Theorem funcestrcsetclem3

Description: Lemma 3 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	funcestrcsetclem3	|- ( ph -> F : B --> C )

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	1 3 5	estrcbasbas	|- ( ( ph /\ x e. B ) -> ( Base ` x ) e. U )
8	2 5	setcbas	|- ( ph -> U = ( Base ` S ) )
9	8	eqcomd	|- ( ph -> ( Base ` S ) = U )
10	9	adantr	|- ( ( ph /\ x e. B ) -> ( Base ` S ) = U )
11	7 10	eleqtrrd	|- ( ( ph /\ x e. B ) -> ( Base ` x ) e. ( Base ` S ) )
12	11 4	eleqtrrdi	|- ( ( ph /\ x e. B ) -> ( Base ` x ) e. C )
13	6 12	fmpt3d	|- ( ph -> F : B --> C )