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Theorem tfri1 6391
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 1460 . . . 4 𝑥(Fun 𝐺 ∧ (𝐺𝑥) ∈ V)
43a1i 9 . . 3 (⊤ → ∀𝑥(Fun 𝐺 ∧ (𝐺𝑥) ∈ V))
51, 4tfri1d 6361 . 2 (⊤ → 𝐹 Fn On)
65mptru 1373 1 𝐹 Fn On
Colors of variables: wff set class
Syntax hints:  wa 104  wal 1362   = wceq 1364  wtru 1365  wcel 2160  Vcvv 2752  Oncon0 4381  Fun wfun 5229   Fn wfn 5230  cfv 5235  recscrecs 6330
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4311  df-iord 4384  df-on 4386  df-suc 4389  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-recs 6331
This theorem is referenced by:  tfri2  6392  tfri3  6393
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