Step |
Hyp |
Ref |
Expression |
1 |
|
in12 |
|- ( A i^i ( ( B X. _V ) i^i ( dom A X. _V ) ) ) = ( ( B X. _V ) i^i ( A i^i ( dom A X. _V ) ) ) |
2 |
|
df-res |
|- ( A |` dom A ) = ( A i^i ( dom A X. _V ) ) |
3 |
|
resdm2 |
|- ( A |` dom A ) = `' `' A |
4 |
2 3
|
eqtr3i |
|- ( A i^i ( dom A X. _V ) ) = `' `' A |
5 |
4
|
ineq2i |
|- ( ( B X. _V ) i^i ( A i^i ( dom A X. _V ) ) ) = ( ( B X. _V ) i^i `' `' A ) |
6 |
|
incom |
|- ( ( B X. _V ) i^i `' `' A ) = ( `' `' A i^i ( B X. _V ) ) |
7 |
1 5 6
|
3eqtri |
|- ( A i^i ( ( B X. _V ) i^i ( dom A X. _V ) ) ) = ( `' `' A i^i ( B X. _V ) ) |
8 |
|
df-res |
|- ( A |` dom ( A |` B ) ) = ( A i^i ( dom ( A |` B ) X. _V ) ) |
9 |
|
dmres |
|- dom ( A |` B ) = ( B i^i dom A ) |
10 |
9
|
xpeq1i |
|- ( dom ( A |` B ) X. _V ) = ( ( B i^i dom A ) X. _V ) |
11 |
|
xpindir |
|- ( ( B i^i dom A ) X. _V ) = ( ( B X. _V ) i^i ( dom A X. _V ) ) |
12 |
10 11
|
eqtri |
|- ( dom ( A |` B ) X. _V ) = ( ( B X. _V ) i^i ( dom A X. _V ) ) |
13 |
12
|
ineq2i |
|- ( A i^i ( dom ( A |` B ) X. _V ) ) = ( A i^i ( ( B X. _V ) i^i ( dom A X. _V ) ) ) |
14 |
8 13
|
eqtri |
|- ( A |` dom ( A |` B ) ) = ( A i^i ( ( B X. _V ) i^i ( dom A X. _V ) ) ) |
15 |
|
df-res |
|- ( `' `' A |` B ) = ( `' `' A i^i ( B X. _V ) ) |
16 |
7 14 15
|
3eqtr4i |
|- ( A |` dom ( A |` B ) ) = ( `' `' A |` B ) |
17 |
|
rescnvcnv |
|- ( `' `' A |` B ) = ( A |` B ) |
18 |
16 17
|
eqtri |
|- ( A |` dom ( A |` B ) ) = ( A |` B ) |