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Theorem tfr1 7085
Description: Principle of Transfinite Recursion, part 1 of 3. Theorem 7.41(1) of [TakeutiZaring] p. 47. We start with an arbitrary class , normally a function, and define a class of all "acceptable" functions. The final function we're interested in is the union of them. is then said to be defined by transfinite recursion. The purpose of the 3 parts of this theorem is to demonstrate properties of . In this first part we show that is a function whose domain is all ordinal numbers. (Contributed by NM, 17-Aug-1994.) (Revised by Mario Carneiro, 18-Jan-2015.)
Hypothesis
Ref Expression
tfr.1
Assertion
Ref Expression
tfr1

Proof of Theorem tfr1
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2457 . . . 4
21tfrlem7 7071 . . 3
31tfrlem14 7079 . . 3
4 df-fn 5596 . . 3
52, 3, 4mpbir2an 920 . 2
6 tfr.1 . . 3
76fneq1i 5680 . 2
85, 7mpbir 209 1
Colors of variables: wff setvar class
Syntax hints:  /\wa 369  =wceq 1395  {cab 2442  A.wral 2807  E.wrex 2808   con0 4883  domcdm 5004  |`cres 5006  Funwfun 5587  Fnwfn 5588  `cfv 5593  recscrecs 7060
This theorem is referenced by:  tfr2  7086  tfr3  7087  recsfnon  7088  rdgfnon  7103  dfac8alem  8431  dfac12lem1  8544  dfac12lem2  8545  zorn2lem1  8897  zorn2lem2  8898  zorn2lem4  8900  zorn2lem5  8901  zorn2lem6  8902  zorn2lem7  8903  ttukeylem3  8912  ttukeylem5  8914  ttukeylem6  8915  dnnumch1  30990  dnnumch3lem  30992  dnnumch3  30993  aomclem6  31005
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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