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Theorem tfri2 5952
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 5951). 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
tfri1.1  |-  F  = recs ( G )
tfri1.2  |-  ( Fun 
G  /\  ( G `  x )  e.  _V )
Assertion
Ref Expression
tfri2  |-  ( A  e.  On  ->  ( F `  A )  =  ( G `  ( F  |`  A ) ) )
Distinct variable group:    x, G
Allowed substitution hints:    A( x)    F( x)

Proof of Theorem tfri2
StepHypRef Expression
1 tfri1.1 . . . . 5  |-  F  = recs ( G )
2 tfri1.2 . . . . 5  |-  ( Fun 
G  /\  ( G `  x )  e.  _V )
31, 2tfri1 5951 . . . 4  |-  F  Fn  On
4 fndm 4998 . . . 4  |-  ( F  Fn  On  ->  dom  F  =  On )
53, 4ax-mp 7 . . 3  |-  dom  F  =  On
65eleq2i 2104 . 2  |-  ( A  e.  dom  F  <->  A  e.  On )
71tfr2a 5936 . 2  |-  ( A  e.  dom  F  -> 
( F `  A
)  =  ( G `
 ( F  |`  A ) ) )
86, 7sylbir 125 1  |-  ( A  e.  On  ->  ( F `  A )  =  ( G `  ( F  |`  A ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 97    = wceq 1243    e. wcel 1393   _Vcvv 2557   Oncon0 4100   dom cdm 4345    |` cres 4347   Fun wfun 4896    Fn wfn 4897   ` cfv 4902  recscrecs 5919
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-id 4030  df-iord 4103  df-on 4105  df-suc 4108  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-recs 5920
This theorem is referenced by:  tfri3  5953
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