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Theorem tfr2 7067
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 7066 . . . 4  |-  F  Fn  On
3 fndm 5679 . . . 4  |-  ( F  Fn  On  ->  dom  F  =  On )
42, 3ax-mp 5 . . 3  |-  dom  F  =  On
54eleq2i 2545 . 2  |-  ( A  e.  dom  F  <->  A  e.  On )
61tfr2a 7064 . 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 1379    e. wcel 1767   Oncon0 4878   dom cdm 4999    |` cres 5001    Fn wfn 5582   ` cfv 5587  recscrecs 7041
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6575
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5550  df-fun 5589  df-fn 5590  df-f 5591  df-f1 5592  df-fo 5593  df-f1o 5594  df-fv 5595  df-recs 7042
This theorem is referenced by:  tfr3  7068  recsval  7070  rdgval  7086  dfac8alem  8409  dfac12lem1  8522  zorn2lem1  8875  ttukeylem3  8890
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