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Theorem tfr2 6410
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 6409 . . . 4  |-  F  Fn  On
3 fndm 5309 . . . 4  |-  ( F  Fn  On  ->  dom  F  =  On )
42, 3ax-mp 10 . . 3  |-  dom  F  =  On
54eleq2i 2349 . 2  |-  ( A  e.  dom  F  <->  A  e.  On )
61tfr2a 6407 . 2  |-  ( A  e.  dom  F  -> 
( F `  A
)  =  ( G `
 ( F  |`  A ) ) )
75, 6sylbir 206 1  |-  ( A  e.  On  ->  ( F `  A )  =  ( G `  ( F  |`  A ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    = wceq 1624    e. wcel 1685   Oncon0 4392   dom cdm 4689    |` cres 4691    Fn wfn 5217   ` cfv 5222  recscrecs 6383
This theorem is referenced by:  tfr3  6411  recsval  6413  rdgval  6429  dfac8alem  7652  dfac12lem1  7765  zorn2lem1  8119  ttukeylem3  8134
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-reu 2552  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-suc 4398  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-fun 5224  df-fn 5225  df-f 5226  df-f1 5227  df-fo 5228  df-f1o 5229  df-fv 5230  df-recs 6384
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