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 ) ) ) }