Theorem tfrlemi14d 6444

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 6429	. . 3 ⊢ Ord dom recs(𝐹)
3		ordsson 4559	. . 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 6443	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑧 ∈ On) → ∃𝑔(𝑔 Fn 𝑧 ∧ ∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐹‘(𝑔 ↾ 𝑢))))
7	5	ad2antrr 488	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ On) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐹‘(𝑔 ↾ 𝑢)))) → ∀𝑥(Fun 𝐹 ∧ (𝐹‘𝑥) ∈ V))
8		simplr 528	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ On) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐹‘(𝑔 ↾ 𝑢)))) → 𝑧 ∈ On)
9		simprl 529	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ On) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐹‘(𝑔 ↾ 𝑢)))) → 𝑔 Fn 𝑧)
10		fneq2 5383	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑧 → (𝑔 Fn 𝑤 ↔ 𝑔 Fn 𝑧))
11		raleq 2706	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑧 → (∀𝑢 ∈ 𝑤 (𝑔‘𝑢) = (𝐹‘(𝑔 ↾ 𝑢)) ↔ ∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐹‘(𝑔 ↾ 𝑢))))
12	10, 11	anbi12d 473	. . . . . . . . . . . 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 2885	. . . . . . . . . . 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 476	. . . . . . . . . 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 2780	. . . . . . . . . . 11 ⊢ 𝑔 ∈ V
16	1, 15	tfrlem3a 6421	. . . . . . . . . 10 ⊢ (𝑔 ∈ 𝐴 ↔ ∃𝑤 ∈ On (𝑔 Fn 𝑤 ∧ ∀𝑢 ∈ 𝑤 (𝑔‘𝑢) = (𝐹‘(𝑔 ↾ 𝑢))))
17	14, 16	sylibr 134	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ On) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐹‘(𝑔 ↾ 𝑢)))) → 𝑔 ∈ 𝐴)
18	1, 7, 8, 9, 17	tfrlemisucaccv 6436	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑧 ∈ On) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐹‘(𝑔 ↾ 𝑢)))) → (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) ∈ 𝐴)
19		vex 2780	. . . . . . . . . . . 12 ⊢ 𝑧 ∈ V
20	5	tfrlem3-2d 6423	. . . . . . . . . . . . 13 ⊢ (𝜑 → (Fun 𝐹 ∧ (𝐹‘𝑔) ∈ V))
21	20	simprd 114	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐹‘𝑔) ∈ V)
22		opexg 4291	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ V ∧ (𝐹‘𝑔) ∈ V) → ⟨𝑧, (𝐹‘𝑔)⟩ ∈ V)
23	19, 21, 22	sylancr 414	. . . . . . . . . . 11 ⊢ (𝜑 → ⟨𝑧, (𝐹‘𝑔)⟩ ∈ V)
24		snidg 3673	. . . . . . . . . . 11 ⊢ (⟨𝑧, (𝐹‘𝑔)⟩ ∈ V → ⟨𝑧, (𝐹‘𝑔)⟩ ∈ {⟨𝑧, (𝐹‘𝑔)⟩})
25		elun2 3350	. . . . . . . . . . 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 488	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 ∈ On) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐹‘(𝑔 ↾ 𝑢)))) → ⟨𝑧, (𝐹‘𝑔)⟩ ∈ (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}))
28		opeldmg 4903	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ V ∧ (𝐹‘𝑔) ∈ V) → (⟨𝑧, (𝐹‘𝑔)⟩ ∈ (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) → 𝑧 ∈ dom (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩})))
29	19, 21, 28	sylancr 414	. . . . . . . . . 10 ⊢ (𝜑 → (⟨𝑧, (𝐹‘𝑔)⟩ ∈ (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) → 𝑧 ∈ dom (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩})))
30	29	ad2antrr 488	. . . . . . . . 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 4898	. . . . . . . . . 10 ⊢ (ℎ = (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) → dom ℎ = dom (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}))
33	32	eleq2d 2277	. . . . . . . . 9 ⊢ (ℎ = (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) → (𝑧 ∈ dom ℎ ↔ 𝑧 ∈ dom (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩})))
34	33	rspcev 2885	. . . . . . . 8 ⊢ (((𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩}) ∈ 𝐴 ∧ 𝑧 ∈ dom (𝑔 ∪ {⟨𝑧, (𝐹‘𝑔)⟩})) → ∃ℎ ∈ 𝐴 𝑧 ∈ dom ℎ)
35	18, 31, 34	syl2anc 411	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑧 ∈ On) ∧ (𝑔 Fn 𝑧 ∧ ∀𝑢 ∈ 𝑧 (𝑔‘𝑢) = (𝐹‘(𝑔 ↾ 𝑢)))) → ∃ℎ ∈ 𝐴 𝑧 ∈ dom ℎ)
36	6, 35	exlimddv 1923	. . . . . 6 ⊢ ((𝜑 ∧ 𝑧 ∈ On) → ∃ℎ ∈ 𝐴 𝑧 ∈ dom ℎ)
37		eliun 3946	. . . . . 6 ⊢ (𝑧 ∈ ∪ ℎ ∈ 𝐴 dom ℎ ↔ ∃ℎ ∈ 𝐴 𝑧 ∈ dom ℎ)
38	36, 37	sylibr 134	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 ∈ On) → 𝑧 ∈ ∪ ℎ ∈ 𝐴 dom ℎ)
39	38	ex 115	. . . 4 ⊢ (𝜑 → (𝑧 ∈ On → 𝑧 ∈ ∪ ℎ ∈ 𝐴 dom ℎ))
40	39	ssrdv 3208	. . 3 ⊢ (𝜑 → On ⊆ ∪ ℎ ∈ 𝐴 dom ℎ)
41	1	recsfval 6426	. . . . 5 ⊢ recs(𝐹) = ∪ 𝐴
42	41	dmeqi 4899	. . . 4 ⊢ dom recs(𝐹) = dom ∪ 𝐴
43		dmuni 4908	. . . 4 ⊢ dom ∪ 𝐴 = ∪ ℎ ∈ 𝐴 dom ℎ
44	42, 43	eqtri 2228	. . 3 ⊢ dom recs(𝐹) = ∪ ℎ ∈ 𝐴 dom ℎ
45	40, 44	sseqtrrdi 3251	. 2 ⊢ (𝜑 → On ⊆ dom recs(𝐹))
46	4, 45	eqssd 3219	1 ⊢ (𝜑 → dom recs(𝐹) = On)

