Theorem rdgon 6439

Description: Evaluating the recursive definition generator produces an ordinal. There is a hypothesis that the characteristic function produces ordinals on ordinal arguments. (Contributed by Jim Kingdon, 26-Jul-2019.) (Revised by Jim Kingdon, 13-Apr-2022.)

Hypotheses

Ref	Expression
rdgon.2	⊢ (𝜑 → 𝐴 ∈ On)
rdgon.3	⊢ (𝜑 → ∀𝑥 ∈ On (𝐹‘𝑥) ∈ On)

Assertion

Ref	Expression
rdgon	⊢ ((𝜑 ∧ 𝐵 ∈ On) → (rec(𝐹, 𝐴)‘𝐵) ∈ On)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐹   𝜑,𝑥

Proof of Theorem rdgon

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

Step	Hyp	Ref	Expression
1		df-irdg 6423	. 2 ⊢ rec(𝐹, 𝐴) = recs((𝑔 ∈ V ↦ (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥)))))
2		funmpt 5292	. . 3 ⊢ Fun (𝑔 ∈ V ↦ (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥))))
3	2	a1i 9	. 2 ⊢ ((𝜑 ∧ 𝐵 ∈ On) → Fun (𝑔 ∈ V ↦ (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥)))))
4		ordon 4518	. . 3 ⊢ Ord On
5	4	a1i 9	. 2 ⊢ ((𝜑 ∧ 𝐵 ∈ On) → Ord On)
6		vex 2763	. . . 4 ⊢ 𝑓 ∈ V
7		rdgon.2	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ On)
8	7	adantr 276	. . . . . 6 ⊢ ((𝜑 ∧ 𝐵 ∈ On) → 𝐴 ∈ On)
9	8	3ad2ant1 1020	. . . . 5 ⊢ (((𝜑 ∧ 𝐵 ∈ On) ∧ 𝑦 ∈ On ∧ 𝑓:𝑦⟶On) → 𝐴 ∈ On)
10	6	dmex 4928	. . . . . 6 ⊢ dom 𝑓 ∈ V
11		fveq2 5554	. . . . . . . . . 10 ⊢ (𝑥 = (𝑓‘𝑧) → (𝐹‘𝑥) = (𝐹‘(𝑓‘𝑧)))
12	11	eleq1d 2262	. . . . . . . . 9 ⊢ (𝑥 = (𝑓‘𝑧) → ((𝐹‘𝑥) ∈ On ↔ (𝐹‘(𝑓‘𝑧)) ∈ On))
13		rdgon.3	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑥 ∈ On (𝐹‘𝑥) ∈ On)
14	13	adantr 276	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝐵 ∈ On) → ∀𝑥 ∈ On (𝐹‘𝑥) ∈ On)
15	14	3ad2ant1 1020	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐵 ∈ On) ∧ 𝑦 ∈ On ∧ 𝑓:𝑦⟶On) → ∀𝑥 ∈ On (𝐹‘𝑥) ∈ On)
16	15	adantr 276	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝐵 ∈ On) ∧ 𝑦 ∈ On ∧ 𝑓:𝑦⟶On) ∧ 𝑧 ∈ dom 𝑓) → ∀𝑥 ∈ On (𝐹‘𝑥) ∈ On)
17		simpl3 1004	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝐵 ∈ On) ∧ 𝑦 ∈ On ∧ 𝑓:𝑦⟶On) ∧ 𝑧 ∈ dom 𝑓) → 𝑓:𝑦⟶On)
18		simpr 110	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝐵 ∈ On) ∧ 𝑦 ∈ On ∧ 𝑓:𝑦⟶On) ∧ 𝑧 ∈ dom 𝑓) → 𝑧 ∈ dom 𝑓)
