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Theorem tfri2d 6199
Description: Principle of Transfinite Recursion, part 2 of 3. Theorem 7.41(2) of [TakeutiZaring] p. 47, with an additional condition on the recursion rule  G ( as described at tfri1 6228). Here we show that the function  F has the property that for any function  G satisfying that condition, the "next" value of  F is  G recursively applied to all "previous" values of  F. (Contributed by Jim Kingdon, 4-May-2019.)
Hypotheses
Ref Expression
tfri1d.1  |-  F  = recs ( G )
tfri1d.2  |-  ( ph  ->  A. x ( Fun 
G  /\  ( G `  x )  e.  _V ) )
Assertion
Ref Expression
tfri2d  |-  ( (
ph  /\  A  e.  On )  ->  ( F `
 A )  =  ( G `  ( F  |`  A ) ) )
Distinct variable group:    x, G
Allowed substitution hints:    ph( x)    A( x)    F( x)

Proof of Theorem tfri2d
StepHypRef Expression
1 tfri1d.1 . . . . . 6  |-  F  = recs ( G )
2 tfri1d.2 . . . . . 6  |-  ( ph  ->  A. x ( Fun 
G  /\  ( G `  x )  e.  _V ) )
31, 2tfri1d 6198 . . . . 5  |-  ( ph  ->  F  Fn  On )
4 fndm 5190 . . . . 5  |-  ( F  Fn  On  ->  dom  F  =  On )
53, 4syl 14 . . . 4  |-  ( ph  ->  dom  F  =  On )
65eleq2d 2185 . . 3  |-  ( ph  ->  ( A  e.  dom  F  <-> 
A  e.  On ) )
76biimpar 293 . 2  |-  ( (
ph  /\  A  e.  On )  ->  A  e. 
dom  F )
81tfr2a 6184 . 2  |-  ( A  e.  dom  F  -> 
( F `  A
)  =  ( G `
 ( F  |`  A ) ) )
97, 8syl 14 1  |-  ( (
ph  /\  A  e.  On )  ->  ( F `
 A )  =  ( G `  ( F  |`  A ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103   A.wal 1312    = wceq 1314    e. wcel 1463   _Vcvv 2658   Oncon0 4253   dom cdm 4507    |` cres 4509   Fun wfun 5085    Fn wfn 5086   ` cfv 5091  recscrecs 6167
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-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4011  ax-sep 4014  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-setind 4420
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-ral 2396  df-rex 2397  df-reu 2398  df-rab 2400  df-v 2660  df-sbc 2881  df-csb 2974  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-iun 3783  df-br 3898  df-opab 3958  df-mpt 3959  df-tr 3995  df-id 4183  df-iord 4256  df-on 4258  df-suc 4261  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098  df-fv 5099  df-recs 6168
This theorem is referenced by:  rdgivallem  6244
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