Theorem tfri1 6420

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 1460	. . . 4 ⊢ ∀𝑥(Fun 𝐺 ∧ (𝐺‘𝑥) ∈ V)
4	3	a1i 9	. . 3 ⊢ (⊤ → ∀𝑥(Fun 𝐺 ∧ (𝐺‘𝑥) ∈ V))
5	1, 4	tfri1d 6390	. 2 ⊢ (⊤ → 𝐹 Fn On)
6	5	mptru 1373	1 ⊢ 𝐹 Fn On

Syntax hints:   ∧ wa 104  ∀wal 1362   = wceq 1364  ⊤wtru 1365   ∈ wcel 2164  Vcvv 2760  Oncon0 4395  Fun wfun 5249   Fn wfn 5250  ‘cfv 5255  recscrecs 6359

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 2166  ax-14 2167  ax-ext 2175  ax-coll 4145  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570

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 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-tr 4129  df-id 4325  df-iord 4398  df-on 4400  df-suc 4403  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-recs 6360

This theorem is referenced by:  tfri2  6421  tfri3  6422