Step |
Hyp |
Ref |
Expression |
1 |
|
snex |
⊢ { 𝑌 } ∈ V |
2 |
|
elrest |
⊢ ( ( { 𝑌 } ∈ V ∧ 𝐴 ∈ 𝑊 ) → ( 𝑥 ∈ ( { 𝑌 } ↾t 𝐴 ) ↔ ∃ 𝑦 ∈ { 𝑌 } 𝑥 = ( 𝑦 ∩ 𝐴 ) ) ) |
3 |
1 2
|
mpan |
⊢ ( 𝐴 ∈ 𝑊 → ( 𝑥 ∈ ( { 𝑌 } ↾t 𝐴 ) ↔ ∃ 𝑦 ∈ { 𝑌 } 𝑥 = ( 𝑦 ∩ 𝐴 ) ) ) |
4 |
|
velsn |
⊢ ( 𝑥 ∈ { ( 𝑦 ∩ 𝐴 ) } ↔ 𝑥 = ( 𝑦 ∩ 𝐴 ) ) |
5 |
|
ineq1 |
⊢ ( 𝑦 = 𝑌 → ( 𝑦 ∩ 𝐴 ) = ( 𝑌 ∩ 𝐴 ) ) |
6 |
5
|
sneqd |
⊢ ( 𝑦 = 𝑌 → { ( 𝑦 ∩ 𝐴 ) } = { ( 𝑌 ∩ 𝐴 ) } ) |
7 |
6
|
eleq2d |
⊢ ( 𝑦 = 𝑌 → ( 𝑥 ∈ { ( 𝑦 ∩ 𝐴 ) } ↔ 𝑥 ∈ { ( 𝑌 ∩ 𝐴 ) } ) ) |
8 |
4 7
|
bitr3id |
⊢ ( 𝑦 = 𝑌 → ( 𝑥 = ( 𝑦 ∩ 𝐴 ) ↔ 𝑥 ∈ { ( 𝑌 ∩ 𝐴 ) } ) ) |
9 |
8
|
rexsng |
⊢ ( 𝑌 ∈ 𝑉 → ( ∃ 𝑦 ∈ { 𝑌 } 𝑥 = ( 𝑦 ∩ 𝐴 ) ↔ 𝑥 ∈ { ( 𝑌 ∩ 𝐴 ) } ) ) |
10 |
3 9
|
sylan9bbr |
⊢ ( ( 𝑌 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊 ) → ( 𝑥 ∈ ( { 𝑌 } ↾t 𝐴 ) ↔ 𝑥 ∈ { ( 𝑌 ∩ 𝐴 ) } ) ) |
11 |
10
|
eqrdv |
⊢ ( ( 𝑌 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊 ) → ( { 𝑌 } ↾t 𝐴 ) = { ( 𝑌 ∩ 𝐴 ) } ) |