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Theorem rdglem1 7091
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 2467 . . 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 7057 . 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 5871 . . . . . . 7  |-  ( v  =  w  ->  (
g `  v )  =  ( g `  w ) )
4 reseq2 5273 . . . . . . . 8  |-  ( v  =  w  ->  (
g  |`  v )  =  ( g  |`  w
) )
54fveq2d 5875 . . . . . . 7  |-  ( v  =  w  ->  ( G `  ( g  |`  v ) )  =  ( G `  (
g  |`  w ) ) )
63, 5eqeq12d 2489 . . . . . 6  |-  ( v  =  w  ->  (
( g `  v
)  =  ( G `
 ( g  |`  v ) )  <->  ( g `  w )  =  ( G `  ( g  |`  w ) ) ) )
76cbvralv 3093 . . . . 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 2969 . . 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 2601 . 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 2496 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 1379   {cab 2452   A.wral 2817   E.wrex 2818   Oncon0 4883    |` cres 5006    Fn wfn 5588   ` cfv 5593
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4251  df-br 4453  df-opab 4511  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-res 5016  df-iota 5556  df-fun 5595  df-fn 5596  df-fv 5601
This theorem is referenced by:  rdgseg  7098
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