Step |
Hyp |
Ref |
Expression |
1 |
|
df-res |
|- ( `' R |` B ) = ( `' R i^i ( B X. _V ) ) |
2 |
1
|
cnveqi |
|- `' ( `' R |` B ) = `' ( `' R i^i ( B X. _V ) ) |
3 |
|
cnvin |
|- `' ( `' R i^i ( B X. _V ) ) = ( `' `' R i^i `' ( B X. _V ) ) |
4 |
|
cnvcnv |
|- `' `' R = ( R i^i ( _V X. _V ) ) |
5 |
|
cnvxp |
|- `' ( B X. _V ) = ( _V X. B ) |
6 |
4 5
|
ineq12i |
|- ( `' `' R i^i `' ( B X. _V ) ) = ( ( R i^i ( _V X. _V ) ) i^i ( _V X. B ) ) |
7 |
|
inass |
|- ( ( R i^i ( _V X. _V ) ) i^i ( _V X. B ) ) = ( R i^i ( ( _V X. _V ) i^i ( _V X. B ) ) ) |
8 |
|
inxp |
|- ( ( _V X. _V ) i^i ( _V X. B ) ) = ( ( _V i^i _V ) X. ( _V i^i B ) ) |
9 |
|
inv1 |
|- ( _V i^i _V ) = _V |
10 |
9
|
eqcomi |
|- _V = ( _V i^i _V ) |
11 |
|
ssv |
|- B C_ _V |
12 |
|
ssid |
|- B C_ B |
13 |
11 12
|
ssini |
|- B C_ ( _V i^i B ) |
14 |
|
inss2 |
|- ( _V i^i B ) C_ B |
15 |
13 14
|
eqssi |
|- B = ( _V i^i B ) |
16 |
10 15
|
xpeq12i |
|- ( _V X. B ) = ( ( _V i^i _V ) X. ( _V i^i B ) ) |
17 |
8 16
|
eqtr4i |
|- ( ( _V X. _V ) i^i ( _V X. B ) ) = ( _V X. B ) |
18 |
17
|
ineq2i |
|- ( R i^i ( ( _V X. _V ) i^i ( _V X. B ) ) ) = ( R i^i ( _V X. B ) ) |
19 |
6 7 18
|
3eqtri |
|- ( `' `' R i^i `' ( B X. _V ) ) = ( R i^i ( _V X. B ) ) |
20 |
2 3 19
|
3eqtri |
|- `' ( `' R |` B ) = ( R i^i ( _V X. B ) ) |