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Theorem tfr1onlem3 6103
Description: Lemma for transfinite recursion. This lemma changes some bound variables in  A (version of tfrlem3 6076 but for tfr1on 6115 related lemmas). (Contributed by Jim Kingdon, 14-Mar-2022.)
Hypothesis
Ref Expression
tfr1onlem3ag.1  |-  A  =  { f  |  E. x  e.  X  (
f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( G `  ( f  |`  y
) ) ) }
Assertion
Ref Expression
tfr1onlem3  |-  A  =  { g  |  E. z  e.  X  (
g  Fn  z  /\  A. w  e.  z  ( g `  w )  =  ( G `  ( g  |`  w
) ) ) }
Distinct variable groups:    f, G, w, x, y, z    f, X, x, z    A, g   
f, g, w, x, y, z
Allowed substitution hints:    A( x, y, z, w, f)    G( g)    X( y, w, g)

Proof of Theorem tfr1onlem3
StepHypRef Expression
1 vex 2622 . . 3  |-  g  e. 
_V
2 tfr1onlem3ag.1 . . . 4  |-  A  =  { f  |  E. x  e.  X  (
f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( G `  ( f  |`  y
) ) ) }
32tfr1onlem3ag 6102 . . 3  |-  ( g  e.  _V  ->  (
g  e.  A  <->  E. z  e.  X  ( g  Fn  z  /\  A. w  e.  z  ( g `  w )  =  ( G `  ( g  |`  w ) ) ) ) )
41, 3ax-mp 7 . 2  |-  ( g  e.  A  <->  E. z  e.  X  ( g  Fn  z  /\  A. w  e.  z  ( g `  w )  =  ( G `  ( g  |`  w ) ) ) )
54abbi2i 2202 1  |-  A  =  { g  |  E. z  e.  X  (
g  Fn  z  /\  A. w  e.  z  ( g `  w )  =  ( G `  ( g  |`  w
) ) ) }
Colors of variables: wff set class
Syntax hints:    /\ wa 102    <-> wb 103    = wceq 1289    e. wcel 1438   {cab 2074   A.wral 2359   E.wrex 2360   _Vcvv 2619    |` cres 4440    Fn wfn 5010   ` cfv 5015
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-opab 3900  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-res 4450  df-iota 4980  df-fun 5017  df-fn 5018  df-fv 5023
This theorem is referenced by:  tfr1on  6115
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