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Theorem tfr1onlem3ag 6227
Description: Lemma for transfinite recursion. This lemma changes some bound variables in  A (version of tfrlem3ag 6199 but for tfr1on 6240 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 5211 . . . 4  |-  ( ( f  =  H  /\  x  =  z )  ->  ( f  Fn  x  <->  H  Fn  z ) )
2 simpll 518 . . . . . . 7  |-  ( ( ( f  =  H  /\  x  =  z )  /\  y  =  w )  ->  f  =  H )
3 simpr 109 . . . . . . 7  |-  ( ( ( f  =  H  /\  x  =  z )  /\  y  =  w )  ->  y  =  w )
42, 3fveq12d 5421 . . . . . 6  |-  ( ( ( f  =  H  /\  x  =  z )  /\  y  =  w )  ->  (
f `  y )  =  ( H `  w ) )
52, 3reseq12d 4815 . . . . . . 7  |-  ( ( ( f  =  H  /\  x  =  z )  /\  y  =  w )  ->  (
f  |`  y )  =  ( H  |`  w
) )
65fveq2d 5418 . . . . . 6  |-  ( ( ( f  =  H  /\  x  =  z )  /\  y  =  w )  ->  ( G `  ( f  |`  y ) )  =  ( G `  ( H  |`  w ) ) )
74, 6eqeq12d 2152 . . . . 5  |-  ( ( ( f  =  H  /\  x  =  z )  /\  y  =  w )  ->  (
( f `  y
)  =  ( G `
 ( f  |`  y ) )  <->  ( H `  w )  =  ( G `  ( H  |`  w ) ) ) )
8 simplr 519 . . . . 5  |-  ( ( ( f  =  H  /\  x  =  z )  /\  y  =  w )  ->  x  =  z )
97, 8cbvraldva2 2656 . . . 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 464 . . 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 2659 . 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 2826 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 103    <-> wb 104    = wceq 1331    e. wcel 1480   {cab 2123   A.wral 2414   E.wrex 2415    |` cres 4536    Fn wfn 5113   ` cfv 5118
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-un 3070  df-in 3072  df-ss 3079  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-opab 3985  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-res 4546  df-iota 5083  df-fun 5120  df-fn 5121  df-fv 5126
This theorem is referenced by:  tfr1onlem3  6228  tfr1onlemsucaccv  6231  tfr1onlembxssdm  6233  tfr1onlemres  6239
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