Theorem rdglem1 7511

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.)

Assertion

Ref	Expression
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 ) ) ) }

Distinct variable groups:   x, y, f, g, z, G   y, w, G, z, g

Proof of Theorem rdglem1

Dummy variable v is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2622	. . 3 |- { 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 7474	. 2 |- { 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 6191	. . . . . . 7 |- ( v = w -> ( g ` v ) = ( g ` w ) )
4		reseq2 5391	. . . . . . . 8 |- ( v = w -> ( g |` v ) = ( g |` w ) )
5	4	fveq2d 6195	. . . . . . 7 |- ( v = w -> ( G ` ( g |` v ) ) = ( G ` ( g |` w ) ) )
6	3, 5	eqeq12d 2637	. . . . . 6 |- ( v = w -> ( ( g ` v ) = ( G ` ( g |` v ) ) <-> ( g ` w ) = ( G ` ( g |` w ) ) ) )
7	6	cbvralv 3171	. . . . 5 |- ( A. v e. z ( g ` v ) = ( G ` ( g |` v ) ) <-> A. w e. z ( g ` w ) = ( G ` ( g |` w ) ) )
8	7	anbi2i 730	. . . 4 |- ( ( 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 3041	. . 3 |- ( 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 2739	. 2 |- { 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 2644	1 |- { 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 ) ) ) }

Syntax hints:   /\ wa 384   = wceq 1483   {cab 2608   A.wral 2912   E.wrex 2913   |` cres 5116   Oncon0 5723   Fn wfn 5883   ` cfv 5888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-res 5126  df-iota 5851  df-fun 5890  df-fn 5891  df-fv 5896

This theorem is referenced by:  rdgseg  7518