Theorem funcestrcsetclem1

Description: Lemma 1 for funcestrcsetc . (Contributed by AV, 22-Mar-2020)

	Ref	Expression
Hypotheses	funcestrcsetc.e	|- E = ( ExtStrCat ` U )
	funcestrcsetc.s	|- S = ( SetCat ` U )
	funcestrcsetc.b	|- B = ( Base ` E )
	funcestrcsetc.c	|- C = ( Base ` S )
	funcestrcsetc.u	|- ( ph -> U e. WUni )
	funcestrcsetc.f	|- ( ph -> F = ( x e. B |-> ( Base ` x ) ) )
Assertion	funcestrcsetclem1	|- ( ( ph /\ X e. B ) -> ( F ` X ) = ( Base ` X ) )

Proof

Step	Hyp	Ref	Expression
1		funcestrcsetc.e	|- E = ( ExtStrCat ` U )
2		funcestrcsetc.s	|- S = ( SetCat ` U )
3		funcestrcsetc.b	|- B = ( Base ` E )
4		funcestrcsetc.c	|- C = ( Base ` S )
5		funcestrcsetc.u	|- ( ph -> U e. WUni )
6		funcestrcsetc.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 ) )