Theorem funcestrcsetclem4

Description: Lemma 4 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 ) ) )
	funcestrcsetc.g	|- ( ph -> G = ( x e. B , y e. B |-> ( _I |` ( ( Base ` y ) ^m ( Base ` x ) ) ) ) )
Assertion	funcestrcsetclem4	|- ( ph -> G Fn ( B X. B ) )

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		funcestrcsetc.g	|- ( ph -> G = ( x e. B , y e. B |-> ( _I |` ( ( Base ` y ) ^m ( Base ` x ) ) ) ) )
8		eqid	|- ( x e. B , y e. B |-> ( _I |` ( ( Base ` y ) ^m ( Base ` x ) ) ) ) = ( x e. B , y e. B |-> ( _I |` ( ( Base ` y ) ^m ( Base ` x ) ) ) )
9		ovex	|- ( ( Base ` y ) ^m ( Base ` x ) ) e. _V
10		resiexg	|- ( ( ( Base ` y ) ^m ( Base ` x ) ) e. _V -> ( _I |` ( ( Base ` y ) ^m ( Base ` x ) ) ) e. _V )
11	9 10	ax-mp	|- ( _I |` ( ( Base ` y ) ^m ( Base ` x ) ) ) e. _V
12	8 11	fnmpoi	|- ( x e. B , y e. B |-> ( _I |` ( ( Base ` y ) ^m ( Base ` x ) ) ) ) Fn ( B X. B )
13	7	fneq1d	|- ( ph -> ( G Fn ( B X. B ) <-> ( x e. B , y e. B |-> ( _I |` ( ( Base ` y ) ^m ( Base ` x ) ) ) ) Fn ( B X. B ) ) )
14	12 13	mpbiri	|- ( ph -> G Fn ( B X. B ) )