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Theorem tfrlem3ag 5952
Description: Lemma for transfinite recursion. This lemma just changes some bound variables in  A for later use. (Contributed by Jim Kingdon, 5-Jul-2019.)
Hypothesis
Ref Expression
tfrlem3.1  |-  A  =  { f  |  E. x  e.  On  (
f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( F `  ( f  |`  y
) ) ) }
Assertion
Ref Expression
tfrlem3ag  |-  ( G  e.  _V  ->  ( G  e.  A  <->  E. z  e.  On  ( G  Fn  z  /\  A. w  e.  z  ( G `  w )  =  ( F `  ( G  |`  w ) ) ) ) )
Distinct variable groups:    w, f, x, y, z, F    f, G, w, x, y, z
Allowed substitution hints:    A( x, y, z, w, f)

Proof of Theorem tfrlem3ag
StepHypRef Expression
1 fneq12 5017 . . . 4  |-  ( ( f  =  G  /\  x  =  z )  ->  ( f  Fn  x  <->  G  Fn  z ) )
2 simpll 496 . . . . . . 7  |-  ( ( ( f  =  G  /\  x  =  z )  /\  y  =  w )  ->  f  =  G )
3 simpr 108 . . . . . . 7  |-  ( ( ( f  =  G  /\  x  =  z )  /\  y  =  w )  ->  y  =  w )
42, 3fveq12d 5209 . . . . . 6  |-  ( ( ( f  =  G  /\  x  =  z )  /\  y  =  w )  ->  (
f `  y )  =  ( G `  w ) )
52, 3reseq12d 4635 . . . . . . 7  |-  ( ( ( f  =  G  /\  x  =  z )  /\  y  =  w )  ->  (
f  |`  y )  =  ( G  |`  w
) )
65fveq2d 5207 . . . . . 6  |-  ( ( ( f  =  G  /\  x  =  z )  /\  y  =  w )  ->  ( F `  ( f  |`  y ) )  =  ( F `  ( G  |`  w ) ) )
74, 6eqeq12d 2096 . . . . 5  |-  ( ( ( f  =  G  /\  x  =  z )  /\  y  =  w )  ->  (
( f `  y
)  =  ( F `
 ( f  |`  y ) )  <->  ( G `  w )  =  ( F `  ( G  |`  w ) ) ) )
8 simplr 497 . . . . 5  |-  ( ( ( f  =  G  /\  x  =  z )  /\  y  =  w )  ->  x  =  z )
97, 8cbvraldva2 2582 . . . 4  |-  ( ( f  =  G  /\  x  =  z )  ->  ( A. y  e.  x  ( f `  y )  =  ( F `  ( f  |`  y ) )  <->  A. w  e.  z  ( G `  w )  =  ( F `  ( G  |`  w ) ) ) )
101, 9anbi12d 457 . . 3  |-  ( ( f  =  G  /\  x  =  z )  ->  ( ( f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( F `  ( f  |`  y ) ) )  <-> 
( G  Fn  z  /\  A. w  e.  z  ( G `  w
)  =  ( F `
 ( G  |`  w ) ) ) ) )
1110cbvrexdva 2585 . 2  |-  ( f  =  G  ->  ( E. x  e.  On  ( f  Fn  x  /\  A. y  e.  x  ( f `  y
)  =  ( F `
 ( f  |`  y ) ) )  <->  E. z  e.  On  ( G  Fn  z  /\  A. w  e.  z  ( G `  w
)  =  ( F `
 ( G  |`  w ) ) ) ) )
12 tfrlem3.1 . 2  |-  A  =  { f  |  E. x  e.  On  (
f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( F `  ( f  |`  y
) ) ) }
1311, 12elab2g 2741 1  |-  ( G  e.  _V  ->  ( G  e.  A  <->  E. z  e.  On  ( G  Fn  z  /\  A. w  e.  z  ( G `  w )  =  ( F `  ( G  |`  w ) ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1285    e. wcel 1434   {cab 2068   A.wral 2349   E.wrex 2350   _Vcvv 2602   Oncon0 4120    |` cres 4367    Fn wfn 4921   ` cfv 4926
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 2064
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-sn 3406  df-pr 3407  df-op 3409  df-uni 3604  df-br 3788  df-opab 3842  df-xp 4371  df-rel 4372  df-cnv 4373  df-co 4374  df-dm 4375  df-res 4377  df-iota 4891  df-fun 4928  df-fn 4929  df-fv 4934
This theorem is referenced by:  tfrlemisucaccv  5968
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