Step |
Hyp |
Ref |
Expression |
1 |
|
resoprab |
⊢ ( { 〈 〈 𝑥 , 𝑦 〉 , 𝑧 〉 ∣ ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) ∧ 𝜑 ) } ↾ ( 𝐶 × 𝐷 ) ) = { 〈 〈 𝑥 , 𝑦 〉 , 𝑧 〉 ∣ ( ( 𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷 ) ∧ ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) ∧ 𝜑 ) ) } |
2 |
|
anass |
⊢ ( ( ( ( 𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷 ) ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) ) ∧ 𝜑 ) ↔ ( ( 𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷 ) ∧ ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) ∧ 𝜑 ) ) ) |
3 |
|
an4 |
⊢ ( ( ( 𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷 ) ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) ) ↔ ( ( 𝑥 ∈ 𝐶 ∧ 𝑥 ∈ 𝐴 ) ∧ ( 𝑦 ∈ 𝐷 ∧ 𝑦 ∈ 𝐵 ) ) ) |
4 |
|
ssel |
⊢ ( 𝐶 ⊆ 𝐴 → ( 𝑥 ∈ 𝐶 → 𝑥 ∈ 𝐴 ) ) |
5 |
4
|
pm4.71d |
⊢ ( 𝐶 ⊆ 𝐴 → ( 𝑥 ∈ 𝐶 ↔ ( 𝑥 ∈ 𝐶 ∧ 𝑥 ∈ 𝐴 ) ) ) |
6 |
5
|
bicomd |
⊢ ( 𝐶 ⊆ 𝐴 → ( ( 𝑥 ∈ 𝐶 ∧ 𝑥 ∈ 𝐴 ) ↔ 𝑥 ∈ 𝐶 ) ) |
7 |
|
ssel |
⊢ ( 𝐷 ⊆ 𝐵 → ( 𝑦 ∈ 𝐷 → 𝑦 ∈ 𝐵 ) ) |
8 |
7
|
pm4.71d |
⊢ ( 𝐷 ⊆ 𝐵 → ( 𝑦 ∈ 𝐷 ↔ ( 𝑦 ∈ 𝐷 ∧ 𝑦 ∈ 𝐵 ) ) ) |
9 |
8
|
bicomd |
⊢ ( 𝐷 ⊆ 𝐵 → ( ( 𝑦 ∈ 𝐷 ∧ 𝑦 ∈ 𝐵 ) ↔ 𝑦 ∈ 𝐷 ) ) |
10 |
6 9
|
bi2anan9 |
⊢ ( ( 𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐵 ) → ( ( ( 𝑥 ∈ 𝐶 ∧ 𝑥 ∈ 𝐴 ) ∧ ( 𝑦 ∈ 𝐷 ∧ 𝑦 ∈ 𝐵 ) ) ↔ ( 𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷 ) ) ) |
11 |
3 10
|
bitrid |
⊢ ( ( 𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐵 ) → ( ( ( 𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷 ) ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) ) ↔ ( 𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷 ) ) ) |
12 |
11
|
anbi1d |
⊢ ( ( 𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐵 ) → ( ( ( ( 𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷 ) ∧ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) ) ∧ 𝜑 ) ↔ ( ( 𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷 ) ∧ 𝜑 ) ) ) |
13 |
2 12
|
bitr3id |
⊢ ( ( 𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐵 ) → ( ( ( 𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷 ) ∧ ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) ∧ 𝜑 ) ) ↔ ( ( 𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷 ) ∧ 𝜑 ) ) ) |
14 |
13
|
oprabbidv |
⊢ ( ( 𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐵 ) → { 〈 〈 𝑥 , 𝑦 〉 , 𝑧 〉 ∣ ( ( 𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷 ) ∧ ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) ∧ 𝜑 ) ) } = { 〈 〈 𝑥 , 𝑦 〉 , 𝑧 〉 ∣ ( ( 𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷 ) ∧ 𝜑 ) } ) |
15 |
1 14
|
eqtrid |
⊢ ( ( 𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐵 ) → ( { 〈 〈 𝑥 , 𝑦 〉 , 𝑧 〉 ∣ ( ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) ∧ 𝜑 ) } ↾ ( 𝐶 × 𝐷 ) ) = { 〈 〈 𝑥 , 𝑦 〉 , 𝑧 〉 ∣ ( ( 𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷 ) ∧ 𝜑 ) } ) |