19		fdm 5409	. . . . . . . . . . . . 13 ⊢ (𝑓:𝑦⟶On → dom 𝑓 = 𝑦)
20	19	eleq2d 2263	. . . . . . . . . . . 12 ⊢ (𝑓:𝑦⟶On → (𝑧 ∈ dom 𝑓 ↔ 𝑧 ∈ 𝑦))
21	17, 20	syl 14	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝐵 ∈ On) ∧ 𝑦 ∈ On ∧ 𝑓:𝑦⟶On) ∧ 𝑧 ∈ dom 𝑓) → (𝑧 ∈ dom 𝑓 ↔ 𝑧 ∈ 𝑦))
22	18, 21	mpbid 147	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝐵 ∈ On) ∧ 𝑦 ∈ On ∧ 𝑓:𝑦⟶On) ∧ 𝑧 ∈ dom 𝑓) → 𝑧 ∈ 𝑦)
23	17, 22	ffvelcdmd 5694	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝐵 ∈ On) ∧ 𝑦 ∈ On ∧ 𝑓:𝑦⟶On) ∧ 𝑧 ∈ dom 𝑓) → (𝑓‘𝑧) ∈ On)
24	12, 16, 23	rspcdva 2869	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐵 ∈ On) ∧ 𝑦 ∈ On ∧ 𝑓:𝑦⟶On) ∧ 𝑧 ∈ dom 𝑓) → (𝐹‘(𝑓‘𝑧)) ∈ On)
25	24	ralrimiva 2567	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐵 ∈ On) ∧ 𝑦 ∈ On ∧ 𝑓:𝑦⟶On) → ∀𝑧 ∈ dom 𝑓(𝐹‘(𝑓‘𝑧)) ∈ On)
26		fveq2 5554	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → (𝑓‘𝑥) = (𝑓‘𝑧))
27	26	fveq2d 5558	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → (𝐹‘(𝑓‘𝑥)) = (𝐹‘(𝑓‘𝑧)))
28	27	eleq1d 2262	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → ((𝐹‘(𝑓‘𝑥)) ∈ On ↔ (𝐹‘(𝑓‘𝑧)) ∈ On))
29	28	cbvralv 2726	. . . . . . 7 ⊢ (∀𝑥 ∈ dom 𝑓(𝐹‘(𝑓‘𝑥)) ∈ On ↔ ∀𝑧 ∈ dom 𝑓(𝐹‘(𝑓‘𝑧)) ∈ On)
30	25, 29	sylibr 134	. . . . . 6 ⊢ (((𝜑 ∧ 𝐵 ∈ On) ∧ 𝑦 ∈ On ∧ 𝑓:𝑦⟶On) → ∀𝑥 ∈ dom 𝑓(𝐹‘(𝑓‘𝑥)) ∈ On)
31		iunon 6337	. . . . . 6 ⊢ ((dom 𝑓 ∈ V ∧ ∀𝑥 ∈ dom 𝑓(𝐹‘(𝑓‘𝑥)) ∈ On) → ∪ 𝑥 ∈ dom 𝑓(𝐹‘(𝑓‘𝑥)) ∈ On)
32	10, 30, 31	sylancr 414	. . . . 5 ⊢ (((𝜑 ∧ 𝐵 ∈ On) ∧ 𝑦 ∈ On ∧ 𝑓:𝑦⟶On) → ∪ 𝑥 ∈ dom 𝑓(𝐹‘(𝑓‘𝑥)) ∈ On)
33		onun2 4522	. . . . 5 ⊢ ((𝐴 ∈ On ∧ ∪ 𝑥 ∈ dom 𝑓(𝐹‘(𝑓‘𝑥)) ∈ On) → (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑓(𝐹‘(𝑓‘𝑥))) ∈ On)
34	9, 32, 33	syl2anc 411	. . . 4 ⊢ (((𝜑 ∧ 𝐵 ∈ On) ∧ 𝑦 ∈ On ∧ 𝑓:𝑦⟶On) → (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑓(𝐹‘(𝑓‘𝑥))) ∈ On)
35		dmeq 4862	. . . . . . 7 ⊢ (𝑔 = 𝑓 → dom 𝑔 = dom 𝑓)
36		fveq1 5553	. . . . . . . 8 ⊢ (𝑔 = 𝑓 → (𝑔‘𝑥) = (𝑓‘𝑥))
37	36	fveq2d 5558	. . . . . . 7 ⊢ (𝑔 = 𝑓 → (𝐹‘(𝑔‘𝑥)) = (𝐹‘(𝑓‘𝑥)))
38	35, 37	iuneq12d 3936	. . . . . 6 ⊢ (𝑔 = 𝑓 → ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥)) = ∪ 𝑥 ∈ dom 𝑓(𝐹‘(𝑓‘𝑥)))
