Metamath Proof Explorer


Theorem funcringcsetclem5ALTV

Description: Lemma 5 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 funcringcsetclem5ALTV
|- ( ( ph /\ ( X e. B /\ Y e. B ) ) -> ( X G Y ) = ( _I |` ( X RingHom Y ) ) )

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 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 ) ) )