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Theorem tfri1 5871
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 (𝐺x) V. Alternatively x Onf(f Fn xf dom 𝐺) would suffice.

Given a function 𝐺 satisfying that condition, we define a class A 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 (𝐺x) V)
Assertion
Ref Expression
tfri1 𝐹 Fn On
Distinct variable group:   x,𝐺
Allowed substitution hint:   𝐹(x)

Proof of Theorem tfri1
Dummy variables f g u w y z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2022 . . . 4 {wy On (w Fn y z y (wz) = (𝐺‘(wz)))} = {wy On (w Fn y z y (wz) = (𝐺‘(wz)))}
21tfrlem7 5853 . . 3 Fun recs(𝐺)
3 eqid 2022 . . . . 5 {gz On (g Fn z u z (gu) = (𝐺‘(gu)))} = {gz On (g Fn z u z (gu) = (𝐺‘(gu)))}
43tfrlem3 5846 . . . 4 {gz On (g Fn z u z (gu) = (𝐺‘(gu)))} = {fx On (f Fn x y x (fy) = (𝐺‘(fy)))}
5 tfri1.2 . . . 4 (Fun 𝐺 (𝐺x) V)
64, 5tfrlemi14 5867 . . 3 dom recs(𝐺) = On
7 df-fn 4830 . . 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 4917 . 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 2282  wrex 2283  Vcvv 2533  Oncon0 4047  dom cdm 4270  cres 4272  Fun wfun 4821   Fn wfn 4822  cfv 4827  recscrecs 5839
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 3844  ax-sep 3847  ax-pow 3899  ax-pr 3916  ax-un 4118  ax-setind 4202
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 2287  df-rex 2288  df-reu 2289  df-rab 2291  df-v 2535  df-sbc 2740  df-csb 2828  df-dif 2895  df-un 2897  df-in 2899  df-ss 2906  df-nul 3200  df-pw 3334  df-sn 3354  df-pr 3355  df-op 3357  df-uni 3553  df-iun 3631  df-br 3737  df-opab 3791  df-mpt 3792  df-tr 3827  df-id 4002  df-iord 4050  df-on 4052  df-suc 4055  df-xp 4276  df-rel 4277  df-cnv 4278  df-co 4279  df-dm 4280  df-rn 4281  df-res 4282  df-ima 4283  df-iota 4792  df-fun 4829  df-fn 4830  df-f 4831  df-f1 4832  df-fo 4833  df-f1o 4834  df-fv 4835  df-recs 5840
This theorem is referenced by:  tfri2  5872  tfri3  5873
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