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Theorem tfri1 5873
 Description: Principle of Transfinite Recursion, part 1 of 3. Theorem 7.41(1) of [TakeutiZaring] p. 47, with an additional condition. The condition is that is defined "everywhere" and here is stated as . Alternatively would suffice. Given a function satisfying that condition, we define a class of all "acceptable" functions. The final function we're interested in is the union recs 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 Jim Kingdon, 4-May-2019.) (Revised by Mario Carneiro, 24-May-2019.)
Hypotheses
Ref Expression
tfri1.1 recs
tfri1.2
Assertion
Ref Expression
tfri1
Distinct variable group:   ,
Allowed substitution hint:   ()

Proof of Theorem tfri1
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2022 . . . 4
21tfrlem7 5855 . . 3 recs
3 eqid 2022 . . . . 5
43tfrlem3 5848 . . . 4
5 tfri1.2 . . . 4
64, 5tfrlemi14 5869 . . 3 recs
7 df-fn 4832 . . 3 recs recs recs
82, 6, 7mpbir2an 837 . 2 recs
9 tfri1.1 . . 3 recs
109fneq1i 4919 . 2 recs
118, 10mpbir 134 1
 Colors of variables: wff set class Syntax hints:   wa 97   wceq 1228   wcel 1374  cab 2008  wral 2284  wrex 2285  cvv 2535  con0 4049   cdm 4272   cres 4274   wfun 4823   wfn 4824  cfv 4829  recscrecs 5841 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 532  ax-in2 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-13 1385  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-coll 3846  ax-sep 3849  ax-pow 3901  ax-pr 3918  ax-un 4120  ax-setind 4204 This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-fal 1234  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ne 2188  df-ral 2289  df-rex 2290  df-reu 2291  df-rab 2293  df-v 2537  df-sbc 2742  df-csb 2830  df-dif 2897  df-un 2899  df-in 2901  df-ss 2908  df-nul 3202  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-uni 3555  df-iun 3633  df-br 3739  df-opab 3793  df-mpt 3794  df-tr 3829  df-id 4004  df-iord 4052  df-on 4054  df-suc 4057  df-xp 4278  df-rel 4279  df-cnv 4280  df-co 4281  df-dm 4282  df-rn 4283  df-res 4284  df-ima 4285  df-iota 4794  df-fun 4831  df-fn 4832  df-f 4833  df-f1 4834  df-fo 4835  df-f1o 4836  df-fv 4837  df-recs 5842 This theorem is referenced by:  tfri2  5874  tfri3  5875
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