Theorem funcsetcestrclem2

Description: Lemma 2 for funcsetcestrc . (Contributed by AV, 27-Mar-2020)

	Ref	Expression
Hypotheses	funcsetcestrc.s	|- S = ( SetCat ` U )
	funcsetcestrc.c	|- C = ( Base ` S )
	funcsetcestrc.f	|- ( ph -> F = ( x e. C |-> { <. ( Base ` ndx ) , x >. } ) )
	funcsetcestrc.u	|- ( ph -> U e. WUni )
	funcsetcestrc.o	|- ( ph -> _om e. U )
Assertion	funcsetcestrclem2	|- ( ( ph /\ X e. C ) -> ( F ` X ) e. U )

Proof

Step	Hyp	Ref	Expression
1		funcsetcestrc.s	|- S = ( SetCat ` U )
2		funcsetcestrc.c	|- C = ( Base ` S )
3		funcsetcestrc.f	|- ( ph -> F = ( x e. C |-> { <. ( Base ` ndx ) , x >. } ) )
4		funcsetcestrc.u	|- ( ph -> U e. WUni )
5		funcsetcestrc.o	|- ( ph -> _om e. U )
6	1 2 3	funcsetcestrclem1	|- ( ( ph /\ X e. C ) -> ( F ` X ) = { <. ( Base ` ndx ) , X >. } )
7	1 2 4 5	setc1strwun	|- ( ( ph /\ X e. C ) -> { <. ( Base ` ndx ) , X >. } e. U )
8	6 7	eqeltrd	|- ( ( ph /\ X e. C ) -> ( F ` X ) e. U )