Theorem tfri1 6530

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 1497	. . . 4 ⊢ ∀𝑥(Fun 𝐺 ∧ (𝐺‘𝑥) ∈ V)
4	3	a1i 9	. . 3 ⊢ (⊤ → ∀𝑥(Fun 𝐺 ∧ (𝐺‘𝑥) ∈ V))
5	1, 4	tfri1d 6500	. 2 ⊢ (⊤ → 𝐹 Fn On)
6	5	mptru 1406	1 ⊢ 𝐹 Fn On

Syntax hints:   ∧ wa 104  ∀wal 1395   = wceq 1397  ⊤wtru 1398   ∈ wcel 2202  Vcvv 2802  Oncon0 4460  Fun wfun 5320   Fn wfn 5321  ‘cfv 5326  recscrecs 6469

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-iord 4463  df-on 4465  df-suc 4468  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-recs 6470

This theorem is referenced by:  tfri2  6531  tfri3  6532