Step |
Hyp |
Ref |
Expression |
1 |
|
elrestd.1 |
⊢ ( 𝜑 → 𝐽 ∈ 𝑉 ) |
2 |
|
elrestd.2 |
⊢ ( 𝜑 → 𝐵 ∈ 𝑊 ) |
3 |
|
elrestd.3 |
⊢ ( 𝜑 → 𝑋 ∈ 𝐽 ) |
4 |
|
elrestd.4 |
⊢ 𝐴 = ( 𝑋 ∩ 𝐵 ) |
5 |
4
|
a1i |
⊢ ( 𝜑 → 𝐴 = ( 𝑋 ∩ 𝐵 ) ) |
6 |
|
ineq1 |
⊢ ( 𝑥 = 𝑋 → ( 𝑥 ∩ 𝐵 ) = ( 𝑋 ∩ 𝐵 ) ) |
7 |
6
|
rspceeqv |
⊢ ( ( 𝑋 ∈ 𝐽 ∧ 𝐴 = ( 𝑋 ∩ 𝐵 ) ) → ∃ 𝑥 ∈ 𝐽 𝐴 = ( 𝑥 ∩ 𝐵 ) ) |
8 |
3 5 7
|
syl2anc |
⊢ ( 𝜑 → ∃ 𝑥 ∈ 𝐽 𝐴 = ( 𝑥 ∩ 𝐵 ) ) |
9 |
|
elrest |
⊢ ( ( 𝐽 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ) → ( 𝐴 ∈ ( 𝐽 ↾t 𝐵 ) ↔ ∃ 𝑥 ∈ 𝐽 𝐴 = ( 𝑥 ∩ 𝐵 ) ) ) |
10 |
1 2 9
|
syl2anc |
⊢ ( 𝜑 → ( 𝐴 ∈ ( 𝐽 ↾t 𝐵 ) ↔ ∃ 𝑥 ∈ 𝐽 𝐴 = ( 𝑥 ∩ 𝐵 ) ) ) |
11 |
8 10
|
mpbird |
⊢ ( 𝜑 → 𝐴 ∈ ( 𝐽 ↾t 𝐵 ) ) |