Theorem tfrlemi14d 5970

Description: The domain of recs is all ordinals (lemma for transfinite recursion). (Contributed by Jim Kingdon, 9-Jul-2019.)

Hypotheses

Ref	Expression
tfrlemi14d.1	⊢ 𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))}
tfrlemi14d.2	⊢ (𝜑 → ∀𝑥(Fun 𝐹 ∧ (𝐹‘𝑥) ∈ V))

Assertion

Ref	Expression
tfrlemi14d	⊢ (𝜑 → dom recs(𝐹) = On)

Distinct variable groups:   𝑥,𝑓,𝑦,𝐴   𝑓,𝐹,𝑥,𝑦   𝜑,𝑓,𝑦

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem tfrlemi14d

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

Step	Hyp	Ref	Expression
1		tfrlemi14d.1	. . . 4 ⊢ 𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))}
2	1	tfrlem8 5957	. . 3 ⊢ Ord dom recs(𝐹)
3		ordsson 4236	. . 3 ⊢ (Ord dom recs(𝐹) → dom recs(𝐹) ⊆ On)
4	2, 3	mp1i 10	. 2 ⊢ (𝜑 → dom recs(𝐹) ⊆ On)
5		tfrlemi14d.2	. . . . . . . 8 ⊢ (𝜑 → ∀𝑥(Fun 𝐹 ∧ (𝐹‘𝑥) ∈ V))
6	1, 5	tfrlemi1 5969	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ On) → ∃𝑔(𝑔 Fn 𝑧 ∧ ∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐹‘(𝑔 ↾ 𝑢))))
7	5	ad2antrr 471	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ On) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐹‘(𝑔 ↾ 𝑢)))) → ∀𝑥(Fun 𝐹 ∧ (𝐹‘𝑥) ∈ V))
8		simplr 496	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ On) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐹‘(𝑔 ↾ 𝑢)))) → 𝑧 ∈ On)
9		simprl 497	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ On) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐹‘(𝑔 ↾ 𝑢)))) → 𝑔 Fn 𝑧)
10		fneq2 5008	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑧 → (𝑔 Fn 𝑤 ↔ 𝑔 Fn 𝑧))
11		raleq 2549	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑧 → (∀𝑢 ∈ 𝑤 (𝑔‘𝑢) = (𝐹‘(𝑔 ↾ 𝑢)) ↔ ∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐹‘(𝑔 ↾ 𝑢))))
12	10, 11	anbi12d 456	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑧 → ((𝑔 Fn 𝑤 ∧ ∀𝑢 ∈ 𝑤 (𝑔‘𝑢) = (𝐹‘(𝑔 ↾ 𝑢))) ↔ (𝑔 Fn 𝑧 ∧ ∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐹‘(𝑔 ↾ 𝑢)))))
13	12	rspcev 2701	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ On ∧ (𝑔 Fn 𝑧 ∧ ∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐹‘(𝑔 ↾ 𝑢)))) → ∃𝑤 ∈ On (𝑔 Fn 𝑤 ∧ ∀𝑢 ∈ 𝑤 (𝑔‘𝑢) = (𝐹‘(𝑔 ↾ 𝑢))))
14	13	adantll 459	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 ∈ On) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐹‘(𝑔 ↾ 𝑢)))) → ∃𝑤 ∈ On (𝑔 Fn 𝑤 ∧ ∀𝑢 ∈ 𝑤 (𝑔‘𝑢) = (𝐹‘(𝑔 ↾ 𝑢))))
15		vex 2604	. . . . . . . . . . 11 ⊢ 𝑔 ∈ V
16	1, 15	tfrlem3a 5948	. . . . . . . . . 10 ⊢ (𝑔 ∈ 𝐴 ↔ ∃𝑤 ∈ On (𝑔 Fn 𝑤 ∧ ∀𝑢 ∈ 𝑤 (𝑔‘𝑢) = (𝐹‘(𝑔 ↾ 𝑢))))
17	14, 16	sylibr 132	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ On) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐹‘(𝑔 ↾ 𝑢)))) → 𝑔 ∈ 𝐴)
18	1, 7, 8, 9, 17	tfrlemisucaccv 5962	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ On) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐹‘(𝑔 ↾ 𝑢)))) → (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) ∈ 𝐴)
19		vex 2604	. . . . . . . . . . . 12 ⊢ 𝑧 ∈ V
20	5	tfrlem3-2d 5951	. . . . . . . . . . . . 13 ⊢ (𝜑 → (Fun 𝐹 ∧ (𝐹‘𝑔) ∈ V))
21	20	simprd 112	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐹‘𝑔) ∈ V)
22		opexg 3983	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ V ∧ (𝐹‘𝑔) ∈ V) → ⟨𝑧, (𝐹‘𝑔)⟩ ∈ V)
23	19, 21, 22	sylancr 405	. . . . . . . . . . 11 ⊢ (𝜑 → ⟨𝑧, (𝐹‘𝑔)⟩ ∈ V)
24		snidg 3423	. . . . . . . . . . 11 ⊢ (⟨𝑧, (𝐹‘𝑔)⟩ ∈ V → ⟨𝑧, (𝐹‘𝑔)⟩ ∈ {⟨𝑧, (𝐹‘𝑔)⟩})
25		elun2 3140	. . . . . . . . . . 11 ⊢ (⟨𝑧, (𝐹‘𝑔)⟩ ∈ {⟨𝑧, (𝐹‘𝑔)⟩} → ⟨𝑧, (𝐹‘𝑔)⟩ ∈ (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}))
