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Metamath Proof Explorer

Theorem homarw

Description: A hom-set is a subset of the collection of all arrows. (Contributed by Mario Carneiro, 11-Jan-2017)

Ref Expression
Hypotheses arwrcl.a 
|- A = ( Arrow ` C )
arwhoma.h 
|- H = ( HomA ` C )
Assertion homarw 
|- ( X H Y ) C_ A

Proof

Step Hyp Ref Expression
1 arwrcl.a 
 |-  A = ( Arrow ` C )
2 arwhoma.h 
 |-  H = ( HomA ` C )
3 ovssunirn 
 |-  ( X H Y ) C_ U. ran H
4 1 2 arwval 
 |-  A = U. ran H
5 3 4 sseqtrri 
 |-  ( X H Y ) C_ A