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Theorem tfri1 6065
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", which is stated here as (𝐺𝑥) ∈ V. Alternately, 𝑥 ∈ On∀𝑓(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
StepHypRef Expression
1 tfri1.1 . . 3 𝐹 = recs(𝐺)
2 tfri1.2 . . . . 5 (Fun 𝐺 ∧ (𝐺𝑥) ∈ V)
32ax-gen 1381 . . . 4 𝑥(Fun 𝐺 ∧ (𝐺𝑥) ∈ V)
43a1i 9 . . 3 (⊤ → ∀𝑥(Fun 𝐺 ∧ (𝐺𝑥) ∈ V))
51, 4tfri1d 6035 . 2 (⊤ → 𝐹 Fn On)
65trud 1296 1 𝐹 Fn On
Colors of variables: wff set class
Syntax hints:  wa 102  wal 1285   = wceq 1287  wtru 1288  wcel 1436  Vcvv 2614  Oncon0 4157  Fun wfun 4966   Fn wfn 4967  cfv 4972  recscrecs 6004
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-coll 3922  ax-sep 3925  ax-pow 3977  ax-pr 4003  ax-un 4227  ax-setind 4319
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-fal 1293  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ne 2252  df-ral 2360  df-rex 2361  df-reu 2362  df-rab 2364  df-v 2616  df-sbc 2829  df-csb 2922  df-dif 2988  df-un 2990  df-in 2992  df-ss 2999  df-nul 3273  df-pw 3411  df-sn 3431  df-pr 3432  df-op 3434  df-uni 3631  df-iun 3709  df-br 3815  df-opab 3869  df-mpt 3870  df-tr 3905  df-id 4087  df-iord 4160  df-on 4162  df-suc 4165  df-xp 4410  df-rel 4411  df-cnv 4412  df-co 4413  df-dm 4414  df-rn 4415  df-res 4416  df-ima 4417  df-iota 4937  df-fun 4974  df-fn 4975  df-f 4976  df-f1 4977  df-fo 4978  df-f1o 4979  df-fv 4980  df-recs 6005
This theorem is referenced by:  tfri2  6066  tfri3  6067
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