Step |
Hyp |
Ref |
Expression |
1 |
|
sseqin2 |
⊢ ( 𝐴 ⊆ 𝑌 ↔ ( 𝑌 ∩ 𝐴 ) = 𝐴 ) |
2 |
|
sneq |
⊢ ( ( 𝑌 ∩ 𝐴 ) = 𝐴 → { ( 𝑌 ∩ 𝐴 ) } = { 𝐴 } ) |
3 |
1 2
|
sylbi |
⊢ ( 𝐴 ⊆ 𝑌 → { ( 𝑌 ∩ 𝐴 ) } = { 𝐴 } ) |
4 |
|
ssexg |
⊢ ( ( 𝐴 ⊆ 𝑌 ∧ 𝑌 ∈ 𝑉 ) → 𝐴 ∈ V ) |
5 |
4
|
ancoms |
⊢ ( ( 𝑌 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑌 ) → 𝐴 ∈ V ) |
6 |
|
bj-restsn |
⊢ ( ( 𝑌 ∈ 𝑉 ∧ 𝐴 ∈ V ) → ( { 𝑌 } ↾t 𝐴 ) = { ( 𝑌 ∩ 𝐴 ) } ) |
7 |
5 6
|
syldan |
⊢ ( ( 𝑌 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑌 ) → ( { 𝑌 } ↾t 𝐴 ) = { ( 𝑌 ∩ 𝐴 ) } ) |
8 |
|
eqeq2 |
⊢ ( { ( 𝑌 ∩ 𝐴 ) } = { 𝐴 } → ( ( { 𝑌 } ↾t 𝐴 ) = { ( 𝑌 ∩ 𝐴 ) } ↔ ( { 𝑌 } ↾t 𝐴 ) = { 𝐴 } ) ) |
9 |
8
|
biimpa |
⊢ ( ( { ( 𝑌 ∩ 𝐴 ) } = { 𝐴 } ∧ ( { 𝑌 } ↾t 𝐴 ) = { ( 𝑌 ∩ 𝐴 ) } ) → ( { 𝑌 } ↾t 𝐴 ) = { 𝐴 } ) |
10 |
3 7 9
|
syl2an2 |
⊢ ( ( 𝑌 ∈ 𝑉 ∧ 𝐴 ⊆ 𝑌 ) → ( { 𝑌 } ↾t 𝐴 ) = { 𝐴 } ) |