Metamath Proof Explorer

Theorem homadmcd

Description: Decompose an arrow into domain, codomain, and morphism. (Contributed by Mario Carneiro, 11-Jan-2017)

Ref Expression
Hypothesis homahom.h 
|- H = ( HomA ` C )
Assertion homadmcd 
|- ( F e. ( X H Y ) -> F = <. X , Y , ( 2nd ` F ) >. )

Proof

Step Hyp Ref Expression
1 homahom.h 
 |-  H = ( HomA ` C )
2 1 homarel 
 |-  Rel ( X H Y )
3 1st2nd 
 |-  ( ( Rel ( X H Y ) /\ F e. ( X H Y ) ) -> F = <. ( 1st ` F ) , ( 2nd ` F ) >. )
4 2 3 mpan 
 |-  ( F e. ( X H Y ) -> F = <. ( 1st ` F ) , ( 2nd ` F ) >. )
5 1st2ndbr 
 |-  ( ( Rel ( X H Y ) /\ F e. ( X H Y ) ) -> ( 1st ` F ) ( X H Y ) ( 2nd ` F ) )
6 2 5 mpan 
 |-  ( F e. ( X H Y ) -> ( 1st ` F ) ( X H Y ) ( 2nd ` F ) )
7 1 homa1 
 |-  ( ( 1st ` F ) ( X H Y ) ( 2nd ` F ) -> ( 1st ` F ) = <. X , Y >. )
8 6 7 syl 
 |-  ( F e. ( X H Y ) -> ( 1st ` F ) = <. X , Y >. )
9 8 opeq1d 
 |-  ( F e. ( X H Y ) -> <. ( 1st ` F ) , ( 2nd ` F ) >. = <. <. X , Y >. , ( 2nd ` F ) >. )
10 4 9 eqtrd 
 |-  ( F e. ( X H Y ) -> F = <. <. X , Y >. , ( 2nd ` F ) >. )
11 df-ot 
 |-  <. X , Y , ( 2nd ` F ) >. = <. <. X , Y >. , ( 2nd ` F ) >.
12 10 11 eqtr4di 
 |-  ( F e. ( X H Y ) -> F = <. X , Y , ( 2nd ` F ) >. )