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Theorem tfri2 6313
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 6312). 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 6312 . . . 4  |-  F  Fn  On
4 fndm 5269 . . . 4  |-  ( F  Fn  On  ->  dom  F  =  On )
53, 4ax-mp 5 . . 3  |-  dom  F  =  On
65eleq2i 2224 . 2  |-  ( A  e.  dom  F  <->  A  e.  On )
71tfr2a 6268 . 2  |-  ( A  e.  dom  F  -> 
( F `  A
)  =  ( G `
 ( F  |`  A ) ) )
86, 7sylbir 134 1  |-  ( A  e.  On  ->  ( F `  A )  =  ( G `  ( F  |`  A ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1335    e. wcel 2128   _Vcvv 2712   Oncon0 4323   dom cdm 4586    |` cres 4588   Fun wfun 5164    Fn wfn 5165   ` cfv 5170  recscrecs 6251
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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4079  ax-sep 4082  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4496
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-rex 2441  df-reu 2442  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-iun 3851  df-br 3966  df-opab 4026  df-mpt 4027  df-tr 4063  df-id 4253  df-iord 4326  df-on 4328  df-suc 4331  df-xp 4592  df-rel 4593  df-cnv 4594  df-co 4595  df-dm 4596  df-rn 4597  df-res 4598  df-ima 4599  df-iota 5135  df-fun 5172  df-fn 5173  df-f 5174  df-f1 5175  df-fo 5176  df-f1o 5177  df-fv 5178  df-recs 6252
This theorem is referenced by:  tfri3  6314
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