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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  G is defined "everywhere" and here is stated as  G `  _V. Alternatively  On  Fn  dom  G would suffice.

Given a function  G satisfying that condition, we define a class of all "acceptable" functions. The final function we're interested in is the union 
F recs G of them.  F is then said to be defined by transfinite recursion. The purpose of the 3 parts of this theorem is to demonstrate properties of  F. In this first part we show that  F 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  F recs G
tfri1.2  Fun 
G  G `  _V
Assertion
Ref Expression
tfri1  F  Fn  On
Distinct variable group:   , G
Allowed substitution hint:    F()

Proof of Theorem tfri1
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2022 . . . 4  {  |  On  Fn  `  G `  |`  }  {  |  On  Fn  `  G `  |`  }
21tfrlem7 5855 . . 3  Fun recs G
3 eqid 2022 . . . . 5  {  |  On  Fn  `  G `  |`  }  {  |  On  Fn  `  G `  |`  }
43tfrlem3 5848 . . . 4  {  |  On  Fn  `  G `  |`  }  {  |  On  Fn  `  G `  |`  }
5 tfri1.2 . . . 4  Fun 
G  G `  _V
64, 5tfrlemi14 5869 . . 3  dom recs G  On
7 df-fn 4832 . . 3 recs G  Fn  On  Fun recs G  dom recs G  On
82, 6, 7mpbir2an 837 . 2 recs G  Fn  On
9 tfri1.1 . . 3  F recs G
109fneq1i 4919 . 2  F  Fn  On recs G  Fn  On
118, 10mpbir 134 1  F  Fn  On
Colors of variables: wff set class
Syntax hints:   wa 97   wceq 1228   wcel 1374   {cab 2008  wral 2284  wrex 2285   _Vcvv 2535   Oncon0 4049   dom cdm 4272    |` cres 4274   Fun wfun 4823    Fn 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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