Theorem elringchom

Description: A morphism of unital rings is a function. (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	elringchom	|- ( ph -> ( F e. ( X H Y ) -> F : ( Base ` X ) --> ( Base ` 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 5 6	ringchom	|- ( ph -> ( X H Y ) = ( X RingHom Y ) )
8	7	eleq2d	|- ( ph -> ( F e. ( X H Y ) <-> F e. ( X RingHom Y ) ) )
9		eqid	|- ( Base ` X ) = ( Base ` X )
10		eqid	|- ( Base ` Y ) = ( Base ` Y )
11	9 10	rhmf	|- ( F e. ( X RingHom Y ) -> F : ( Base ` X ) --> ( Base ` Y ) )
12	8 11	biimtrdi	|- ( ph -> ( F e. ( X H Y ) -> F : ( Base ` X ) --> ( Base ` Y ) ) )