Step |
Hyp |
Ref |
Expression |
1 |
|
vex |
⊢ 𝑦 ∈ V |
2 |
|
vex |
⊢ 𝑥 ∈ V |
3 |
1 2
|
breldm |
⊢ ( 𝑦 𝑅 𝑥 → 𝑦 ∈ dom 𝑅 ) |
4 |
3
|
pm4.71ri |
⊢ ( 𝑦 𝑅 𝑥 ↔ ( 𝑦 ∈ dom 𝑅 ∧ 𝑦 𝑅 𝑥 ) ) |
5 |
4
|
rexbii |
⊢ ( ∃ 𝑦 ∈ 𝐴 𝑦 𝑅 𝑥 ↔ ∃ 𝑦 ∈ 𝐴 ( 𝑦 ∈ dom 𝑅 ∧ 𝑦 𝑅 𝑥 ) ) |
6 |
|
rexin |
⊢ ( ∃ 𝑦 ∈ ( 𝐴 ∩ dom 𝑅 ) 𝑦 𝑅 𝑥 ↔ ∃ 𝑦 ∈ 𝐴 ( 𝑦 ∈ dom 𝑅 ∧ 𝑦 𝑅 𝑥 ) ) |
7 |
5 6
|
bitr4i |
⊢ ( ∃ 𝑦 ∈ 𝐴 𝑦 𝑅 𝑥 ↔ ∃ 𝑦 ∈ ( 𝐴 ∩ dom 𝑅 ) 𝑦 𝑅 𝑥 ) |
8 |
2
|
elima |
⊢ ( 𝑥 ∈ ( 𝑅 “ 𝐴 ) ↔ ∃ 𝑦 ∈ 𝐴 𝑦 𝑅 𝑥 ) |
9 |
2
|
elima |
⊢ ( 𝑥 ∈ ( 𝑅 “ ( 𝐴 ∩ dom 𝑅 ) ) ↔ ∃ 𝑦 ∈ ( 𝐴 ∩ dom 𝑅 ) 𝑦 𝑅 𝑥 ) |
10 |
7 8 9
|
3bitr4i |
⊢ ( 𝑥 ∈ ( 𝑅 “ 𝐴 ) ↔ 𝑥 ∈ ( 𝑅 “ ( 𝐴 ∩ dom 𝑅 ) ) ) |
11 |
10
|
eqriv |
⊢ ( 𝑅 “ 𝐴 ) = ( 𝑅 “ ( 𝐴 ∩ dom 𝑅 ) ) |