Theorem elrngchom

Description: A morphism of non-unital rings is a function. (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	elrngchom	|- ( ph -> ( F e. ( X H Y ) -> F : ( Base ` X ) --> ( Base ` 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 5 6	rngchom	|- ( 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 ) ) )