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Theorem rdglem1 6876
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.
StepHypRef Expression
1 eqid 2443 . . 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 6842 . 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 5696 . . . . . . 7  |-  ( v  =  w  ->  (
g `  v )  =  ( g `  w ) )
4 reseq2 5110 . . . . . . . 8  |-  ( v  =  w  ->  (
g  |`  v )  =  ( g  |`  w
) )
54fveq2d 5700 . . . . . . 7  |-  ( v  =  w  ->  ( G `  ( g  |`  v ) )  =  ( G `  (
g  |`  w ) ) )
63, 5eqeq12d 2457 . . . . . 6  |-  ( v  =  w  ->  (
( g `  v
)  =  ( G `
 ( g  |`  v ) )  <->  ( g `  w )  =  ( G `  ( g  |`  w ) ) ) )
76cbvralv 2952 . . . . 5  |-  ( A. v  e.  z  (
g `  v )  =  ( G `  ( g  |`  v
) )  <->  A. w  e.  z  ( g `  w )  =  ( G `  ( g  |`  w ) ) )
87anbi2i 694 . . . 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 ) ) ) )
98rexbii 2745 . . 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 ) ) ) )
109abbii 2560 . 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 ) ) ) }
112, 10eqtri 2463 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 setvar class
Syntax hints:    /\ wa 369    = wceq 1369   {cab 2429   A.wral 2720   E.wrex 2721   Oncon0 4724    |` cres 4847    Fn wfn 5418   ` cfv 5423
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ral 2725  df-rex 2726  df-rab 2729  df-v 2979  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-br 4298  df-opab 4356  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-res 4857  df-iota 5386  df-fun 5425  df-fn 5426  df-fv 5431
This theorem is referenced by:  rdgseg  6883
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