Step |
Hyp |
Ref |
Expression |
1 |
|
df-fn |
|- ( ( F |` B ) Fn B <-> ( Fun ( F |` B ) /\ dom ( F |` B ) = B ) ) |
2 |
|
fnfun |
|- ( F Fn A -> Fun F ) |
3 |
|
funres |
|- ( Fun F -> Fun ( F |` B ) ) |
4 |
2 3
|
syl |
|- ( F Fn A -> Fun ( F |` B ) ) |
5 |
4
|
biantrurd |
|- ( F Fn A -> ( dom ( F |` B ) = B <-> ( Fun ( F |` B ) /\ dom ( F |` B ) = B ) ) ) |
6 |
|
ssdmres |
|- ( B C_ dom F <-> dom ( F |` B ) = B ) |
7 |
|
fndm |
|- ( F Fn A -> dom F = A ) |
8 |
7
|
sseq2d |
|- ( F Fn A -> ( B C_ dom F <-> B C_ A ) ) |
9 |
6 8
|
bitr3id |
|- ( F Fn A -> ( dom ( F |` B ) = B <-> B C_ A ) ) |
10 |
5 9
|
bitr3d |
|- ( F Fn A -> ( ( Fun ( F |` B ) /\ dom ( F |` B ) = B ) <-> B C_ A ) ) |
11 |
1 10
|
syl5bb |
|- ( F Fn A -> ( ( F |` B ) Fn B <-> B C_ A ) ) |