Theorem elrngchomALTV

Description: A morphism of non-unital rings is a function. (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	elrngchomALTV	|- ( ph -> ( F e. ( X H Y ) -> F : ( Base ` X ) --> ( Base ` 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 5 6	rngchomALTV	|- ( ph -> ( X H Y ) = ( X RngHomo Y ) )
8	7	eleq2d	|- ( ph -> ( F e. ( X H Y ) <-> F e. ( X RngHomo Y ) ) )
9		eqid	|- ( Base ` X ) = ( Base ` X )
10		eqid	|- ( Base ` Y ) = ( Base ` Y )
11	9 10	rnghmf	|- ( F e. ( X RngHomo Y ) -> F : ( Base ` X ) --> ( Base ` Y ) )
12	8 11	syl6bi	|- ( ph -> ( F e. ( X H Y ) -> F : ( Base ` X ) --> ( Base ` Y ) ) )