Description: The composition of an onto function and its converse. (Contributed by Stefan O'Rear, 12-Feb-2015)
Ref | Expression | ||
---|---|---|---|
Assertion | fococnv2 | |- ( F : A -onto-> B -> ( F o. `' F ) = ( _I |` B ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fofun | |- ( F : A -onto-> B -> Fun F ) |
|
2 | funcocnv2 | |- ( Fun F -> ( F o. `' F ) = ( _I |` ran F ) ) |
|
3 | 1 2 | syl | |- ( F : A -onto-> B -> ( F o. `' F ) = ( _I |` ran F ) ) |
4 | forn | |- ( F : A -onto-> B -> ran F = B ) |
|
5 | 4 | reseq2d | |- ( F : A -onto-> B -> ( _I |` ran F ) = ( _I |` B ) ) |
6 | 3 5 | eqtrd | |- ( F : A -onto-> B -> ( F o. `' F ) = ( _I |` B ) ) |