Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  tfri1d Structured version   GIF version

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 (𝐺‘x) ∈ V. Alternatively ∀x ∈ On∀f(f Fn x → f ∈ 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
 Copyright terms: Public domain W3C validator