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Metamath Proof Explorer

Theorem rdglem1

Description: Lemma used with the recursive definition generator. This is a trivial lemma that just changes bound variables for later use. (Contributed by NM, 9-Apr-1995)

		Ref	Expression
	Assertion	rdglem1	\|- { f \| E. x e. On ( f Fn x /\ A. y e. x ( f ` y ) = ( G ` ( f \|` y ) ) ) } = { g \| E. z e. On ( g Fn z /\ A. w e. z ( g ` w ) = ( G ` ( g \|` w ) ) ) }

Proof

Step	Hyp	Ref	Expression
1		eqid	\|- { f \| E. x e. On ( f Fn x /\ A. y e. x ( f ` y ) = ( G ` ( f \|` y ) ) ) } = { f \| E. x e. On ( f Fn x /\ A. y e. x ( f ` y ) = ( G ` ( f \|` y ) ) ) }
2	1	tfrlem3	\|- { f \| E. x e. On ( f Fn x /\ A. y e. x ( f ` y ) = ( G ` ( f \|` y ) ) ) } = { g \| E. z e. On ( g Fn z /\ A. v e. z ( g ` v ) = ( G ` ( g \|` v ) ) ) }
3		fveq2	\|- ( v = w -> ( g ` v ) = ( g ` w ) )
4		reseq2	\|- ( v = w -> ( g \|` v ) = ( g \|` w ) )
5	4	fveq2d	\|- ( v = w -> ( G ` ( g \|` v ) ) = ( G ` ( g \|` w ) ) )
6	3 5	eqeq12d	\|- ( v = w -> ( ( g ` v ) = ( G ` ( g \|` v ) ) <-> ( g ` w ) = ( G ` ( g \|` w ) ) ) )
7	6	cbvralvw	\|- ( A. v e. z ( g ` v ) = ( G ` ( g \|` v ) ) <-> A. w e. z ( g ` w ) = ( G ` ( g \|` w ) ) )
8	7	anbi2i	\|- ( ( g Fn z /\ A. v e. z ( g ` v ) = ( G ` ( g \|` v ) ) ) <-> ( g Fn z /\ A. w e. z ( g ` w ) = ( G ` ( g \|` w ) ) ) )
9	8	rexbii	\|- ( E. z e. On ( g Fn z /\ A. v e. z ( g ` v ) = ( G ` ( g \|` v ) ) ) <-> E. z e. On ( g Fn z /\ A. w e. z ( g ` w ) = ( G ` ( g \|` w ) ) ) )
10	9	abbii	\|- { g \| E. z e. On ( g Fn z /\ A. v e. z ( g ` v ) = ( G ` ( g \|` v ) ) ) } = { g \| E. z e. On ( g Fn z /\ A. w e. z ( g ` w ) = ( G ` ( g \|` w ) ) ) }
11	2 10	eqtri	\|- { f \| E. x e. On ( f Fn x /\ A. y e. x ( f ` y ) = ( G ` ( f \|` y ) ) ) } = { g \| E. z e. On ( g Fn z /\ A. w e. z ( g ` w ) = ( G ` ( g \|` w ) ) ) }