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Theorem tfr2 6853
Description: Principle of Transfinite Recursion, part 2 of 3. Theorem 7.41(2) of [TakeutiZaring] p. 47. Here we show that the function  F has the property that for any function  G whatsoever, the "next" value of  F is  G recursively applied to all "previous" values of  F. (Contributed by NM, 9-Apr-1995.) (Revised by Stefan O'Rear, 18-Jan-2015.)
Hypothesis
Ref Expression
tfr.1  |-  F  = recs ( G )
Assertion
Ref Expression
tfr2  |-  ( A  e.  On  ->  ( F `  A )  =  ( G `  ( F  |`  A ) ) )

Proof of Theorem tfr2
StepHypRef Expression
1 tfr.1 . . . . 5  |-  F  = recs ( G )
21tfr1 6852 . . . 4  |-  F  Fn  On
3 fndm 5507 . . . 4  |-  ( F  Fn  On  ->  dom  F  =  On )
42, 3ax-mp 5 . . 3  |-  dom  F  =  On
54eleq2i 2505 . 2  |-  ( A  e.  dom  F  <->  A  e.  On )
61tfr2a 6850 . 2  |-  ( A  e.  dom  F  -> 
( F `  A
)  =  ( G `
 ( F  |`  A ) ) )
75, 6sylbir 213 1  |-  ( A  e.  On  ->  ( F `  A )  =  ( G `  ( F  |`  A ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1364    e. wcel 1761   Oncon0 4715   dom cdm 4836    |` cres 4838    Fn wfn 5410   ` cfv 5415  recscrecs 6827
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-recs 6828
This theorem is referenced by:  tfr3  6854  recsval  6856  rdgval  6872  dfac8alem  8195  dfac12lem1  8308  zorn2lem1  8661  ttukeylem3  8676
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