Syntax hints:   → wi 4   ∧ wa 104  ∀wal 1371   = wceq 1373   ∈ wcel 2178  {cab 2193  ∀wral 2486  ∃wrex 2487  Vcvv 2777   ∪ cun 3173   ⊆ wss 3175  {csn 3644  ⟨cop 3647  ∪ cuni 3865  ∪ ciun 3942  Ord word 4428  Oncon0 4429  dom cdm 4694   ↾ cres 4696  Fun wfun 5285   Fn wfn 5286  ‘cfv 5291  recscrecs 6415

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-coll 4176  ax-sep 4179  ax-pow 4235  ax-pr 4270  ax-un 4499  ax-setind 4604

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-ral 2491  df-rex 2492  df-reu 2493  df-rab 2495  df-v 2779  df-sbc 3007  df-csb 3103  df-dif 3177  df-un 3179  df-in 3181  df-ss 3188  df-nul 3470  df-pw 3629  df-sn 3650  df-pr 3651  df-op 3653  df-uni 3866  df-iun 3944  df-br 4061  df-opab 4123  df-mpt 4124  df-tr 4160  df-id 4359  df-iord 4432  df-on 4434  df-suc 4437  df-xp 4700  df-rel 4701  df-cnv 4702  df-co 4703  df-dm 4704  df-rn 4705  df-res 4706  df-ima 4707  df-iota 5252  df-fun 5293  df-fn 5294  df-f 5295  df-f1 5296  df-fo 5297  df-f1o 5298  df-fv 5299  df-recs 6416

This theorem is referenced by:  tfri1d  6446