Step |
Hyp |
Ref |
Expression |
1 |
|
brimage.1 |
⊢ 𝐴 ∈ V |
2 |
|
brimage.2 |
⊢ 𝐵 ∈ V |
3 |
|
df-image |
⊢ Image 𝑅 = ( ( V × V ) ∖ ran ( ( V ⊗ E ) △ ( ( E ∘ ◡ 𝑅 ) ⊗ V ) ) ) |
4 |
|
brxp |
⊢ ( 𝐴 ( V × V ) 𝐵 ↔ ( 𝐴 ∈ V ∧ 𝐵 ∈ V ) ) |
5 |
1 2 4
|
mpbir2an |
⊢ 𝐴 ( V × V ) 𝐵 |
6 |
|
vex |
⊢ 𝑥 ∈ V |
7 |
|
vex |
⊢ 𝑦 ∈ V |
8 |
6 7
|
brcnv |
⊢ ( 𝑥 ◡ 𝑅 𝑦 ↔ 𝑦 𝑅 𝑥 ) |
9 |
8
|
rexbii |
⊢ ( ∃ 𝑦 ∈ 𝐴 𝑥 ◡ 𝑅 𝑦 ↔ ∃ 𝑦 ∈ 𝐴 𝑦 𝑅 𝑥 ) |
10 |
6 1
|
coep |
⊢ ( 𝑥 ( E ∘ ◡ 𝑅 ) 𝐴 ↔ ∃ 𝑦 ∈ 𝐴 𝑥 ◡ 𝑅 𝑦 ) |
11 |
6
|
elima |
⊢ ( 𝑥 ∈ ( 𝑅 “ 𝐴 ) ↔ ∃ 𝑦 ∈ 𝐴 𝑦 𝑅 𝑥 ) |
12 |
9 10 11
|
3bitr4ri |
⊢ ( 𝑥 ∈ ( 𝑅 “ 𝐴 ) ↔ 𝑥 ( E ∘ ◡ 𝑅 ) 𝐴 ) |
13 |
1 2 3 5 12
|
brtxpsd3 |
⊢ ( 𝐴 Image 𝑅 𝐵 ↔ 𝐵 = ( 𝑅 “ 𝐴 ) ) |