31 |
8 28 30
|
mpbir2and |
⊢ ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑥 ⊆ ( unifTop ‘ 𝑈 ) ) → ∪ 𝑥 ∈ ( unifTop ‘ 𝑈 ) ) |
32 |
31
|
ex |
⊢ ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) → ( 𝑥 ⊆ ( unifTop ‘ 𝑈 ) → ∪ 𝑥 ∈ ( unifTop ‘ 𝑈 ) ) ) |
33 |
32
|
alrimiv |
⊢ ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) → ∀ 𝑥 ( 𝑥 ⊆ ( unifTop ‘ 𝑈 ) → ∪ 𝑥 ∈ ( unifTop ‘ 𝑈 ) ) ) |
34 |
|
elutop |
⊢ ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) → ( 𝑥 ∈ ( unifTop ‘ 𝑈 ) ↔ ( 𝑥 ⊆ 𝑋 ∧ ∀ 𝑝 ∈ 𝑥 ∃ 𝑢 ∈ 𝑈 ( 𝑢 “ { 𝑝 } ) ⊆ 𝑥 ) ) ) |
35 |
34
|
biimpa |
⊢ ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑥 ∈ ( unifTop ‘ 𝑈 ) ) → ( 𝑥 ⊆ 𝑋 ∧ ∀ 𝑝 ∈ 𝑥 ∃ 𝑢 ∈ 𝑈 ( 𝑢 “ { 𝑝 } ) ⊆ 𝑥 ) ) |
36 |
35
|
simpld |
⊢ ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑥 ∈ ( unifTop ‘ 𝑈 ) ) → 𝑥 ⊆ 𝑋 ) |
37 |
36
|
adantrr |
⊢ ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ ( 𝑥 ∈ ( unifTop ‘ 𝑈 ) ∧ 𝑦 ∈ ( unifTop ‘ 𝑈 ) ) ) → 𝑥 ⊆ 𝑋 ) |
38 |
|
ssinss1 |
⊢ ( 𝑥 ⊆ 𝑋 → ( 𝑥 ∩ 𝑦 ) ⊆ 𝑋 ) |
39 |
37 38
|
syl |
⊢ ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ ( 𝑥 ∈ ( unifTop ‘ 𝑈 ) ∧ 𝑦 ∈ ( unifTop ‘ 𝑈 ) ) ) → ( 𝑥 ∩ 𝑦 ) ⊆ 𝑋 ) |
40 |
|
simpl |
⊢ ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ ( ( 𝑥 ∈ ( unifTop ‘ 𝑈 ) ∧ 𝑦 ∈ ( unifTop ‘ 𝑈 ) ) ∧ 𝑝 ∈ ( 𝑥 ∩ 𝑦 ) ∧ ( 𝑢 ∈ 𝑈 ∧ 𝑣 ∈ 𝑈 ∧ ( ( 𝑢 “ { 𝑝 } ) ⊆ 𝑥 ∧ ( 𝑣 “ { 𝑝 } ) ⊆ 𝑦 ) ) ) ) → 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ) |
41 |
|
simpr31 |
