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Theorem tfr1onlem3ag 6102
Description: Lemma for transfinite recursion. This lemma changes some bound variables in  A (version of tfrlem3ag 6074 but for tfr1on 6115 related lemmas). (Contributed by Jim Kingdon, 13-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
tfr1onlem3ag  |-  ( H  e.  V  ->  ( H  e.  A  <->  E. z  e.  X  ( H  Fn  z  /\  A. w  e.  z  ( H `  w )  =  ( G `  ( H  |`  w ) ) ) ) )
Distinct variable groups:    f, G, w, x, y, z    f, H, w, x, y, z   
f, X, x, z
Allowed substitution hints:    A( x, y, z, w, f)    V( x, y, z, w, f)    X( y, w)

Proof of Theorem tfr1onlem3ag
StepHypRef Expression
1 fneq12 5107 . . . 4  |-  ( ( f  =  H  /\  x  =  z )  ->  ( f  Fn  x  <->  H  Fn  z ) )
2 simpll 496 . . . . . . 7  |-  ( ( ( f  =  H  /\  x  =  z )  /\  y  =  w )  ->  f  =  H )
3 simpr 108 . . . . . . 7  |-  ( ( ( f  =  H  /\  x  =  z )  /\  y  =  w )  ->  y  =  w )
42, 3fveq12d 5312 . . . . . 6  |-  ( ( ( f  =  H  /\  x  =  z )  /\  y  =  w )  ->  (
f `  y )  =  ( H `  w ) )
52, 3reseq12d 4714 . . . . . . 7  |-  ( ( ( f  =  H  /\  x  =  z )  /\  y  =  w )  ->  (
f  |`  y )  =  ( H  |`  w
) )
65fveq2d 5309 . . . . . 6  |-  ( ( ( f  =  H  /\  x  =  z )  /\  y  =  w )  ->  ( G `  ( f  |`  y ) )  =  ( G `  ( H  |`  w ) ) )
74, 6eqeq12d 2102 . . . . 5  |-  ( ( ( f  =  H  /\  x  =  z )  /\  y  =  w )  ->  (
( f `  y
)  =  ( G `
 ( f  |`  y ) )  <->  ( H `  w )  =  ( G `  ( H  |`  w ) ) ) )
8 simplr 497 . . . . 5  |-  ( ( ( f  =  H  /\  x  =  z )  /\  y  =  w )  ->  x  =  z )
97, 8cbvraldva2 2594 . . . 4  |-  ( ( f  =  H  /\  x  =  z )  ->  ( A. y  e.  x  ( f `  y )  =  ( G `  ( f  |`  y ) )  <->  A. w  e.  z  ( H `  w )  =  ( G `  ( H  |`  w ) ) ) )
101, 9anbi12d 457 . . 3  |-  ( ( f  =  H  /\  x  =  z )  ->  ( ( f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( G `  ( f  |`  y ) ) )  <-> 
( H  Fn  z  /\  A. w  e.  z  ( H `  w
)  =  ( G `
 ( H  |`  w ) ) ) ) )
1110cbvrexdva 2597 . 2  |-  ( f  =  H  ->  ( E. x  e.  X  ( f  Fn  x  /\  A. y  e.  x  ( f `  y
)  =  ( G `
 ( f  |`  y ) ) )  <->  E. z  e.  X  ( H  Fn  z  /\  A. w  e.  z  ( H `  w
)  =  ( G `
 ( H  |`  w ) ) ) ) )
12 tfr1onlem3ag.1 . 2  |-  A  =  { f  |  E. x  e.  X  (
f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( G `  ( f  |`  y
) ) ) }
1311, 12elab2g 2762 1  |-  ( H  e.  V  ->  ( H  e.  A  <->  E. z  e.  X  ( H  Fn  z  /\  A. w  e.  z  ( H `  w )  =  ( G `  ( H  |`  w ) ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1289    e. wcel 1438   {cab 2074   A.wral 2359   E.wrex 2360    |` 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:  tfr1onlem3  6103  tfr1onlemsucaccv  6106  tfr1onlembxssdm  6108  tfr1onlemres  6114
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