Step |
Hyp |
Ref |
Expression |
1 |
|
ussval.1 |
⊢ 𝐵 = ( Base ‘ 𝑊 ) |
2 |
|
ussval.2 |
⊢ 𝑈 = ( UnifSet ‘ 𝑊 ) |
3 |
|
oveq2 |
⊢ ( ( 𝐵 × 𝐵 ) = ∪ 𝑈 → ( 𝑈 ↾t ( 𝐵 × 𝐵 ) ) = ( 𝑈 ↾t ∪ 𝑈 ) ) |
4 |
|
id |
⊢ ( ( 𝐵 × 𝐵 ) = ∪ 𝑈 → ( 𝐵 × 𝐵 ) = ∪ 𝑈 ) |
5 |
1
|
fvexi |
⊢ 𝐵 ∈ V |
6 |
5 5
|
xpex |
⊢ ( 𝐵 × 𝐵 ) ∈ V |
7 |
4 6
|
eqeltrrdi |
⊢ ( ( 𝐵 × 𝐵 ) = ∪ 𝑈 → ∪ 𝑈 ∈ V ) |
8 |
|
uniexb |
⊢ ( 𝑈 ∈ V ↔ ∪ 𝑈 ∈ V ) |
9 |
7 8
|
sylibr |
⊢ ( ( 𝐵 × 𝐵 ) = ∪ 𝑈 → 𝑈 ∈ V ) |
10 |
|
eqid |
⊢ ∪ 𝑈 = ∪ 𝑈 |
11 |
10
|
restid |
⊢ ( 𝑈 ∈ V → ( 𝑈 ↾t ∪ 𝑈 ) = 𝑈 ) |
12 |
9 11
|
syl |
⊢ ( ( 𝐵 × 𝐵 ) = ∪ 𝑈 → ( 𝑈 ↾t ∪ 𝑈 ) = 𝑈 ) |
13 |
3 12
|
eqtr2d |
⊢ ( ( 𝐵 × 𝐵 ) = ∪ 𝑈 → 𝑈 = ( 𝑈 ↾t ( 𝐵 × 𝐵 ) ) ) |
14 |
1 2
|
ussval |
⊢ ( 𝑈 ↾t ( 𝐵 × 𝐵 ) ) = ( UnifSt ‘ 𝑊 ) |
15 |
13 14
|
eqtrdi |
⊢ ( ( 𝐵 × 𝐵 ) = ∪ 𝑈 → 𝑈 = ( UnifSt ‘ 𝑊 ) ) |