⊢ ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ ( ( 𝑥 ∈ ( unifTop ‘ 𝑈 ) ∧ 𝑦 ∈ ( unifTop ‘ 𝑈 ) ) ∧ 𝑝 ∈ ( 𝑥 ∩ 𝑦 ) ∧ ( 𝑢 ∈ 𝑈 ∧ 𝑣 ∈ 𝑈 ∧ ( ( 𝑢 “ { 𝑝 } ) ⊆ 𝑥 ∧ ( 𝑣 “ { 𝑝 } ) ⊆ 𝑦 ) ) ) ) → 𝑢 ∈ 𝑈 ) |
42 |
|
simpr32 |
⊢ ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ ( ( 𝑥 ∈ ( unifTop ‘ 𝑈 ) ∧ 𝑦 ∈ ( unifTop ‘ 𝑈 ) ) ∧ 𝑝 ∈ ( 𝑥 ∩ 𝑦 ) ∧ ( 𝑢 ∈ 𝑈 ∧ 𝑣 ∈ 𝑈 ∧ ( ( 𝑢 “ { 𝑝 } ) ⊆ 𝑥 ∧ ( 𝑣 “ { 𝑝 } ) ⊆ 𝑦 ) ) ) ) → 𝑣 ∈ 𝑈 ) |
43 |
|
ustincl |
⊢ ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑢 ∈ 𝑈 ∧ 𝑣 ∈ 𝑈 ) → ( 𝑢 ∩ 𝑣 ) ∈ 𝑈 ) |
44 |
40 41 42 43
|
syl3anc |
⊢ ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ ( ( 𝑥 ∈ ( unifTop ‘ 𝑈 ) ∧ 𝑦 ∈ ( unifTop ‘ 𝑈 ) ) ∧ 𝑝 ∈ ( 𝑥 ∩ 𝑦 ) ∧ ( 𝑢 ∈ 𝑈 ∧ 𝑣 ∈ 𝑈 ∧ ( ( 𝑢 “ { 𝑝 } ) ⊆ 𝑥 ∧ ( 𝑣 “ { 𝑝 } ) ⊆ 𝑦 ) ) ) ) → ( 𝑢 ∩ 𝑣 ) ∈ 𝑈 ) |
45 |
|
inss1 |
⊢ ( 𝑢 ∩ 𝑣 ) ⊆ 𝑢 |
46 |
|
imass1 |
⊢ ( ( 𝑢 ∩ 𝑣 ) ⊆ 𝑢 → ( ( 𝑢 ∩ 𝑣 ) “ { 𝑝 } ) ⊆ ( 𝑢 “ { 𝑝 } ) ) |
47 |
45 46
|
ax-mp |
⊢ ( ( 𝑢 ∩ 𝑣 ) “ { 𝑝 } ) ⊆ ( 𝑢 “ { 𝑝 } ) |
48 |
|
simpr33 |
⊢ ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ ( ( 𝑥 ∈ ( unifTop ‘ 𝑈 ) ∧ 𝑦 ∈ ( unifTop ‘ 𝑈 ) ) ∧ 𝑝 ∈ ( 𝑥 ∩ 𝑦 ) ∧ ( 𝑢 ∈ 𝑈 ∧ 𝑣 ∈ 𝑈 ∧ ( ( 𝑢 “ { 𝑝 } ) ⊆ 𝑥 ∧ ( 𝑣 “ { 𝑝 } ) ⊆ 𝑦 ) ) ) ) → ( ( 𝑢 “ { 𝑝 } ) ⊆ 𝑥 ∧ ( 𝑣 “ { 𝑝 } ) ⊆ 𝑦 ) ) |
49 |
48
|
simpld |
⊢ ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ ( ( 𝑥 ∈ ( unifTop ‘ 𝑈 ) ∧ 𝑦 ∈ ( unifTop ‘ 𝑈 ) ) ∧ 𝑝 ∈ ( 𝑥 ∩ 𝑦 ) ∧ ( 𝑢 ∈ 𝑈 ∧ 𝑣 ∈ 𝑈 ∧ ( ( 𝑢 “ { 𝑝 } ) ⊆ 𝑥 ∧ ( 𝑣 “ { 𝑝 } ) ⊆ 𝑦 ) ) ) ) → ( 𝑢 “ { 𝑝 } ) ⊆ 𝑥 ) |
50 |
47 49
|
sstrid |
⊢ ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ ( ( 𝑥 ∈ ( unifTop ‘ 𝑈 ) ∧ 𝑦 ∈ ( unifTop ‘ 𝑈 ) ) ∧ 𝑝 ∈ ( 𝑥 ∩ 𝑦 ) ∧ ( 𝑢 ∈ 𝑈 ∧ 𝑣 ∈ 𝑈 ∧ ( ( 𝑢 “ { 𝑝 } ) ⊆ 𝑥 ∧ ( 𝑣 “ { 𝑝 } ) ⊆ 𝑦 ) ) ) ) → ( ( 𝑢 ∩ 𝑣 ) “ { 𝑝 } ) ⊆ 𝑥 ) |
