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Theorem rdglem1 6428
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 2283 . . 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 6393 . 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 5525 . . . . . . 7  |-  ( y  =  w  ->  (
g `  y )  =  ( g `  w ) )
4 reseq2 4950 . . . . . . . 8  |-  ( y  =  w  ->  (
g  |`  y )  =  ( g  |`  w
) )
54fveq2d 5529 . . . . . . 7  |-  ( y  =  w  ->  ( G `  ( g  |`  y ) )  =  ( G `  (
g  |`  w ) ) )
63, 5eqeq12d 2297 . . . . . 6  |-  ( y  =  w  ->  (
( g `  y
)  =  ( G `
 ( g  |`  y ) )  <->  ( g `  w )  =  ( G `  ( g  |`  w ) ) ) )
76cbvralv 2764 . . . . 5  |-  ( A. y  e.  z  (
g `  y )  =  ( G `  ( g  |`  y
) )  <->  A. w  e.  z  ( g `  w )  =  ( G `  ( g  |`  w ) ) )
87anbi2i 675 . . . 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 2568 . . 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 2395 . 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 2303 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 358    = wceq 1623   {cab 2269   A.wral 2543   E.wrex 2544   Oncon0 4392    |` cres 4691    Fn wfn 5250   ` cfv 5255
This theorem is referenced by:  rdgseg  6435
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-res 4701  df-iota 5219  df-fun 5257  df-fn 5258  df-fv 5263
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