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Theorem tfr1onlem3 6007
Description: Lemma for transfinite recursion. This lemma changes some bound variables in  A (version of tfrlem3 5980 but for tfr1on 6019 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 2613 . . 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 6006 . . 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 2197 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 1285    e. wcel 1434   {cab 2069   A.wral 2353   E.wrex 2354   _Vcvv 2610    |` cres 4393    Fn wfn 4947   ` cfv 4952
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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2612  df-un 2986  df-in 2988  df-ss 2995  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-br 3806  df-opab 3860  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-res 4403  df-iota 4917  df-fun 4954  df-fn 4955  df-fv 4960
This theorem is referenced by:  tfr1on  6019
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