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Theorem tfri1d 5890
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 2037 . . . . . 6 {gz On (g Fn z u z (gu) = (𝐺‘(gu)))} = {gz On (g Fn z u z (gu) = (𝐺‘(gu)))}
21tfrlem3 5867 . . . . 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 5888 . . . 4 (φ → dom recs(𝐺) = On)
5 eqid 2037 . . . . 5 {wy On (w Fn y z y (wz) = (𝐺‘(wz)))} = {wy On (w Fn y z y (wz) = (𝐺‘(wz)))}
65tfrlem7 5874 . . . 4 Fun recs(𝐺)
74, 6jctil 295 . . 3 (φ → (Fun recs(𝐺) dom recs(𝐺) = On))
8 df-fn 4848 . . 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 4936 . 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 1240   = wceq 1242   wcel 1390  {cab 2023  wral 2300  wrex 2301  Vcvv 2551  Oncon0 4066  dom cdm 4288  cres 4290  Fun wfun 4839   Fn wfn 4840  cfv 4845  recscrecs 5860
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 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-tr 3846  df-id 4021  df-iord 4069  df-on 4071  df-suc 4074  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-recs 5861
This theorem is referenced by:  tfri2d  5891  tfri1  5892  rdgifnon  5906  rdgifnon2  5907  frecfnom  5925  frecsuclemdm  5927  frecsuclem3  5929
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