Theorem funcringcsetcALTV2lem1

Description: Lemma 1 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 ) ) )
Assertion	funcringcsetcALTV2lem1	|- ( ( ph /\ X e. B ) -> ( F ` X ) = ( Base ` X ) )

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	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 ) )