26	23, 24, 25	3syl 17	. . . . . . . . . 10 ⊢ (𝜑 → ⟨𝑧, (𝐹‘𝑔)⟩ ∈ (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}))
27	26	ad2antrr 471	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ On) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐹‘(𝑔 ↾ 𝑢)))) → ⟨𝑧, (𝐹‘𝑔)⟩ ∈ (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}))
28		opeldmg 4558	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ V ∧ (𝐹‘𝑔) ∈ V) → (⟨𝑧, (𝐹‘𝑔)⟩ ∈ (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) → 𝑧 ∈ dom (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩})))
29	19, 21, 28	sylancr 405	. . . . . . . . . 10 ⊢ (𝜑 → (⟨𝑧, (𝐹‘𝑔)⟩ ∈ (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) → 𝑧 ∈ dom (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩})))
30	29	ad2antrr 471	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ On) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐹‘(𝑔 ↾ 𝑢)))) → (⟨𝑧, (𝐹‘𝑔)⟩ ∈ (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) → 𝑧 ∈ dom (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩})))
31	27, 30	mpd 13	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ On) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐹‘(𝑔 ↾ 𝑢)))) → 𝑧 ∈ dom (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}))
32		dmeq 4553	. . . . . . . . . 10 ⊢ (ℎ = (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) → dom ℎ = dom (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}))
33	32	eleq2d 2148	. . . . . . . . 9 ⊢ (ℎ = (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) → (𝑧 ∈ dom ℎ ↔ 𝑧 ∈ dom (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩})))
34	33	rspcev 2701	. . . . . . . 8 ⊢ (((𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) ∈ 𝐴 ∧ 𝑧 ∈ dom (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩})) → ∃ℎ ∈ 𝐴 𝑧 ∈ dom ℎ)
35	18, 31, 34	syl2anc 403	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ On) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐹‘(𝑔 ↾ 𝑢)))) → ∃ℎ ∈ 𝐴 𝑧 ∈ dom ℎ)
36	6, 35	exlimddv 1819	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ On) → ∃ℎ ∈ 𝐴 𝑧 ∈ dom ℎ)
37		eliun 3682	. . . . . 6 ⊢ (𝑧 ∈ ∪ ℎ ∈ 𝐴 dom ℎ ↔ ∃ℎ ∈ 𝐴 𝑧 ∈ dom ℎ)
38	36, 37	sylibr 132	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ On) → 𝑧 ∈ ∪ ℎ ∈ 𝐴 dom ℎ)
39	38	ex 113	. . . 4 ⊢ (𝜑 → (𝑧 ∈ On → 𝑧 ∈ ∪ ℎ ∈ 𝐴 dom ℎ))
40	39	ssrdv 3005	. . 3 ⊢ (𝜑 → On ⊆ ∪ ℎ ∈ 𝐴 dom ℎ)
41	1	recsfval 5954	. . . . 5 ⊢ recs(𝐹) = ∪ 𝐴
42	41	dmeqi 4554	. . . 4 ⊢ dom recs(𝐹) = dom ∪ 𝐴
43		dmuni 4563	. . . 4 ⊢ dom ∪ 𝐴 = ∪ ℎ ∈ 𝐴 dom ℎ
44	42, 43	eqtri 2101	. . 3 ⊢ dom recs(𝐹) = ∪ ℎ ∈ 𝐴 dom ℎ
45	40, 44	syl6sseqr 3046	. 2 ⊢ (𝜑 → On ⊆ dom recs(𝐹))
46	4, 45	eqssd 3016	1 ⊢ (𝜑 → dom recs(𝐹) = On)

Syntax hints:   → wi 4   ∧ wa 102  ∀wal 1282   = wceq 1284   ∈ wcel 1433  {cab 2067  ∀wral 2348  ∃wrex 2349  Vcvv 2601   ∪ cun 2971   ⊆ wss 2973  {csn 3398  ⟨cop 3401  ∪ cuni 3601  ∪ ciun 3678  Ord word 4117  Oncon0 4118  dom cdm 4363   ↾ cres 4365  Fun wfun 4916   Fn wfn 4917  ‘cfv 4922  recscrecs 5942

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-coll 3893  ax-sep 3896  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-ral 2353  df-rex 2354  df-reu 2355  df-rab 2357  df-v 2603  df-sbc 2816  df-csb 2909  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-iun 3680  df-br 3786  df-opab 3840  df-mpt 3841  df-tr 3876  df-id 4048  df-iord 4121  df-on 4123  df-suc 4126  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-fv 4930  df-recs 5943

This theorem is referenced by:  tfri1d  5972