51 |
|
inss2 |
⊢ ( 𝑢 ∩ 𝑣 ) ⊆ 𝑣 |
52 |
|
imass1 |
⊢ ( ( 𝑢 ∩ 𝑣 ) ⊆ 𝑣 → ( ( 𝑢 ∩ 𝑣 ) “ { 𝑝 } ) ⊆ ( 𝑣 “ { 𝑝 } ) ) |
53 |
51 52
|
ax-mp |
⊢ ( ( 𝑢 ∩ 𝑣 ) “ { 𝑝 } ) ⊆ ( 𝑣 “ { 𝑝 } ) |
54 |
48
|
simprd |
⊢ ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ ( ( 𝑥 ∈ ( unifTop ‘ 𝑈 ) ∧ 𝑦 ∈ ( unifTop ‘ 𝑈 ) ) ∧ 𝑝 ∈ ( 𝑥 ∩ 𝑦 ) ∧ ( 𝑢 ∈ 𝑈 ∧ 𝑣 ∈ 𝑈 ∧ ( ( 𝑢 “ { 𝑝 } ) ⊆ 𝑥 ∧ ( 𝑣 “ { 𝑝 } ) ⊆ 𝑦 ) ) ) ) → ( 𝑣 “ { 𝑝 } ) ⊆ 𝑦 ) |
55 |
53 54
|
sstrid |
⊢ ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ ( ( 𝑥 ∈ ( unifTop ‘ 𝑈 ) ∧ 𝑦 ∈ ( unifTop ‘ 𝑈 ) ) ∧ 𝑝 ∈ ( 𝑥 ∩ 𝑦 ) ∧ ( 𝑢 ∈ 𝑈 ∧ 𝑣 ∈ 𝑈 ∧ ( ( 𝑢 “ { 𝑝 } ) ⊆ 𝑥 ∧ ( 𝑣 “ { 𝑝 } ) ⊆ 𝑦 ) ) ) ) → ( ( 𝑢 ∩ 𝑣 ) “ { 𝑝 } ) ⊆ 𝑦 ) |
56 |
50 55
|
ssind |
⊢ ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ ( ( 𝑥 ∈ ( unifTop ‘ 𝑈 ) ∧ 𝑦 ∈ ( unifTop ‘ 𝑈 ) ) ∧ 𝑝 ∈ ( 𝑥 ∩ 𝑦 ) ∧ ( 𝑢 ∈ 𝑈 ∧ 𝑣 ∈ 𝑈 ∧ ( ( 𝑢 “ { 𝑝 } ) ⊆ 𝑥 ∧ ( 𝑣 “ { 𝑝 } ) ⊆ 𝑦 ) ) ) ) → ( ( 𝑢 ∩ 𝑣 ) “ { 𝑝 } ) ⊆ ( 𝑥 ∩ 𝑦 ) ) |
57 |
|
imaeq1 |
⊢ ( 𝑤 = ( 𝑢 ∩ 𝑣 ) → ( 𝑤 “ { 𝑝 } ) = ( ( 𝑢 ∩ 𝑣 ) “ { 𝑝 } ) ) |
58 |
57
|
sseq1d |
⊢ ( 𝑤 = ( 𝑢 ∩ 𝑣 ) → ( ( 𝑤 “ { 𝑝 } ) ⊆ ( 𝑥 ∩ 𝑦 ) ↔ ( ( 𝑢 ∩ 𝑣 ) “ { 𝑝 } ) ⊆ ( 𝑥 ∩ 𝑦 ) ) ) |
59 |
58
|
rspcev |
⊢ ( ( ( 𝑢 ∩ 𝑣 ) ∈ 𝑈 ∧ ( ( 𝑢 ∩ 𝑣 ) “ { 𝑝 } ) ⊆ ( 𝑥 ∩ 𝑦 ) ) → ∃ 𝑤 ∈ 𝑈 ( 𝑤 “ { 𝑝 } ) ⊆ ( 𝑥 ∩ 𝑦 ) ) |
60 |
44 56 59
|
syl2anc |
⊢ ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ ( ( 𝑥 ∈ ( unifTop ‘ 𝑈 ) ∧ 𝑦 ∈ ( unifTop ‘ 𝑈 ) ) ∧ 𝑝 ∈ ( 𝑥 ∩ 𝑦 ) ∧ ( 𝑢 ∈ 𝑈 ∧ 𝑣 ∈ 𝑈 ∧ ( ( 𝑢 “ { 𝑝 } ) ⊆ 𝑥 ∧ ( 𝑣 “ { 𝑝 } ) ⊆ 𝑦 ) ) ) ) → ∃ 𝑤 ∈ 𝑈 ( 𝑤 “ { 𝑝 } ) ⊆ ( 𝑥 ∩ 𝑦 ) ) |
