Metamath Proof Explorer


Theorem funcringcsetclem6ALTV

Description: Lemma 6 for funcringcsetcALTV . (Contributed by AV, 15-Feb-2020) (New usage is discouraged.)

Ref Expression
Hypotheses funcringcsetcALTV.r
|- R = ( RingCatALTV ` U )
funcringcsetcALTV.s
|- S = ( SetCat ` U )
funcringcsetcALTV.b
|- B = ( Base ` R )
funcringcsetcALTV.c
|- C = ( Base ` S )
funcringcsetcALTV.u
|- ( ph -> U e. WUni )
funcringcsetcALTV.f
|- ( ph -> F = ( x e. B |-> ( Base ` x ) ) )
funcringcsetcALTV.g
|- ( ph -> G = ( x e. B , y e. B |-> ( _I |` ( x RingHom y ) ) ) )
Assertion funcringcsetclem6ALTV
|- ( ( ph /\ ( X e. B /\ Y e. B ) /\ H e. ( X RingHom Y ) ) -> ( ( X G Y ) ` H ) = H )

Proof

Step Hyp Ref Expression
1 funcringcsetcALTV.r
 |-  R = ( RingCatALTV ` U )
2 funcringcsetcALTV.s
 |-  S = ( SetCat ` U )
3 funcringcsetcALTV.b
 |-  B = ( Base ` R )
4 funcringcsetcALTV.c
 |-  C = ( Base ` S )
5 funcringcsetcALTV.u
 |-  ( ph -> U e. WUni )
6 funcringcsetcALTV.f
 |-  ( ph -> F = ( x e. B |-> ( Base ` x ) ) )
7 funcringcsetcALTV.g
 |-  ( ph -> G = ( x e. B , y e. B |-> ( _I |` ( x RingHom y ) ) ) )
8 1 2 3 4 5 6 7 funcringcsetclem5ALTV
 |-  ( ( 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 )