Step |
Hyp |
Ref |
Expression |
1 |
|
df-res |
|- ( C |` B ) = ( C i^i ( B X. _V ) ) |
2 |
1
|
difeq2i |
|- ( A \ ( C |` B ) ) = ( A \ ( C i^i ( B X. _V ) ) ) |
3 |
|
difindi |
|- ( A \ ( C i^i ( B X. _V ) ) ) = ( ( A \ C ) u. ( A \ ( B X. _V ) ) ) |
4 |
|
ssdif |
|- ( A C_ ( B X. _V ) -> ( A \ ( B X. _V ) ) C_ ( ( B X. _V ) \ ( B X. _V ) ) ) |
5 |
|
difid |
|- ( ( B X. _V ) \ ( B X. _V ) ) = (/) |
6 |
4 5
|
sseqtrdi |
|- ( A C_ ( B X. _V ) -> ( A \ ( B X. _V ) ) C_ (/) ) |
7 |
|
ss0 |
|- ( ( A \ ( B X. _V ) ) C_ (/) -> ( A \ ( B X. _V ) ) = (/) ) |
8 |
6 7
|
syl |
|- ( A C_ ( B X. _V ) -> ( A \ ( B X. _V ) ) = (/) ) |
9 |
8
|
uneq2d |
|- ( A C_ ( B X. _V ) -> ( ( A \ C ) u. ( A \ ( B X. _V ) ) ) = ( ( A \ C ) u. (/) ) ) |
10 |
3 9
|
syl5eq |
|- ( A C_ ( B X. _V ) -> ( A \ ( C i^i ( B X. _V ) ) ) = ( ( A \ C ) u. (/) ) ) |
11 |
|
un0 |
|- ( ( A \ C ) u. (/) ) = ( A \ C ) |
12 |
10 11
|
eqtrdi |
|- ( A C_ ( B X. _V ) -> ( A \ ( C i^i ( B X. _V ) ) ) = ( A \ C ) ) |
13 |
2 12
|
syl5eq |
|- ( A C_ ( B X. _V ) -> ( A \ ( C |` B ) ) = ( A \ C ) ) |