Theorem homarcl2

Description: Reverse closure for the domain and codomain of an arrow. (Contributed by Mario Carneiro, 11-Jan-2017)

	Ref	Expression
Hypotheses	homahom.h	|- H = ( HomA ` C )
	homarcl2.b	|- B = ( Base ` C )
Assertion	homarcl2	|- ( F e. ( X H Y ) -> ( X e. B /\ Y e. B ) )

Proof

Step	Hyp	Ref	Expression
1		homahom.h	|- H = ( HomA ` C )
2		homarcl2.b	|- B = ( Base ` C )
3		elfvdm	|- ( F e. ( H ` <. X , Y >. ) -> <. X , Y >. e. dom H )
4		df-ov	|- ( X H Y ) = ( H ` <. X , Y >. )
5	3 4	eleq2s	|- ( F e. ( X H Y ) -> <. X , Y >. e. dom H )
6	1	homarcl	|- ( F e. ( X H Y ) -> C e. Cat )
7	1 2 6	homaf	|- ( F e. ( X H Y ) -> H : ( B X. B ) --> ~P ( ( B X. B ) X. _V ) )
8	7	fdmd	|- ( F e. ( X H Y ) -> dom H = ( B X. B ) )
9	5 8	eleqtrd	|- ( F e. ( X H Y ) -> <. X , Y >. e. ( B X. B ) )
10		opelxp	|- ( <. X , Y >. e. ( B X. B ) <-> ( X e. B /\ Y e. B ) )
11	9 10	sylib	|- ( F e. ( X H Y ) -> ( X e. B /\ Y e. B ) )