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Theorem tfri1d 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
tfri1d.1 𝐹 = recs(𝐺)
tfri1d.2 (φx(Fun 𝐺 (𝐺x) V))
Assertion
Ref Expression
tfri1d (φ𝐹 Fn On)
Distinct variable group:   x,𝐺
Allowed substitution hints:   φ(x)   𝐹(x)

Proof of Theorem tfri1d
Dummy variables f g u w y z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2022 . . . . . 6 {gz On (g Fn z u z (gu) = (𝐺‘(gu)))} = {gz On (g Fn z u z (gu) = (𝐺‘(gu)))}
21tfrlem3 5848 . . . . 5 {gz On (g Fn z u z (gu) = (𝐺‘(gu)))} = {fx On (f Fn x y x (fy) = (𝐺‘(fy)))}
3 tfri1d.2 . . . . 5 (φx(Fun 𝐺 (𝐺x) V))
42, 3tfrlemi14d 5868 . . . 4 (φ → dom recs(𝐺) = On)
5 eqid 2022 . . . . 5 {wy On (w Fn y z y (wz) = (𝐺‘(wz)))} = {wy On (w Fn y z y (wz) = (𝐺‘(wz)))}
65tfrlem7 5855 . . . 4 Fun recs(𝐺)
74, 6jctil 295 . . 3 (φ → (Fun recs(𝐺) dom recs(𝐺) = On))
8 df-fn 4832 . . 3 (recs(𝐺) Fn On ↔ (Fun recs(𝐺) dom recs(𝐺) = On))
97, 8sylibr 137 . 2 (φ → recs(𝐺) Fn On)
10 tfri1d.1 . . 3 𝐹 = recs(𝐺)
1110fneq1i 4919 . 2 (𝐹 Fn On ↔ recs(𝐺) Fn On)
129, 11sylibr 137 1 (φ𝐹 Fn On)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wal 1226   = 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:  tfri2d  5872  rdgifnon  5887  frecfnom  5901  frecsuclemdm  5904  frecsuclem3  5906
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