Metamath Proof Explorer


Theorem funcringcsetclem1ALTV

Description: Lemma 1 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 ) ) )
Assertion funcringcsetclem1ALTV
|- ( ( ph /\ X e. B ) -> ( F ` X ) = ( Base ` X ) )

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 6 adantr
 |-  ( ( ph /\ X e. B ) -> F = ( x e. B |-> ( Base ` x ) ) )
8 fveq2
 |-  ( x = X -> ( Base ` x ) = ( Base ` X ) )
9 8 adantl
 |-  ( ( ( ph /\ X e. B ) /\ x = X ) -> ( Base ` x ) = ( Base ` X ) )
10 simpr
 |-  ( ( ph /\ X e. B ) -> X e. B )
11 fvexd
 |-  ( ( ph /\ X e. B ) -> ( Base ` X ) e. _V )
12 7 9 10 11 fvmptd
 |-  ( ( ph /\ X e. B ) -> ( F ` X ) = ( Base ` X ) )