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Theorem tfri2 5890
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 5889). 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 `  _V
Assertion
Ref Expression
tfri2  On  F `  G `  F  |`
Distinct variable group:   , G
Allowed substitution hints:   ()    F()

Proof of Theorem tfri2
StepHypRef Expression
1 tfri1.1 . . . . 5  F recs G
2 tfri1.2 . . . . 5  Fun 
G  G `  _V
31, 2tfri1 5889 . . . 4  F  Fn  On
4 fndm 4939 . . . 4  F  Fn  On  dom  F  On
53, 4ax-mp 7 . . 3  dom  F  On
65eleq2i 2101 . 2  dom  F  On
71tfr2a 5874 . 2  dom  F  F `  G `  F  |`
86, 7sylbir 125 1  On  F `  G `  F  |`
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wceq 1242   wcel 1390   _Vcvv 2551   Oncon0 4065   dom cdm 4287    |` cres 4289   Fun wfun 4838    Fn wfn 4839   ` cfv 4844  recscrecs 5857
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3862  ax-sep 3865  ax-pow 3917  ax-pr 3934  ax-un 4135  ax-setind 4219
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3352  df-sn 3372  df-pr 3373  df-op 3375  df-uni 3571  df-iun 3649  df-br 3755  df-opab 3809  df-mpt 3810  df-tr 3845  df-id 4020  df-iord 4068  df-on 4070  df-suc 4073  df-xp 4293  df-rel 4294  df-cnv 4295  df-co 4296  df-dm 4297  df-rn 4298  df-res 4299  df-ima 4300  df-iota 4809  df-fun 4846  df-fn 4847  df-f 4848  df-f1 4849  df-fo 4850  df-f1o 4851  df-fv 4852  df-recs 5858
This theorem is referenced by:  tfri3  5891
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