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Metamath Proof Explorer

Theorem homahom2

Description: The second component of an arrow is the corresponding morphism (without the domain/codomain tag). (Contributed by Mario Carneiro, 11-Jan-2017)

Ref Expression
Hypotheses homahom.h 
|- H = ( HomA ` C )
homahom.j 
|- J = ( Hom ` C )
Assertion homahom2 
|- ( Z ( X H Y ) F -> F e. ( X J Y ) )

Proof

Step Hyp Ref Expression
1 homahom.h 
 |-  H = ( HomA ` C )
2 homahom.j 
 |-  J = ( Hom ` C )
3 df-br 
 |-  ( Z ( X H Y ) F <-> <. Z , F >. e. ( X H Y ) )
4 eqid 
 |-  ( Base ` C ) = ( Base ` C )
5 1 homarcl 
 |-  ( <. Z , F >. e. ( X H Y ) -> C e. Cat )
6 1 4 homarcl2 
 |-  ( <. Z , F >. e. ( X H Y ) -> ( X e. ( Base ` C ) /\ Y e. ( Base ` C ) ) )
7 6 simpld 
 |-  ( <. Z , F >. e. ( X H Y ) -> X e. ( Base ` C ) )
8 6 simprd 
 |-  ( <. Z , F >. e. ( X H Y ) -> Y e. ( Base ` C ) )
9 1 4 5 2 7 8 elhoma 
 |-  ( <. Z , F >. e. ( X H Y ) -> ( Z ( X H Y ) F <-> ( Z = <. X , Y >. /\ F e. ( X J Y ) ) ) )
10 3 9 sylbi 
 |-  ( Z ( X H Y ) F -> ( Z ( X H Y ) F <-> ( Z = <. X , Y >. /\ F e. ( X J Y ) ) ) )
11 10 ibi 
 |-  ( Z ( X H Y ) F -> ( Z = <. X , Y >. /\ F e. ( X J Y ) ) )
12 11 simprd 
 |-  ( Z ( X H Y ) F -> F e. ( X J Y ) )