39	38	uneq2d 3313	. . . . 5 ⊢ (𝑔 = 𝑓 → (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥))) = (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑓(𝐹‘(𝑓‘𝑥))))
40		eqid 2193	. . . . 5 ⊢ (𝑔 ∈ V ↦ (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥)))) = (𝑔 ∈ V ↦ (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥))))
41	39, 40	fvmptg 5633	. . . 4 ⊢ ((𝑓 ∈ V ∧ (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑓(𝐹‘(𝑓‘𝑥))) ∈ On) → ((𝑔 ∈ V ↦ (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥))))‘𝑓) = (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑓(𝐹‘(𝑓‘𝑥))))
42	6, 34, 41	sylancr 414	. . 3 ⊢ (((𝜑 ∧ 𝐵 ∈ On) ∧ 𝑦 ∈ On ∧ 𝑓:𝑦⟶On) → ((𝑔 ∈ V ↦ (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥))))‘𝑓) = (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑓(𝐹‘(𝑓‘𝑥))))
43	42, 34	eqeltrd 2270	. 2 ⊢ (((𝜑 ∧ 𝐵 ∈ On) ∧ 𝑦 ∈ On ∧ 𝑓:𝑦⟶On) → ((𝑔 ∈ V ↦ (𝐴 ∪ ∪ 𝑥 ∈ dom 𝑔(𝐹‘(𝑔‘𝑥))))‘𝑓) ∈ On)
44		unon 4543	. . . . . 6 ⊢ ∪ On = On
45	44	eleq2i 2260	. . . . 5 ⊢ (𝑦 ∈ ∪ On ↔ 𝑦 ∈ On)
46	45	biimpi 120	. . . 4 ⊢ (𝑦 ∈ ∪ On → 𝑦 ∈ On)
47	46	adantl 277	. . 3 ⊢ (((𝜑 ∧ 𝐵 ∈ On) ∧ 𝑦 ∈ ∪ On) → 𝑦 ∈ On)
48		onsuc 4533	. . 3 ⊢ (𝑦 ∈ On → suc 𝑦 ∈ On)
49	47, 48	syl 14	. 2 ⊢ (((𝜑 ∧ 𝐵 ∈ On) ∧ 𝑦 ∈ ∪ On) → suc 𝑦 ∈ On)
50	44	eleq2i 2260	. . . 4 ⊢ (𝐵 ∈ ∪ On ↔ 𝐵 ∈ On)
51	50	biimpri 133	. . 3 ⊢ (𝐵 ∈ On → 𝐵 ∈ ∪ On)
52	51	adantl 277	. 2 ⊢ ((𝜑 ∧ 𝐵 ∈ On) → 𝐵 ∈ ∪ On)
53	1, 3, 5, 43, 49, 52	tfrcl 6417	1 ⊢ ((𝜑 ∧ 𝐵 ∈ On) → (rec(𝐹, 𝐴)‘𝐵) ∈ On)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 980   = wceq 1364   ∈ wcel 2164  ∀wral 2472  Vcvv 2760   ∪ cun 3151  ∪ cuni 3835  ∪ ciun 3912   ↦ cmpt 4090  Ord word 4393  Oncon0 4394  suc csuc 4396  dom cdm 4659  Fun wfun 5248  ⟶wf 5250  ‘cfv 5254  reccrdg 6422

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 4144  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569

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 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-id 4324  df-iord 4397  df-on 4399  df-suc 4402  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-recs 6358  df-irdg 6423

This theorem is referenced by:  oacl  6513  omcl  6514  oeicl  6515