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 ) ) ) }