Metamath Proof Explorer


Theorem arwdm

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 )

Proof

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 )