Theorem rngchomALTV

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

	Ref	Expression
Hypotheses	rngcbasALTV.c	|- C = ( RngCatALTV ` U )
	rngcbasALTV.b	|- B = ( Base ` C )
	rngcbasALTV.u	|- ( ph -> U e. V )
	rngchomfvalALTV.h	|- H = ( Hom ` C )
	rngchomALTV.x	|- ( ph -> X e. B )
	rngchomALTV.y	|- ( ph -> Y e. B )
Assertion	rngchomALTV	|- ( ph -> ( X H Y ) = ( X RngHom Y ) )

Proof

Step	Hyp	Ref	Expression
1		rngcbasALTV.c	|- C = ( RngCatALTV ` U )
2		rngcbasALTV.b	|- B = ( Base ` C )
3		rngcbasALTV.u	|- ( ph -> U e. V )
4		rngchomfvalALTV.h	|- H = ( Hom ` C )
5		rngchomALTV.x	|- ( ph -> X e. B )
6		rngchomALTV.y	|- ( ph -> Y e. B )
7	1 2 3 4	rngchomfvalALTV	|- ( ph -> H = ( x e. B , y e. B |-> ( x RngHom y ) ) )
8		oveq12	|- ( ( x = X /\ y = Y ) -> ( x RngHom y ) = ( X RngHom Y ) )
9	8	adantl	|- ( ( ph /\ ( x = X /\ y = Y ) ) -> ( x RngHom y ) = ( X RngHom Y ) )
10		ovexd	|- ( ph -> ( X RngHom Y ) e. _V )
11	7 9 5 6 10	ovmpod	|- ( ph -> ( X H Y ) = ( X RngHom Y ) )