61 |
60
|
3anassrs |
⊢ ( ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ ( 𝑥 ∈ ( unifTop ‘ 𝑈 ) ∧ 𝑦 ∈ ( unifTop ‘ 𝑈 ) ) ) ∧ 𝑝 ∈ ( 𝑥 ∩ 𝑦 ) ) ∧ ( 𝑢 ∈ 𝑈 ∧ 𝑣 ∈ 𝑈 ∧ ( ( 𝑢 “ { 𝑝 } ) ⊆ 𝑥 ∧ ( 𝑣 “ { 𝑝 } ) ⊆ 𝑦 ) ) ) → ∃ 𝑤 ∈ 𝑈 ( 𝑤 “ { 𝑝 } ) ⊆ ( 𝑥 ∩ 𝑦 ) ) |
62 |
61
|
3anassrs |
⊢ ( ( ( ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ ( 𝑥 ∈ ( unifTop ‘ 𝑈 ) ∧ 𝑦 ∈ ( unifTop ‘ 𝑈 ) ) ) ∧ 𝑝 ∈ ( 𝑥 ∩ 𝑦 ) ) ∧ 𝑢 ∈ 𝑈 ) ∧ 𝑣 ∈ 𝑈 ) ∧ ( ( 𝑢 “ { 𝑝 } ) ⊆ 𝑥 ∧ ( 𝑣 “ { 𝑝 } ) ⊆ 𝑦 ) ) → ∃ 𝑤 ∈ 𝑈 ( 𝑤 “ { 𝑝 } ) ⊆ ( 𝑥 ∩ 𝑦 ) ) |
63 |
|
simpll |
⊢ ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ ( 𝑥 ∈ ( unifTop ‘ 𝑈 ) ∧ 𝑦 ∈ ( unifTop ‘ 𝑈 ) ) ) ∧ 𝑝 ∈ ( 𝑥 ∩ 𝑦 ) ) → 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ) |
64 |
|
simplrl |
⊢ ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ ( 𝑥 ∈ ( unifTop ‘ 𝑈 ) ∧ 𝑦 ∈ ( unifTop ‘ 𝑈 ) ) ) ∧ 𝑝 ∈ ( 𝑥 ∩ 𝑦 ) ) → 𝑥 ∈ ( unifTop ‘ 𝑈 ) ) |
65 |
|
simpr |
⊢ ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ ( 𝑥 ∈ ( unifTop ‘ 𝑈 ) ∧ 𝑦 ∈ ( unifTop ‘ 𝑈 ) ) ) ∧ 𝑝 ∈ ( 𝑥 ∩ 𝑦 ) ) → 𝑝 ∈ ( 𝑥 ∩ 𝑦 ) ) |
66 |
|
elin |
⊢ ( 𝑝 ∈ ( 𝑥 ∩ 𝑦 ) ↔ ( 𝑝 ∈ 𝑥 ∧ 𝑝 ∈ 𝑦 ) ) |
67 |
65 66
|
sylib |
⊢ ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ ( 𝑥 ∈ ( unifTop ‘ 𝑈 ) ∧ 𝑦 ∈ ( unifTop ‘ 𝑈 ) ) ) ∧ 𝑝 ∈ ( 𝑥 ∩ 𝑦 ) ) → ( 𝑝 ∈ 𝑥 ∧ 𝑝 ∈ 𝑦 ) ) |
68 |
67
|
simpld |
⊢ ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ ( 𝑥 ∈ ( unifTop ‘ 𝑈 ) ∧ 𝑦 ∈ ( unifTop ‘ 𝑈 ) ) ) ∧ 𝑝 ∈ ( 𝑥 ∩ 𝑦 ) ) → 𝑝 ∈ 𝑥 ) |
