Theorem ringchom

Description: Set of arrows of the category of unital rings (in a universe). (Contributed by AV, 14-Feb-2020)

	Ref	Expression
Hypotheses	ringcbas.c	|- C = ( RingCat ` U )
	ringcbas.b	|- B = ( Base ` C )
	ringcbas.u	|- ( ph -> U e. V )
	ringchomfval.h	|- H = ( Hom ` C )
	ringchom.x	|- ( ph -> X e. B )
	ringchom.y	|- ( ph -> Y e. B )
Assertion	ringchom	|- ( ph -> ( X H Y ) = ( X RingHom Y ) )

Proof

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