Step |
Hyp |
Ref |
Expression |
1 |
|
vex |
⊢ 𝑥 ∈ V |
2 |
1
|
eldm |
⊢ ( 𝑥 ∈ dom ( 𝐴 × 𝐵 ) ↔ ∃ 𝑦 𝑥 ( 𝐴 × 𝐵 ) 𝑦 ) |
3 |
|
brxp |
⊢ ( 𝑥 ( 𝐴 × 𝐵 ) 𝑦 ↔ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) ) |
4 |
3
|
exbii |
⊢ ( ∃ 𝑦 𝑥 ( 𝐴 × 𝐵 ) 𝑦 ↔ ∃ 𝑦 ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) ) |
5 |
|
19.42v |
⊢ ( ∃ 𝑦 ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) ↔ ( 𝑥 ∈ 𝐴 ∧ ∃ 𝑦 𝑦 ∈ 𝐵 ) ) |
6 |
2 4 5
|
3bitri |
⊢ ( 𝑥 ∈ dom ( 𝐴 × 𝐵 ) ↔ ( 𝑥 ∈ 𝐴 ∧ ∃ 𝑦 𝑦 ∈ 𝐵 ) ) |
7 |
|
n0 |
⊢ ( 𝐵 ≠ ∅ ↔ ∃ 𝑦 𝑦 ∈ 𝐵 ) |
8 |
7
|
biimpi |
⊢ ( 𝐵 ≠ ∅ → ∃ 𝑦 𝑦 ∈ 𝐵 ) |
9 |
8
|
biantrud |
⊢ ( 𝐵 ≠ ∅ → ( 𝑥 ∈ 𝐴 ↔ ( 𝑥 ∈ 𝐴 ∧ ∃ 𝑦 𝑦 ∈ 𝐵 ) ) ) |
10 |
6 9
|
bitr4id |
⊢ ( 𝐵 ≠ ∅ → ( 𝑥 ∈ dom ( 𝐴 × 𝐵 ) ↔ 𝑥 ∈ 𝐴 ) ) |
11 |
10
|
eqrdv |
⊢ ( 𝐵 ≠ ∅ → dom ( 𝐴 × 𝐵 ) = 𝐴 ) |