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Theorem tfr2 7086
Description: Principle of Transfinite Recursion, part 2 of 3. Theorem 7.41(2) of [TakeutiZaring] p. 47. Here we show that the function has the property that for any function whatsoever, the "next" value of is recursively applied to all "previous" values of . (Contributed by NM, 9-Apr-1995.) (Revised by Stefan O'Rear, 18-Jan-2015.)
Hypothesis
Ref Expression
tfr.1
Assertion
Ref Expression
tfr2

Proof of Theorem tfr2
StepHypRef Expression
1 tfr.1 . . . . 5
21tfr1 7085 . . . 4
3 fndm 5685 . . . 4
42, 3ax-mp 5 . . 3
54eleq2i 2535 . 2
61tfr2a 7083 . 2
75, 6sylbir 213 1
Colors of variables: wff setvar class
Syntax hints:  ->wi 4  =wceq 1395  e.wcel 1818   con0 4883  domcdm 5004  |`cres 5006  Fnwfn 5588  `cfv 5593  recscrecs 7060
This theorem is referenced by:  tfr3  7087  recsval  7089  rdgval  7105  dfac8alem  8431  dfac12lem1  8544  zorn2lem1  8897  ttukeylem3  8912
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-8 1820  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-rep 4563  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691  ax-un 6592
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3435  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-pss 3491  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-tp 4034  df-op 4036  df-uni 4250  df-iun 4332  df-br 4453  df-opab 4511  df-mpt 4512  df-tr 4546  df-eprel 4796  df-id 4800  df-po 4805  df-so 4806  df-fr 4843  df-we 4845  df-ord 4886  df-on 4887  df-suc 4889  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-fv 5601  df-recs 7061
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