Description: The domain of an arrow is an object. (Contributed by Mario Carneiro, 11-Jan-2017)
Ref | Expression | ||
---|---|---|---|
Hypotheses | arwrcl.a | |- A = ( Arrow ` C ) |
|
arwdm.b | |- B = ( Base ` C ) |
||
Assertion | arwdm | |- ( F e. A -> ( domA ` F ) e. B ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | arwrcl.a | |- A = ( Arrow ` C ) |
|
2 | arwdm.b | |- B = ( Base ` C ) |
|
3 | eqid | |- ( HomA ` C ) = ( HomA ` C ) |
|
4 | 1 3 | arwhoma | |- ( F e. A -> F e. ( ( domA ` F ) ( HomA ` C ) ( codA ` F ) ) ) |
5 | 3 2 | homarcl2 | |- ( F e. ( ( domA ` F ) ( HomA ` C ) ( codA ` F ) ) -> ( ( domA ` F ) e. B /\ ( codA ` F ) e. B ) ) |
6 | 4 5 | syl | |- ( F e. A -> ( ( domA ` F ) e. B /\ ( codA ` F ) e. B ) ) |
7 | 6 | simpld | |- ( F e. A -> ( domA ` F ) e. B ) |