69 |
35
|
simprd |
⊢ ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑥 ∈ ( unifTop ‘ 𝑈 ) ) → ∀ 𝑝 ∈ 𝑥 ∃ 𝑢 ∈ 𝑈 ( 𝑢 “ { 𝑝 } ) ⊆ 𝑥 ) |
70 |
69
|
r19.21bi |
⊢ ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑥 ∈ ( unifTop ‘ 𝑈 ) ) ∧ 𝑝 ∈ 𝑥 ) → ∃ 𝑢 ∈ 𝑈 ( 𝑢 “ { 𝑝 } ) ⊆ 𝑥 ) |
71 |
63 64 68 70
|
syl21anc |
⊢ ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ ( 𝑥 ∈ ( unifTop ‘ 𝑈 ) ∧ 𝑦 ∈ ( unifTop ‘ 𝑈 ) ) ) ∧ 𝑝 ∈ ( 𝑥 ∩ 𝑦 ) ) → ∃ 𝑢 ∈ 𝑈 ( 𝑢 “ { 𝑝 } ) ⊆ 𝑥 ) |
72 |
|
simplrr |
⊢ ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ ( 𝑥 ∈ ( unifTop ‘ 𝑈 ) ∧ 𝑦 ∈ ( unifTop ‘ 𝑈 ) ) ) ∧ 𝑝 ∈ ( 𝑥 ∩ 𝑦 ) ) → 𝑦 ∈ ( unifTop ‘ 𝑈 ) ) |
73 |
67
|
simprd |
⊢ ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ ( 𝑥 ∈ ( unifTop ‘ 𝑈 ) ∧ 𝑦 ∈ ( unifTop ‘ 𝑈 ) ) ) ∧ 𝑝 ∈ ( 𝑥 ∩ 𝑦 ) ) → 𝑝 ∈ 𝑦 ) |
74 |
63 72 73 17
|
syl21anc |
⊢ ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ ( 𝑥 ∈ ( unifTop ‘ 𝑈 ) ∧ 𝑦 ∈ ( unifTop ‘ 𝑈 ) ) ) ∧ 𝑝 ∈ ( 𝑥 ∩ 𝑦 ) ) → ∃ 𝑣 ∈ 𝑈 ( 𝑣 “ { 𝑝 } ) ⊆ 𝑦 ) |
75 |
|
reeanv |
⊢ ( ∃ 𝑢 ∈ 𝑈 ∃ 𝑣 ∈ 𝑈 ( ( 𝑢 “ { 𝑝 } ) ⊆ 𝑥 ∧ ( 𝑣 “ { 𝑝 } ) ⊆ 𝑦 ) ↔ ( ∃ 𝑢 ∈ 𝑈 ( 𝑢 “ { 𝑝 } ) ⊆ 𝑥 ∧ ∃ 𝑣 ∈ 𝑈 ( 𝑣 “ { 𝑝 } ) ⊆ 𝑦 ) ) |
76 |
71 74 75
|
sylanbrc |
⊢ ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ ( 𝑥 ∈ ( unifTop ‘ 𝑈 ) ∧ 𝑦 ∈ ( unifTop ‘ 𝑈 ) ) ) ∧ 𝑝 ∈ ( 𝑥 ∩ 𝑦 ) ) → ∃ 𝑢 ∈ 𝑈 ∃ 𝑣 ∈ 𝑈 ( ( 𝑢 “ { 𝑝 } ) ⊆ 𝑥 ∧ ( 𝑣 “ { 𝑝 } ) ⊆ 𝑦 ) ) |
77 |
62 76
|
r19.29vva |
⊢ ( ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ ( 𝑥 ∈ ( unifTop ‘ 𝑈 ) ∧ 𝑦 ∈ ( unifTop ‘ 𝑈 ) ) ) ∧ 𝑝 ∈ ( 𝑥 ∩ 𝑦 ) ) → ∃ 𝑤 ∈ 𝑈 ( 𝑤 “ { 𝑝 } ) ⊆ ( 𝑥 ∩ 𝑦 ) ) |
