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Theorem rdglem1 6609
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    x, z,
y, g    f, G, g, x    z, G    y, w, G, z, g

Proof of Theorem rdglem1
StepHypRef Expression
1 eqid 2387 . . 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 ) ) ) }
21tfrlem3 6574 . 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. y  e.  z  ( g `  y )  =  ( G `  ( g  |`  y ) ) ) }
3 fveq2 5668 . . . . . . 7  |-  ( y  =  w  ->  (
g `  y )  =  ( g `  w ) )
4 reseq2 5081 . . . . . . . 8  |-  ( y  =  w  ->  (
g  |`  y )  =  ( g  |`  w
) )
54fveq2d 5672 . . . . . . 7  |-  ( y  =  w  ->  ( G `  ( g  |`  y ) )  =  ( G `  (
g  |`  w ) ) )
63, 5eqeq12d 2401 . . . . . 6  |-  ( y  =  w  ->  (
( g `  y
)  =  ( G `
 ( g  |`  y ) )  <->  ( g `  w )  =  ( G `  ( g  |`  w ) ) ) )
76cbvralv 2875 . . . . 5  |-  ( A. y  e.  z  (
g `  y )  =  ( G `  ( g  |`  y
) )  <->  A. w  e.  z  ( g `  w )  =  ( G `  ( g  |`  w ) ) )
87anbi2i 676 . . . 4  |-  ( ( g  Fn  z  /\  A. y  e.  z  ( g `  y )  =  ( G `  ( g  |`  y
) ) )  <->  ( g  Fn  z  /\  A. w  e.  z  ( g `  w )  =  ( G `  ( g  |`  w ) ) ) )
98rexbii 2674 . . 3  |-  ( E. z  e.  On  (
g  Fn  z  /\  A. y  e.  z  ( g `  y )  =  ( G `  ( g  |`  y
) ) )  <->  E. z  e.  On  ( g  Fn  z  /\  A. w  e.  z  ( g `  w )  =  ( G `  ( g  |`  w ) ) ) )
109abbii 2499 . 2  |-  { g  |  E. z  e.  On  ( g  Fn  z  /\  A. y  e.  z  ( g `  y )  =  ( G `  ( g  |`  y ) ) ) }  =  { g  |  E. z  e.  On  ( g  Fn  z  /\  A. w  e.  z  ( g `  w )  =  ( G `  ( g  |`  w ) ) ) }
112, 10eqtri 2407 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 ) ) ) }
Colors of variables: wff set class
Syntax hints:    /\ wa 359    = wceq 1649   {cab 2373   A.wral 2649   E.wrex 2650   Oncon0 4522    |` cres 4820    Fn wfn 5389   ` cfv 5394
This theorem is referenced by:  rdgseg  6616
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-br 4154  df-opab 4208  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-res 4830  df-iota 5358  df-fun 5396  df-fn 5397  df-fv 5402
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