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Metamath Proof Explorer

Theorem homa1

Description: The first component of an arrow is the ordered pair of domain and codomain. (Contributed by Mario Carneiro, 11-Jan-2017)

Ref Expression
Hypothesis homahom.h 
|- H = ( HomA ` C )
Assertion homa1 
|- ( Z ( X H Y ) F -> Z = <. X , Y >. )

Proof

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