Theorem tfrlemi14d 6184

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 6169	. . 3 ⊢ Ord dom recs(𝐹)
3		ordsson 4368	. . 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 6183	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ On) → ∃𝑔(𝑔 Fn 𝑧 ∧ ∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐹‘(𝑔 ↾ 𝑢))))
7	5	ad2antrr 477	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ On) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐹‘(𝑔 ↾ 𝑢)))) → ∀𝑥(Fun 𝐹 ∧ (𝐹‘𝑥) ∈ V))
8		simplr 502	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ On) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐹‘(𝑔 ↾ 𝑢)))) → 𝑧 ∈ On)
9		simprl 503	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ On) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐹‘(𝑔 ↾ 𝑢)))) → 𝑔 Fn 𝑧)
10		fneq2 5170	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑧 → (𝑔 Fn 𝑤 ↔ 𝑔 Fn 𝑧))
11		raleq 2600	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑧 → (∀𝑢 ∈ 𝑤 (𝑔‘𝑢) = (𝐹‘(𝑔 ↾ 𝑢)) ↔ ∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐹‘(𝑔 ↾ 𝑢))))
12	10, 11	anbi12d 462	. . . . . . . . . . . 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 2760	. . . . . . . . . . 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 465	. . . . . . . . . 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 2660	. . . . . . . . . . 11 ⊢ 𝑔 ∈ V
16	1, 15	tfrlem3a 6161	. . . . . . . . . 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 6176	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ On) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐹‘(𝑔 ↾ 𝑢)))) → (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) ∈ 𝐴)
19		vex 2660	. . . . . . . . . . . 12 ⊢ 𝑧 ∈ V
20	5	tfrlem3-2d 6163	. . . . . . . . . . . . 13 ⊢ (𝜑 → (Fun 𝐹 ∧ (𝐹‘𝑔) ∈ V))
21	20	simprd 113	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐹‘𝑔) ∈ V)
22		opexg 4110	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ V ∧ (𝐹‘𝑔) ∈ V) → ⟨𝑧, (𝐹‘𝑔)⟩ ∈ V)
23	19, 21, 22	sylancr 408	. . . . . . . . . . 11 ⊢ (𝜑 → ⟨𝑧, (𝐹‘𝑔)⟩ ∈ V)
24		snidg 3520	. . . . . . . . . . 11 ⊢ (⟨𝑧, (𝐹‘𝑔)⟩ ∈ V → ⟨𝑧, (𝐹‘𝑔)⟩ ∈ {⟨𝑧, (𝐹‘𝑔)⟩})
25		elun2 3210	. . . . . . . . . . 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 477	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ On) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐹‘(𝑔 ↾ 𝑢)))) → ⟨𝑧, (𝐹‘𝑔)⟩ ∈ (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}))
28		opeldmg 4704	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ V ∧ (𝐹‘𝑔) ∈ V) → (⟨𝑧, (𝐹‘𝑔)⟩ ∈ (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) → 𝑧 ∈ dom (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩})))
29	19, 21, 28	sylancr 408	. . . . . . . . . 10 ⊢ (𝜑 → (⟨𝑧, (𝐹‘𝑔)⟩ ∈ (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) → 𝑧 ∈ dom (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩})))
30	29	ad2antrr 477	. . . . . . . . 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 4699	. . . . . . . . . 10 ⊢ (ℎ = (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) → dom ℎ = dom (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}))
33	32	eleq2d 2184	. . . . . . . . 9 ⊢ (ℎ = (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) → (𝑧 ∈ dom ℎ ↔ 𝑧 ∈ dom (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩})))
34	33	rspcev 2760	. . . . . . . 8 ⊢ (((𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) ∈ 𝐴 ∧ 𝑧 ∈ dom (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩})) → ∃ℎ ∈ 𝐴 𝑧 ∈ dom ℎ)
35	18, 31, 34	syl2anc 406	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ On) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐹‘(𝑔 ↾ 𝑢)))) → ∃ℎ ∈ 𝐴 𝑧 ∈ dom ℎ)
36	6, 35	exlimddv 1852	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ On) → ∃ℎ ∈ 𝐴 𝑧 ∈ dom ℎ)
37		eliun 3783	. . . . . 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 3069	. . 3 ⊢ (𝜑 → On ⊆ ∪ ℎ ∈ 𝐴 dom ℎ)
41	1	recsfval 6166	. . . . 5 ⊢ recs(𝐹) = ∪ 𝐴
42	41	dmeqi 4700	. . . 4 ⊢ dom recs(𝐹) = dom ∪ 𝐴
43		dmuni 4709	. . . 4 ⊢ dom ∪ 𝐴 = ∪ ℎ ∈ 𝐴 dom ℎ
44	42, 43	eqtri 2135	. . 3 ⊢ dom recs(𝐹) = ∪ ℎ ∈ 𝐴 dom ℎ
45	40, 44	syl6sseqr 3112	. 2 ⊢ (𝜑 → On ⊆ dom recs(𝐹))
46	4, 45	eqssd 3080	1 ⊢ (𝜑 → dom recs(𝐹) = On)

Syntax hints:   → wi 4   ∧ wa 103  ∀wal 1312   = wceq 1314   ∈ wcel 1463  {cab 2101  ∀wral 2390  ∃wrex 2391  Vcvv 2657   ∪ cun 3035   ⊆ wss 3037  {csn 3493  ⟨cop 3496  ∪ cuni 3702  ∪ ciun 3779  Ord word 4244  Oncon0 4245  dom cdm 4499   ↾ cres 4501  Fun wfun 5075   Fn wfn 5076  ‘cfv 5081  recscrecs 6155

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4003  ax-sep 4006  ax-pow 4058  ax-pr 4091  ax-un 4315  ax-setind 4412

This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ne 2283  df-ral 2395  df-rex 2396  df-reu 2397  df-rab 2399  df-v 2659  df-sbc 2879  df-csb 2972  df-dif 3039  df-un 3041  df-in 3043  df-ss 3050  df-nul 3330  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-uni 3703  df-iun 3781  df-br 3896  df-opab 3950  df-mpt 3951  df-tr 3987  df-id 4175  df-iord 4248  df-on 4250  df-suc 4253  df-xp 4505  df-rel 4506  df-cnv 4507  df-co 4508  df-dm 4509  df-rn 4510  df-res 4511  df-ima 4512  df-iota 5046  df-fun 5083  df-fn 5084  df-f 5085  df-f1 5086  df-fo 5087  df-f1o 5088  df-fv 5089  df-recs 6156

This theorem is referenced by:  tfri1d  6186