Theorem tfri1 6270

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 (𝐺‘𝑥) ∈ V. Alternately, ∀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 ∧ (𝐺‘𝑥) ∈ V)

Assertion

Ref	Expression
tfri1	⊢ 𝐹 Fn On

Distinct variable group:   𝑥,𝐺

Allowed substitution hint:   𝐹(𝑥)

Proof of Theorem tfri1

Step	Hyp	Ref	Expression
1		tfri1.1	. . 3 ⊢ 𝐹 = recs(𝐺)
2		tfri1.2	. . . . 5 ⊢ (Fun 𝐺 ∧ (𝐺‘𝑥) ∈ V)
3	2	ax-gen 1426	. . . 4 ⊢ ∀𝑥(Fun 𝐺 ∧ (𝐺‘𝑥) ∈ V)
4	3	a1i 9	. . 3 ⊢ (⊤ → ∀𝑥(Fun 𝐺 ∧ (𝐺‘𝑥) ∈ V))
5	1, 4	tfri1d 6240	. 2 ⊢ (⊤ → 𝐹 Fn On)
6	5	mptru 1341	1 ⊢ 𝐹 Fn On

Syntax hints:   ∧ wa 103  ∀wal 1330   = wceq 1332  ⊤wtru 1333   ∈ wcel 1481  Vcvv 2689  Oncon0 4293  Fun wfun 5125   Fn wfn 5126  ‘cfv 5131  recscrecs 6209

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4051  ax-sep 4054  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2914  df-csb 3008  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-iun 3823  df-br 3938  df-opab 3998  df-mpt 3999  df-tr 4035  df-id 4223  df-iord 4296  df-on 4298  df-suc 4301  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-f1 5136  df-fo 5137  df-f1o 5138  df-fv 5139  df-recs 6210

This theorem is referenced by:  tfri2  6271  tfri3  6272