Theorem funcringcsetcALTV2lem3

Description: Lemma 3 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	funcringcsetcALTV2lem3	|- ( ph -> F : B --> C )

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	1 3 5	ringcbasbas	|- ( ( 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	12	fmpttd	|- ( ph -> ( x e. B |-> ( Base ` x ) ) : B --> C )
14	6	feq1d	|- ( ph -> ( F : B --> C <-> ( x e. B |-> ( Base ` x ) ) : B --> C ) )
15	13 14	mpbird	|- ( ph -> F : B --> C )