Theorem tfri1d 6336

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
tfri1d.1	⊢ 𝐹 = recs(𝐺)
tfri1d.2	⊢ (𝜑 → ∀𝑥(Fun 𝐺 ∧ (𝐺‘𝑥) ∈ V))

Assertion

Ref	Expression
tfri1d	⊢ (𝜑 → 𝐹 Fn On)

Distinct variable group:   𝑥,𝐺

Allowed substitution hints:   𝜑(𝑥)   𝐹(𝑥)

Proof of Theorem tfri1d

Dummy variables 𝑓 𝑔 𝑢 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2177	. . . . . 6 ⊢ {𝑔 ∣ ∃𝑧 ∈ On (𝑔 Fn 𝑧 ∧ ∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢)))} = {𝑔 ∣ ∃𝑧 ∈ On (𝑔 Fn 𝑧 ∧ ∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢)))}
2	1	tfrlem3 6312	. . . . 5 ⊢ {𝑔 ∣ ∃𝑧 ∈ On (𝑔 Fn 𝑧 ∧ ∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐺‘(𝑔 ↾ 𝑢)))} = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)))}
3		tfri1d.2	. . . . 5 ⊢ (𝜑 → ∀𝑥(Fun 𝐺 ∧ (𝐺‘𝑥) ∈ V))
4	2, 3	tfrlemi14d 6334	. . . 4 ⊢ (𝜑 → dom recs(𝐺) = On)
5		eqid 2177	. . . . 5 ⊢ {𝑤 ∣ ∃𝑦 ∈ On (𝑤 Fn 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝑤‘𝑧) = (𝐺‘(𝑤 ↾ 𝑧)))} = {𝑤 ∣ ∃𝑦 ∈ On (𝑤 Fn 𝑦 ∧ ∀𝑧 ∈ 𝑦 (𝑤‘𝑧) = (𝐺‘(𝑤 ↾ 𝑧)))}
6	5	tfrlem7 6318	. . . 4 ⊢ Fun recs(𝐺)
7	4, 6	jctil 312	. . 3 ⊢ (𝜑 → (Fun recs(𝐺) ∧ dom recs(𝐺) = On))
8		df-fn 5220	. . 3 ⊢ (recs(𝐺) Fn On ↔ (Fun recs(𝐺) ∧ dom recs(𝐺) = On))
9	7, 8	sylibr 134	. 2 ⊢ (𝜑 → recs(𝐺) Fn On)
10		tfri1d.1	. . 3 ⊢ 𝐹 = recs(𝐺)
11	10	fneq1i 5311	. 2 ⊢ (𝐹 Fn On ↔ recs(𝐺) Fn On)
12	9, 11	sylibr 134	1 ⊢ (𝜑 → 𝐹 Fn On)

Syntax hints:   → wi 4   ∧ wa 104  ∀wal 1351   = wceq 1353   ∈ wcel 2148  {cab 2163  ∀wral 2455  ∃wrex 2456  Vcvv 2738  Oncon0 4364  dom cdm 4627   ↾ cres 4629  Fun wfun 5211   Fn wfn 5212  ‘cfv 5217  recscrecs 6305

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4119  ax-sep 4122  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2740  df-sbc 2964  df-csb 3059  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  df-tr 4103  df-id 4294  df-iord 4367  df-on 4369  df-suc 4372  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-recs 6306

This theorem is referenced by:  tfri2d  6337  tfri1  6366  rdgifnon  6380  rdgifnon2  6381  frecfnom  6402