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Theorem tfrlem3a 6304
Description: Lemma for transfinite recursion. Let  A be the class of "acceptable" functions. The final thing we're interested in is the union of all these acceptable functions. This lemma just changes some bound variables in  A for later use. (Contributed by NM, 9-Apr-1995.)
Hypotheses
Ref Expression
tfrlem3.1  |-  A  =  { f  |  E. x  e.  On  (
f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( F `  ( f  |`  y
) ) ) }
tfrlem3.2  |-  G  e. 
_V
Assertion
Ref Expression
tfrlem3a  |-  ( 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 tfrlem3a
StepHypRef Expression
1 tfrlem3.2 . 2  |-  G  e. 
_V
2 fneq12 5304 . . . 4  |-  ( ( f  =  G  /\  x  =  z )  ->  ( f  Fn  x  <->  G  Fn  z ) )
3 simpll 527 . . . . . . 7  |-  ( ( ( f  =  G  /\  x  =  z )  /\  y  =  w )  ->  f  =  G )
4 simpr 110 . . . . . . 7  |-  ( ( ( f  =  G  /\  x  =  z )  /\  y  =  w )  ->  y  =  w )
53, 4fveq12d 5517 . . . . . 6  |-  ( ( ( f  =  G  /\  x  =  z )  /\  y  =  w )  ->  (
f `  y )  =  ( G `  w ) )
63, 4reseq12d 4903 . . . . . . 7  |-  ( ( ( f  =  G  /\  x  =  z )  /\  y  =  w )  ->  (
f  |`  y )  =  ( G  |`  w
) )
76fveq2d 5514 . . . . . 6  |-  ( ( ( f  =  G  /\  x  =  z )  /\  y  =  w )  ->  ( F `  ( f  |`  y ) )  =  ( F `  ( G  |`  w ) ) )
85, 7eqeq12d 2192 . . . . 5  |-  ( ( ( f  =  G  /\  x  =  z )  /\  y  =  w )  ->  (
( f `  y
)  =  ( F `
 ( f  |`  y ) )  <->  ( G `  w )  =  ( F `  ( G  |`  w ) ) ) )
9 simpr 110 . . . . . 6  |-  ( ( f  =  G  /\  x  =  z )  ->  x  =  z )
109adantr 276 . . . . 5  |-  ( ( ( f  =  G  /\  x  =  z )  /\  y  =  w )  ->  x  =  z )
118, 10cbvraldva2 2710 . . . 4  |-  ( ( f  =  G  /\  x  =  z )  ->  ( A. y  e.  x  ( f `  y )  =  ( F `  ( f  |`  y ) )  <->  A. w  e.  z  ( G `  w )  =  ( F `  ( G  |`  w ) ) ) )
122, 11anbi12d 473 . . 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 ) ) ) ) )
1312cbvrexdva 2713 . 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 ) ) ) ) )
14 tfrlem3.1 . 2  |-  A  =  { f  |  E. x  e.  On  (
f  Fn  x  /\  A. y  e.  x  ( f `  y )  =  ( F `  ( f  |`  y
) ) ) }
151, 13, 14elab2 2885 1  |-  ( 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:    /\ wa 104    <-> wb 105    = wceq 1353    e. wcel 2148   {cab 2163   A.wral 2455   E.wrex 2456   _Vcvv 2737   Oncon0 4359    |` cres 4624    Fn wfn 5206   ` cfv 5211
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-br 4001  df-opab 4062  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-res 4634  df-iota 5173  df-fun 5213  df-fn 5214  df-fv 5219
This theorem is referenced by:  tfrlem3  6305  tfrlem5  6308  tfrlemisucaccv  6319  tfrlemibxssdm  6321  tfrlemi14d  6327  tfrexlem  6328
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