Step |
Hyp |
Ref |
Expression |
1 |
|
utopval |
⊢ ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) → ( unifTop ‘ 𝑈 ) = { 𝑎 ∈ 𝒫 𝑋 ∣ ∀ 𝑥 ∈ 𝑎 ∃ 𝑣 ∈ 𝑈 ( 𝑣 “ { 𝑥 } ) ⊆ 𝑎 } ) |
2 |
|
ssrab2 |
⊢ { 𝑎 ∈ 𝒫 𝑋 ∣ ∀ 𝑥 ∈ 𝑎 ∃ 𝑣 ∈ 𝑈 ( 𝑣 “ { 𝑥 } ) ⊆ 𝑎 } ⊆ 𝒫 𝑋 |
3 |
1 2
|
eqsstrdi |
⊢ ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) → ( unifTop ‘ 𝑈 ) ⊆ 𝒫 𝑋 ) |
4 |
|
ssidd |
⊢ ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) → 𝑋 ⊆ 𝑋 ) |
5 |
|
ustssxp |
⊢ ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑣 ∈ 𝑈 ) → 𝑣 ⊆ ( 𝑋 × 𝑋 ) ) |
6 |
|
imassrn |
⊢ ( 𝑣 “ { 𝑥 } ) ⊆ ran 𝑣 |
7 |
|
rnss |
⊢ ( 𝑣 ⊆ ( 𝑋 × 𝑋 ) → ran 𝑣 ⊆ ran ( 𝑋 × 𝑋 ) ) |
8 |
|
rnxpid |
⊢ ran ( 𝑋 × 𝑋 ) = 𝑋 |
9 |
7 8
|
sseqtrdi |
⊢ ( 𝑣 ⊆ ( 𝑋 × 𝑋 ) → ran 𝑣 ⊆ 𝑋 ) |
10 |
6 9
|
sstrid |
⊢ ( 𝑣 ⊆ ( 𝑋 × 𝑋 ) → ( 𝑣 “ { 𝑥 } ) ⊆ 𝑋 ) |
11 |
5 10
|
syl |
⊢ ( ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) ∧ 𝑣 ∈ 𝑈 ) → ( 𝑣 “ { 𝑥 } ) ⊆ 𝑋 ) |
12 |
11
|
ralrimiva |
⊢ ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) → ∀ 𝑣 ∈ 𝑈 ( 𝑣 “ { 𝑥 } ) ⊆ 𝑋 ) |
13 |
|
ustne0 |
⊢ ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) → 𝑈 ≠ ∅ ) |
14 |
|
r19.2zb |
⊢ ( 𝑈 ≠ ∅ ↔ ( ∀ 𝑣 ∈ 𝑈 ( 𝑣 “ { 𝑥 } ) ⊆ 𝑋 → ∃ 𝑣 ∈ 𝑈 ( 𝑣 “ { 𝑥 } ) ⊆ 𝑋 ) ) |
15 |
13 14
|
sylib |
⊢ ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) → ( ∀ 𝑣 ∈ 𝑈 ( 𝑣 “ { 𝑥 } ) ⊆ 𝑋 → ∃ 𝑣 ∈ 𝑈 ( 𝑣 “ { 𝑥 } ) ⊆ 𝑋 ) ) |
16 |
12 15
|
mpd |
⊢ ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) → ∃ 𝑣 ∈ 𝑈 ( 𝑣 “ { 𝑥 } ) ⊆ 𝑋 ) |
17 |
16
|
ralrimivw |
⊢ ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) → ∀ 𝑥 ∈ 𝑋 ∃ 𝑣 ∈ 𝑈 ( 𝑣 “ { 𝑥 } ) ⊆ 𝑋 ) |
18 |
|
elutop |
⊢ ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) → ( 𝑋 ∈ ( unifTop ‘ 𝑈 ) ↔ ( 𝑋 ⊆ 𝑋 ∧ ∀ 𝑥 ∈ 𝑋 ∃ 𝑣 ∈ 𝑈 ( 𝑣 “ { 𝑥 } ) ⊆ 𝑋 ) ) ) |
19 |
4 17 18
|
mpbir2and |
⊢ ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) → 𝑋 ∈ ( unifTop ‘ 𝑈 ) ) |
20 |
|
elpwuni |
⊢ ( 𝑋 ∈ ( unifTop ‘ 𝑈 ) → ( ( unifTop ‘ 𝑈 ) ⊆ 𝒫 𝑋 ↔ ∪ ( unifTop ‘ 𝑈 ) = 𝑋 ) ) |
21 |
19 20
|
syl |
⊢ ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) → ( ( unifTop ‘ 𝑈 ) ⊆ 𝒫 𝑋 ↔ ∪ ( unifTop ‘ 𝑈 ) = 𝑋 ) ) |
22 |
3 21
|
mpbid |
⊢ ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) → ∪ ( unifTop ‘ 𝑈 ) = 𝑋 ) |
23 |
22
|
eqcomd |
⊢ ( 𝑈 ∈ ( UnifOn ‘ 𝑋 ) → 𝑋 = ∪ ( unifTop ‘ 𝑈 ) ) |