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Theorem tfri1 6036
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 1379 . . . 4 𝑥(Fun 𝐺 ∧ (𝐺𝑥) ∈ V)
43a1i 9 . . 3 (⊤ → ∀𝑥(Fun 𝐺 ∧ (𝐺𝑥) ∈ V))
51, 4tfri1d 6006 . 2 (⊤ → 𝐹 Fn On)
65trud 1294 1 𝐹 Fn On
Colors of variables: wff set class
Syntax hints:  wa 102  wal 1283   = wceq 1285  wtru 1286  wcel 1434  Vcvv 2611  Oncon0 4147  Fun wfun 4947   Fn wfn 4948  cfv 4953  recscrecs 5975
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 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3914  ax-sep 3917  ax-pow 3969  ax-pr 3993  ax-un 4217  ax-setind 4309
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2613  df-sbc 2826  df-csb 2919  df-dif 2985  df-un 2987  df-in 2989  df-ss 2996  df-nul 3269  df-pw 3403  df-sn 3423  df-pr 3424  df-op 3426  df-uni 3623  df-iun 3701  df-br 3807  df-opab 3861  df-mpt 3862  df-tr 3897  df-id 4077  df-iord 4150  df-on 4152  df-suc 4155  df-xp 4398  df-rel 4399  df-cnv 4400  df-co 4401  df-dm 4402  df-rn 4403  df-res 4404  df-ima 4405  df-iota 4918  df-fun 4955  df-fn 4956  df-f 4957  df-f1 4958  df-fo 4959  df-f1o 4960  df-fv 4961  df-recs 5976
This theorem is referenced by:  tfri2  6037  tfri3  6038
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