Step |
Hyp |
Ref |
Expression |
1 |
|
resco |
⊢ ( ( ◡ ( 1st ↾ ( V × V ) ) ∘ 𝑅 ) ↾ 𝐴 ) = ( ◡ ( 1st ↾ ( V × V ) ) ∘ ( 𝑅 ↾ 𝐴 ) ) |
2 |
1
|
ineq1i |
⊢ ( ( ( ◡ ( 1st ↾ ( V × V ) ) ∘ 𝑅 ) ↾ 𝐴 ) ∩ ( ◡ ( 2nd ↾ ( V × V ) ) ∘ 𝑆 ) ) = ( ( ◡ ( 1st ↾ ( V × V ) ) ∘ ( 𝑅 ↾ 𝐴 ) ) ∩ ( ◡ ( 2nd ↾ ( V × V ) ) ∘ 𝑆 ) ) |
3 |
|
df-xrn |
⊢ ( 𝑅 ⋉ 𝑆 ) = ( ( ◡ ( 1st ↾ ( V × V ) ) ∘ 𝑅 ) ∩ ( ◡ ( 2nd ↾ ( V × V ) ) ∘ 𝑆 ) ) |
4 |
3
|
reseq1i |
⊢ ( ( 𝑅 ⋉ 𝑆 ) ↾ 𝐴 ) = ( ( ( ◡ ( 1st ↾ ( V × V ) ) ∘ 𝑅 ) ∩ ( ◡ ( 2nd ↾ ( V × V ) ) ∘ 𝑆 ) ) ↾ 𝐴 ) |
5 |
|
inres2 |
⊢ ( ( ( ◡ ( 1st ↾ ( V × V ) ) ∘ 𝑅 ) ↾ 𝐴 ) ∩ ( ◡ ( 2nd ↾ ( V × V ) ) ∘ 𝑆 ) ) = ( ( ( ◡ ( 1st ↾ ( V × V ) ) ∘ 𝑅 ) ∩ ( ◡ ( 2nd ↾ ( V × V ) ) ∘ 𝑆 ) ) ↾ 𝐴 ) |
6 |
4 5
|
eqtr4i |
⊢ ( ( 𝑅 ⋉ 𝑆 ) ↾ 𝐴 ) = ( ( ( ◡ ( 1st ↾ ( V × V ) ) ∘ 𝑅 ) ↾ 𝐴 ) ∩ ( ◡ ( 2nd ↾ ( V × V ) ) ∘ 𝑆 ) ) |
7 |
|
df-xrn |
⊢ ( ( 𝑅 ↾ 𝐴 ) ⋉ 𝑆 ) = ( ( ◡ ( 1st ↾ ( V × V ) ) ∘ ( 𝑅 ↾ 𝐴 ) ) ∩ ( ◡ ( 2nd ↾ ( V × V ) ) ∘ 𝑆 ) ) |
8 |
2 6 7
|
3eqtr4i |
⊢ ( ( 𝑅 ⋉ 𝑆 ) ↾ 𝐴 ) = ( ( 𝑅 ↾ 𝐴 ) ⋉ 𝑆 ) |