Theorem funcestrcsetclem6

Description: Lemma 6 for funcestrcsetc . (Contributed by AV, 23-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 ) ) ) ) )
	funcestrcsetc.m	|- M = ( Base ` X )
	funcestrcsetc.n	|- N = ( Base ` Y )
Assertion	funcestrcsetclem6	|- ( ( ph /\ ( X e. B /\ Y e. B ) /\ H e. ( N ^m M ) ) -> ( ( X G Y ) ` H ) = H )

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		funcestrcsetc.m	|- M = ( Base ` X )
9		funcestrcsetc.n	|- N = ( Base ` Y )
10	1 2 3 4 5 6 7 8 9	funcestrcsetclem5	|- ( ( ph /\ ( X e. B /\ Y e. B ) ) -> ( X G Y ) = ( _I |` ( N ^m M ) ) )
11	10	3adant3	|- ( ( ph /\ ( X e. B /\ Y e. B ) /\ H e. ( N ^m M ) ) -> ( X G Y ) = ( _I |` ( N ^m M ) ) )
12	11	fveq1d	|- ( ( ph /\ ( X e. B /\ Y e. B ) /\ H e. ( N ^m M ) ) -> ( ( X G Y ) ` H ) = ( ( _I |` ( N ^m M ) ) ` H ) )
13		fvresi	|- ( H e. ( N ^m M ) -> ( ( _I |` ( N ^m M ) ) ` H ) = H )
14	13	3ad2ant3	|- ( ( ph /\ ( X e. B /\ Y e. B ) /\ H e. ( N ^m M ) ) -> ( ( _I |` ( N ^m M ) ) ` H ) = H )
15	12 14	eqtrd	|- ( ( ph /\ ( X e. B /\ Y e. B ) /\ H e. ( N ^m M ) ) -> ( ( X G Y ) ` H ) = H )