78 |
77
|
ralrimiva |
⊢ ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ ( 𝑥 ∈ ( unifTop ‘ 𝑈 ) ∧ 𝑦 ∈ ( unifTop ‘ 𝑈 ) ) ) → ∀ 𝑝 ∈ ( 𝑥 ∩ 𝑦 ) ∃ 𝑤 ∈ 𝑈 ( 𝑤 “ { 𝑝 } ) ⊆ ( 𝑥 ∩ 𝑦 ) ) |
79 |
|
elutop |
⊢ ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) → ( ( 𝑥 ∩ 𝑦 ) ∈ ( unifTop ‘ 𝑈 ) ↔ ( ( 𝑥 ∩ 𝑦 ) ⊆ 𝑋 ∧ ∀ 𝑝 ∈ ( 𝑥 ∩ 𝑦 ) ∃ 𝑤 ∈ 𝑈 ( 𝑤 “ { 𝑝 } ) ⊆ ( 𝑥 ∩ 𝑦 ) ) ) ) |
80 |
79
|
adantr |
⊢ ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ ( 𝑥 ∈ ( unifTop ‘ 𝑈 ) ∧ 𝑦 ∈ ( unifTop ‘ 𝑈 ) ) ) → ( ( 𝑥 ∩ 𝑦 ) ∈ ( unifTop ‘ 𝑈 ) ↔ ( ( 𝑥 ∩ 𝑦 ) ⊆ 𝑋 ∧ ∀ 𝑝 ∈ ( 𝑥 ∩ 𝑦 ) ∃ 𝑤 ∈ 𝑈 ( 𝑤 “ { 𝑝 } ) ⊆ ( 𝑥 ∩ 𝑦 ) ) ) ) |
81 |
39 78 80
|
mpbir2and |
⊢ ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ ( 𝑥 ∈ ( unifTop ‘ 𝑈 ) ∧ 𝑦 ∈ ( unifTop ‘ 𝑈 ) ) ) → ( 𝑥 ∩ 𝑦 ) ∈ ( unifTop ‘ 𝑈 ) ) |
82 |
81
|
ralrimivva |
⊢ ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) → ∀ 𝑥 ∈ ( unifTop ‘ 𝑈 ) ∀ 𝑦 ∈ ( unifTop ‘ 𝑈 ) ( 𝑥 ∩ 𝑦 ) ∈ ( unifTop ‘ 𝑈 ) ) |
83 |
|
fvex |
⊢ ( unifTop ‘ 𝑈 ) ∈ V |
84 |
|
istopg |
⊢ ( ( unifTop ‘ 𝑈 ) ∈ V → ( ( unifTop ‘ 𝑈 ) ∈ Top ↔ ( ∀ 𝑥 ( 𝑥 ⊆ ( unifTop ‘ 𝑈 ) → ∪ 𝑥 ∈ ( unifTop ‘ 𝑈 ) ) ∧ ∀ 𝑥 ∈ ( unifTop ‘ 𝑈 ) ∀ 𝑦 ∈ ( unifTop ‘ 𝑈 ) ( 𝑥 ∩ 𝑦 ) ∈ ( unifTop ‘ 𝑈 ) ) ) ) |
85 |
83 84
|
ax-mp |
⊢ ( ( unifTop ‘ 𝑈 ) ∈ Top ↔ ( ∀ 𝑥 ( 𝑥 ⊆ ( unifTop ‘ 𝑈 ) → ∪ 𝑥 ∈ ( unifTop ‘ 𝑈 ) ) ∧ ∀ 𝑥 ∈ ( unifTop ‘ 𝑈 ) ∀ 𝑦 ∈ ( unifTop ‘ 𝑈 ) ( 𝑥 ∩ 𝑦 ) ∈ ( unifTop ‘ 𝑈 ) ) ) |
86 |
33 82 85
|
sylanbrc |
⊢ ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) → ( unifTop ‘ 𝑈 ) ∈ Top ) |