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Theorem rdglem1 6444
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 2296 . . 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 6409 . 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 5541 . . . . . . 7  |-  ( y  =  w  ->  (
g `  y )  =  ( g `  w ) )
4 reseq2 4966 . . . . . . . 8  |-  ( y  =  w  ->  (
g  |`  y )  =  ( g  |`  w
) )
54fveq2d 5545 . . . . . . 7  |-  ( y  =  w  ->  ( G `  ( g  |`  y ) )  =  ( G `  (
g  |`  w ) ) )
63, 5eqeq12d 2310 . . . . . 6  |-  ( y  =  w  ->  (
( g `  y
)  =  ( G `
 ( g  |`  y ) )  <->  ( g `  w )  =  ( G `  ( g  |`  w ) ) ) )
76cbvralv 2777 . . . . 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 2581 . . 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 2408 . 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 2316 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 1632   {cab 2282   A.wral 2556   E.wrex 2557   Oncon0 4408    |` cres 4707    Fn wfn 5266   ` cfv 5271
This theorem is referenced by:  rdgseg  6451
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-res 4717  df-iota 5235  df-fun 5273  df-fn 5274  df-fv 5279
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