Step |
Hyp |
Ref |
Expression |
1 |
|
id |
⊢ ( 𝐴 ⊆ 𝐵 → 𝐴 ⊆ 𝐵 ) |
2 |
|
ssidd |
⊢ ( 𝐴 ⊆ 𝐵 → 𝐴 ⊆ 𝐴 ) |
3 |
1 2
|
ssind |
⊢ ( 𝐴 ⊆ 𝐵 → 𝐴 ⊆ ( 𝐵 ∩ 𝐴 ) ) |
4 |
|
inss2 |
⊢ ( 𝐵 ∩ 𝐴 ) ⊆ 𝐴 |
5 |
4
|
a1i |
⊢ ( 𝐴 ⊆ 𝐵 → ( 𝐵 ∩ 𝐴 ) ⊆ 𝐴 ) |
6 |
3 5
|
eqssd |
⊢ ( 𝐴 ⊆ 𝐵 → 𝐴 = ( 𝐵 ∩ 𝐴 ) ) |
7 |
|
eleq1 |
⊢ ( 𝑦 = 𝐵 → ( 𝑦 ∈ 𝑋 ↔ 𝐵 ∈ 𝑋 ) ) |
8 |
|
ineq1 |
⊢ ( 𝑦 = 𝐵 → ( 𝑦 ∩ 𝐴 ) = ( 𝐵 ∩ 𝐴 ) ) |
9 |
8
|
eqeq2d |
⊢ ( 𝑦 = 𝐵 → ( 𝐴 = ( 𝑦 ∩ 𝐴 ) ↔ 𝐴 = ( 𝐵 ∩ 𝐴 ) ) ) |
10 |
7 9
|
anbi12d |
⊢ ( 𝑦 = 𝐵 → ( ( 𝑦 ∈ 𝑋 ∧ 𝐴 = ( 𝑦 ∩ 𝐴 ) ) ↔ ( 𝐵 ∈ 𝑋 ∧ 𝐴 = ( 𝐵 ∩ 𝐴 ) ) ) ) |
11 |
10
|
spcegv |
⊢ ( 𝐵 ∈ 𝑋 → ( ( 𝐵 ∈ 𝑋 ∧ 𝐴 = ( 𝐵 ∩ 𝐴 ) ) → ∃ 𝑦 ( 𝑦 ∈ 𝑋 ∧ 𝐴 = ( 𝑦 ∩ 𝐴 ) ) ) ) |
12 |
11
|
expd |
⊢ ( 𝐵 ∈ 𝑋 → ( 𝐵 ∈ 𝑋 → ( 𝐴 = ( 𝐵 ∩ 𝐴 ) → ∃ 𝑦 ( 𝑦 ∈ 𝑋 ∧ 𝐴 = ( 𝑦 ∩ 𝐴 ) ) ) ) ) |
13 |
12
|
pm2.43i |
⊢ ( 𝐵 ∈ 𝑋 → ( 𝐴 = ( 𝐵 ∩ 𝐴 ) → ∃ 𝑦 ( 𝑦 ∈ 𝑋 ∧ 𝐴 = ( 𝑦 ∩ 𝐴 ) ) ) ) |
14 |
6 13
|
mpan9 |
⊢ ( ( 𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝑋 ) → ∃ 𝑦 ( 𝑦 ∈ 𝑋 ∧ 𝐴 = ( 𝑦 ∩ 𝐴 ) ) ) |
15 |
|
df-rex |
⊢ ( ∃ 𝑦 ∈ 𝑋 𝐴 = ( 𝑦 ∩ 𝐴 ) ↔ ∃ 𝑦 ( 𝑦 ∈ 𝑋 ∧ 𝐴 = ( 𝑦 ∩ 𝐴 ) ) ) |
16 |
14 15
|
sylibr |
⊢ ( ( 𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝑋 ) → ∃ 𝑦 ∈ 𝑋 𝐴 = ( 𝑦 ∩ 𝐴 ) ) |
17 |
16
|
adantl |
⊢ ( ( 𝑋 ∈ 𝑉 ∧ ( 𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝑋 ) ) → ∃ 𝑦 ∈ 𝑋 𝐴 = ( 𝑦 ∩ 𝐴 ) ) |
18 |
|
ssexg |
⊢ ( ( 𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝑋 ) → 𝐴 ∈ V ) |
19 |
|
elrest |
⊢ ( ( 𝑋 ∈ 𝑉 ∧ 𝐴 ∈ V ) → ( 𝐴 ∈ ( 𝑋 ↾t 𝐴 ) ↔ ∃ 𝑦 ∈ 𝑋 𝐴 = ( 𝑦 ∩ 𝐴 ) ) ) |
20 |
18 19
|
sylan2 |
⊢ ( ( 𝑋 ∈ 𝑉 ∧ ( 𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝑋 ) ) → ( 𝐴 ∈ ( 𝑋 ↾t 𝐴 ) ↔ ∃ 𝑦 ∈ 𝑋 𝐴 = ( 𝑦 ∩ 𝐴 ) ) ) |
21 |
17 20
|
mpbird |
⊢ ( ( 𝑋 ∈ 𝑉 ∧ ( 𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝑋 ) ) → 𝐴 ∈ ( 𝑋 ↾t 𝐴 ) ) |
22 |
21
|
ex |
⊢ ( 𝑋 ∈ 𝑉 → ( ( 𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝑋 ) → 𝐴 ∈ ( 𝑋 ↾t 𝐴 ) ) ) |