Theorem funcringcsetcALTV2lem5

Description: Lemma 5 for funcringcsetcALTV2 . (Contributed by AV, 15-Feb-2020) (New usage is discouraged.)

	Ref	Expression
Hypotheses	funcringcsetcALTV2.r	|- R = ( RingCat ` U )
	funcringcsetcALTV2.s	|- S = ( SetCat ` U )
	funcringcsetcALTV2.b	|- B = ( Base ` R )
	funcringcsetcALTV2.c	|- C = ( Base ` S )
	funcringcsetcALTV2.u	|- ( ph -> U e. WUni )
	funcringcsetcALTV2.f	|- ( ph -> F = ( x e. B |-> ( Base ` x ) ) )
	funcringcsetcALTV2.g	|- ( ph -> G = ( x e. B , y e. B |-> ( _I |` ( x RingHom y ) ) ) )
Assertion	funcringcsetcALTV2lem5	|- ( ( ph /\ ( X e. B /\ Y e. B ) ) -> ( X G Y ) = ( _I |` ( X RingHom Y ) ) )

Proof

Step	Hyp	Ref	Expression
1		funcringcsetcALTV2.r	|- R = ( RingCat ` U )
2		funcringcsetcALTV2.s	|- S = ( SetCat ` U )
3		funcringcsetcALTV2.b	|- B = ( Base ` R )
4		funcringcsetcALTV2.c	|- C = ( Base ` S )
5		funcringcsetcALTV2.u	|- ( ph -> U e. WUni )
6		funcringcsetcALTV2.f	|- ( ph -> F = ( x e. B |-> ( Base ` x ) ) )
7		funcringcsetcALTV2.g	|- ( ph -> G = ( x e. B , y e. B |-> ( _I |` ( x RingHom y ) ) ) )
8	7	adantr	|- ( ( ph /\ ( X e. B /\ Y e. B ) ) -> G = ( x e. B , y e. B |-> ( _I |` ( x RingHom y ) ) ) )
9		oveq12	|- ( ( x = X /\ y = Y ) -> ( x RingHom y ) = ( X RingHom Y ) )
10	9	adantl	|- ( ( ( ph /\ ( X e. B /\ Y e. B ) ) /\ ( x = X /\ y = Y ) ) -> ( x RingHom y ) = ( X RingHom Y ) )
11	10	reseq2d	|- ( ( ( ph /\ ( X e. B /\ Y e. B ) ) /\ ( x = X /\ y = Y ) ) -> ( _I |` ( x RingHom y ) ) = ( _I |` ( X RingHom Y ) ) )
12		simprl	|- ( ( ph /\ ( X e. B /\ Y e. B ) ) -> X e. B )
13		simprr	|- ( ( ph /\ ( X e. B /\ Y e. B ) ) -> Y e. B )
14		ovexd	|- ( ( ph /\ ( X e. B /\ Y e. B ) ) -> ( X RingHom Y ) e. _V )
15	14	resiexd	|- ( ( ph /\ ( X e. B /\ Y e. B ) ) -> ( _I |` ( X RingHom Y ) ) e. _V )
16	8 11 12 13 15	ovmpod	|- ( ( ph /\ ( X e. B /\ Y e. B ) ) -> ( X G Y ) = ( _I |` ( X RingHom Y ) ) )