Theorem tfrlemi14d 6230

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 6215	. . 3 ⊢ Ord dom recs(𝐹)
3		ordsson 4408	. . 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 6229	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ On) → ∃𝑔(𝑔 Fn 𝑧 ∧ ∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐹‘(𝑔 ↾ 𝑢))))
7	5	ad2antrr 479	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ On) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐹‘(𝑔 ↾ 𝑢)))) → ∀𝑥(Fun 𝐹 ∧ (𝐹‘𝑥) ∈ V))
8		simplr 519	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ On) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐹‘(𝑔 ↾ 𝑢)))) → 𝑧 ∈ On)
9		simprl 520	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ On) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐹‘(𝑔 ↾ 𝑢)))) → 𝑔 Fn 𝑧)
10		fneq2 5212	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑧 → (𝑔 Fn 𝑤 ↔ 𝑔 Fn 𝑧))
11		raleq 2626	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑧 → (∀𝑢 ∈ 𝑤 (𝑔‘𝑢) = (𝐹‘(𝑔 ↾ 𝑢)) ↔ ∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐹‘(𝑔 ↾ 𝑢))))
12	10, 11	anbi12d 464	. . . . . . . . . . . 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 2789	. . . . . . . . . . 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 467	. . . . . . . . . 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 2689	. . . . . . . . . . 11 ⊢ 𝑔 ∈ V
16	1, 15	tfrlem3a 6207	. . . . . . . . . 10 ⊢ (𝑔 ∈ 𝐴 ↔ ∃𝑤 ∈ On (𝑔 Fn 𝑤 ∧ ∀𝑢 ∈ 𝑤 (𝑔‘𝑢) = (𝐹‘(𝑔 ↾ 𝑢))))
17	14, 16	sylibr 133	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ On) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐹‘(𝑔 ↾ 𝑢)))) → 𝑔 ∈ 𝐴)
18	1, 7, 8, 9, 17	tfrlemisucaccv 6222	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ On) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐹‘(𝑔 ↾ 𝑢)))) → (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) ∈ 𝐴)
19		vex 2689	. . . . . . . . . . . 12 ⊢ 𝑧 ∈ V
20	5	tfrlem3-2d 6209	. . . . . . . . . . . . 13 ⊢ (𝜑 → (Fun 𝐹 ∧ (𝐹‘𝑔) ∈ V))
21	20	simprd 113	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐹‘𝑔) ∈ V)
22		opexg 4150	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ V ∧ (𝐹‘𝑔) ∈ V) → ⟨𝑧, (𝐹‘𝑔)⟩ ∈ V)
23	19, 21, 22	sylancr 410	. . . . . . . . . . 11 ⊢ (𝜑 → ⟨𝑧, (𝐹‘𝑔)⟩ ∈ V)
24		snidg 3554	. . . . . . . . . . 11 ⊢ (⟨𝑧, (𝐹‘𝑔)⟩ ∈ V → ⟨𝑧, (𝐹‘𝑔)⟩ ∈ {⟨𝑧, (𝐹‘𝑔)⟩})
25		elun2 3244	. . . . . . . . . . 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 479	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ On) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐹‘(𝑔 ↾ 𝑢)))) → ⟨𝑧, (𝐹‘𝑔)⟩ ∈ (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}))
28		opeldmg 4744	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ V ∧ (𝐹‘𝑔) ∈ V) → (⟨𝑧, (𝐹‘𝑔)⟩ ∈ (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) → 𝑧 ∈ dom (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩})))
29	19, 21, 28	sylancr 410	. . . . . . . . . 10 ⊢ (𝜑 → (⟨𝑧, (𝐹‘𝑔)⟩ ∈ (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) → 𝑧 ∈ dom (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩})))
30	29	ad2antrr 479	. . . . . . . . 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 4739	. . . . . . . . . 10 ⊢ (ℎ = (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) → dom ℎ = dom (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}))
33	32	eleq2d 2209	. . . . . . . . 9 ⊢ (ℎ = (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) → (𝑧 ∈ dom ℎ ↔ 𝑧 ∈ dom (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩})))
34	33	rspcev 2789	. . . . . . . 8 ⊢ (((𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) ∈ 𝐴 ∧ 𝑧 ∈ dom (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩})) → ∃ℎ ∈ 𝐴 𝑧 ∈ dom ℎ)
35	18, 31, 34	syl2anc 408	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ On) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐹‘(𝑔 ↾ 𝑢)))) → ∃ℎ ∈ 𝐴 𝑧 ∈ dom ℎ)
36	6, 35	exlimddv 1870	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ On) → ∃ℎ ∈ 𝐴 𝑧 ∈ dom ℎ)
37		eliun 3817	. . . . . 6 ⊢ (𝑧 ∈ ∪ ℎ ∈ 𝐴 dom ℎ ↔ ∃ℎ ∈ 𝐴 𝑧 ∈ dom ℎ)
38	36, 37	sylibr 133	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ On) → 𝑧 ∈ ∪ ℎ ∈ 𝐴 dom ℎ)
39	38	ex 114	. . . 4 ⊢ (𝜑 → (𝑧 ∈ On → 𝑧 ∈ ∪ ℎ ∈ 𝐴 dom ℎ))
40	39	ssrdv 3103	. . 3 ⊢ (𝜑 → On ⊆ ∪ ℎ ∈ 𝐴 dom ℎ)
41	1	recsfval 6212	. . . . 5 ⊢ recs(𝐹) = ∪ 𝐴
42	41	dmeqi 4740	. . . 4 ⊢ dom recs(𝐹) = dom ∪ 𝐴
43		dmuni 4749	. . . 4 ⊢ dom ∪ 𝐴 = ∪ ℎ ∈ 𝐴 dom ℎ
44	42, 43	eqtri 2160	. . 3 ⊢ dom recs(𝐹) = ∪ ℎ ∈ 𝐴 dom ℎ
45	40, 44	sseqtrrdi 3146	. 2 ⊢ (𝜑 → On ⊆ dom recs(𝐹))
46	4, 45	eqssd 3114	1 ⊢ (𝜑 → dom recs(𝐹) = On)

Syntax hints:   → wi 4   ∧ wa 103  ∀wal 1329   = wceq 1331   ∈ wcel 1480  {cab 2125  ∀wral 2416  ∃wrex 2417  Vcvv 2686   ∪ cun 3069   ⊆ wss 3071  {csn 3527  ⟨cop 3530  ∪ cuni 3736  ∪ ciun 3813  Ord word 4284  Oncon0 4285  dom cdm 4539   ↾ cres 4541  Fun wfun 5117   Fn wfn 5118  ‘cfv 5123  recscrecs 6201

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-iord 4288  df-on 4290  df-suc 4293  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-recs 6202

This theorem is referenced by:  tfri1d  6232