Theorem funcringcsetcALTV2lem4

Description: Lemma 4 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 ) ) )
	funcringcsetcALTV2.g	|- ( ph -> G = ( x e. B , y e. B |-> ( _I |` ( x RingHom y ) ) ) )
Assertion	funcringcsetcALTV2lem4	|- ( ph -> G Fn ( B X. B ) )

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		funcringcsetcALTV2.g	|- ( ph -> G = ( x e. B , y e. B |-> ( _I |` ( x RingHom y ) ) ) )
8		eqid	|- ( x e. B , y e. B |-> ( _I |` ( x RingHom y ) ) ) = ( x e. B , y e. B |-> ( _I |` ( x RingHom y ) ) )
9		ovex	|- ( x RingHom y ) e. _V
10		id	|- ( ( x RingHom y ) e. _V -> ( x RingHom y ) e. _V )
11	10	resiexd	|- ( ( x RingHom y ) e. _V -> ( _I |` ( x RingHom y ) ) e. _V )
12	9 11	ax-mp	|- ( _I |` ( x RingHom y ) ) e. _V
13	8 12	fnmpoi	|- ( x e. B , y e. B |-> ( _I |` ( x RingHom y ) ) ) Fn ( B X. B )
14	7	fneq1d	|- ( ph -> ( G Fn ( B X. B ) <-> ( x e. B , y e. B |-> ( _I |` ( x RingHom y ) ) ) Fn ( B X. B ) ) )
15	13 14	mpbiri	|- ( ph -> G Fn ( B X. B ) )