Theorem rngchom

Description: Set of arrows of the category of non-unital rings (in a universe). (Contributed by AV, 27-Feb-2020)

	Ref	Expression
Hypotheses	rngcbas.c	|- C = ( RngCat ` U )
	rngcbas.b	|- B = ( Base ` C )
	rngcbas.u	|- ( ph -> U e. V )
	rngchomfval.h	|- H = ( Hom ` C )
	rngchom.x	|- ( ph -> X e. B )
	rngchom.y	|- ( ph -> Y e. B )
Assertion	rngchom	|- ( ph -> ( X H Y ) = ( X RngHom Y ) )

Proof

Step	Hyp	Ref	Expression
1		rngcbas.c	|- C = ( RngCat ` U )
2		rngcbas.b	|- B = ( Base ` C )
3		rngcbas.u	|- ( ph -> U e. V )
4		rngchomfval.h	|- H = ( Hom ` C )
5		rngchom.x	|- ( ph -> X e. B )
6		rngchom.y	|- ( ph -> Y e. B )
7	1 2 3 4	rngchomfval	|- ( ph -> H = ( RngHom |` ( B X. B ) ) )
8	7	oveqd	|- ( ph -> ( X H Y ) = ( X ( RngHom |` ( B X. B ) ) Y ) )
9	5 6	ovresd	|- ( ph -> ( X ( RngHom |` ( B X. B ) ) Y ) = ( X RngHom Y ) )
10	8 9	eqtrd	|- ( ph -> ( X H Y ) = ( X RngHom Y ) )