Step |
Hyp |
Ref |
Expression |
1 |
|
resundi |
|- ( A |` ( B u. C ) ) = ( ( A |` B ) u. ( A |` C ) ) |
2 |
1
|
rneqi |
|- ran ( A |` ( B u. C ) ) = ran ( ( A |` B ) u. ( A |` C ) ) |
3 |
|
rnun |
|- ran ( ( A |` B ) u. ( A |` C ) ) = ( ran ( A |` B ) u. ran ( A |` C ) ) |
4 |
2 3
|
eqtri |
|- ran ( A |` ( B u. C ) ) = ( ran ( A |` B ) u. ran ( A |` C ) ) |
5 |
|
df-ima |
|- ( A " ( B u. C ) ) = ran ( A |` ( B u. C ) ) |
6 |
|
df-ima |
|- ( A " B ) = ran ( A |` B ) |
7 |
|
df-ima |
|- ( A " C ) = ran ( A |` C ) |
8 |
6 7
|
uneq12i |
|- ( ( A " B ) u. ( A " C ) ) = ( ran ( A |` B ) u. ran ( A |` C ) ) |
9 |
4 5 8
|
3eqtr4i |
|- ( A " ( B u. C ) ) = ( ( A " B ) u. ( A " C ) ) |