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Theorem tfr1onlem3ag 6316
Description: Lemma for transfinite recursion. This lemma changes some bound variables in  A (version of tfrlem3ag 6288 but for tfr1on 6329 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 5291 . . . 4  |-  ( ( f  =  H  /\  x  =  z )  ->  ( f  Fn  x  <->  H  Fn  z ) )
2 simpll 524 . . . . . . 7  |-  ( ( ( f  =  H  /\  x  =  z )  /\  y  =  w )  ->  f  =  H )
3 simpr 109 . . . . . . 7  |-  ( ( ( f  =  H  /\  x  =  z )  /\  y  =  w )  ->  y  =  w )
42, 3fveq12d 5503 . . . . . 6  |-  ( ( ( f  =  H  /\  x  =  z )  /\  y  =  w )  ->  (
f `  y )  =  ( H `  w ) )
52, 3reseq12d 4892 . . . . . . 7  |-  ( ( ( f  =  H  /\  x  =  z )  /\  y  =  w )  ->  (
f  |`  y )  =  ( H  |`  w
) )
65fveq2d 5500 . . . . . 6  |-  ( ( ( f  =  H  /\  x  =  z )  /\  y  =  w )  ->  ( G `  ( f  |`  y ) )  =  ( G `  ( H  |`  w ) ) )
74, 6eqeq12d 2185 . . . . 5  |-  ( ( ( f  =  H  /\  x  =  z )  /\  y  =  w )  ->  (
( f `  y
)  =  ( G `
 ( f  |`  y ) )  <->  ( H `  w )  =  ( G `  ( H  |`  w ) ) ) )
8 simplr 525 . . . . 5  |-  ( ( ( f  =  H  /\  x  =  z )  /\  y  =  w )  ->  x  =  z )
97, 8cbvraldva2 2703 . . . 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 470 . . 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 2706 . 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 2877 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 1348    e. wcel 2141   {cab 2156   A.wral 2448   E.wrex 2449    |` cres 4613    Fn wfn 5193   ` cfv 5198
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-res 4623  df-iota 5160  df-fun 5200  df-fn 5201  df-fv 5206
This theorem is referenced by:  tfr1onlem3  6317  tfr1onlemsucaccv  6320  tfr1onlembxssdm  6322  tfr1onlemres  6328
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