Theorem funcringcsetcALTV2lem6

Description: Lemma 6 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	funcringcsetcALTV2lem6	|- ( ( ph /\ ( X e. B /\ Y e. B ) /\ H e. ( X RingHom Y ) ) -> ( ( X G Y ) ` H ) = H )

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	1 2 3 4 5 6 7	funcringcsetcALTV2lem5	|- ( ( ph /\ ( X e. B /\ Y e. B ) ) -> ( X G Y ) = ( _I |` ( X RingHom Y ) ) )
9	8	3adant3	|- ( ( ph /\ ( X e. B /\ Y e. B ) /\ H e. ( X RingHom Y ) ) -> ( X G Y ) = ( _I |` ( X RingHom Y ) ) )
10	9	fveq1d	|- ( ( ph /\ ( X e. B /\ Y e. B ) /\ H e. ( X RingHom Y ) ) -> ( ( X G Y ) ` H ) = ( ( _I |` ( X RingHom Y ) ) ` H ) )
11		fvresi	|- ( H e. ( X RingHom Y ) -> ( ( _I |` ( X RingHom Y ) ) ` H ) = H )
12	11	3ad2ant3	|- ( ( ph /\ ( X e. B /\ Y e. B ) /\ H e. ( X RingHom Y ) ) -> ( ( _I |` ( X RingHom Y ) ) ` H ) = H )
13	10 12	eqtrd	|- ( ( ph /\ ( X e. B /\ Y e. B ) /\ H e. ( X RingHom Y ) ) -> ( ( X G Y ) ` H ) = H )