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Theorem tfri1 5892
 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
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
StepHypRef Expression
1 tfri1.1 . . 3 𝐹 = recs(𝐺)
2 tfri1.2 . . . . 5 (Fun 𝐺 (𝐺x) V)
32ax-gen 1335 . . . 4 x(Fun 𝐺 (𝐺x) V)
43a1i 9 . . 3 ( ⊤ → x(Fun 𝐺 (𝐺x) V))
51, 4tfri1d 5890 . 2 ( ⊤ → 𝐹 Fn On)
65trud 1251 1 𝐹 Fn On
 Colors of variables: wff set class Syntax hints:   ∧ wa 97  ∀wal 1240   = wceq 1242   ⊤ wtru 1243   ∈ wcel 1390  Vcvv 2551  Oncon0 4066  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:  tfri2  5893  tfri3  5894
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