Theorem tfr0dm 6553

Description: Transfinite recursion is defined at the empty set. (Contributed by Jim Kingdon, 8-Mar-2022.)

Hypothesis

Ref	Expression
tfr.1	⊢ 𝐹 = recs(𝐺)

Assertion

Ref	Expression
tfr0dm	⊢ ((𝐺‘∅) ∈ 𝑉 → ∅ ∈ dom 𝐹)

Proof of Theorem tfr0dm

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

Step	Hyp	Ref	Expression
1		0ex 4237	. . . . 5 ⊢ ∅ ∈ V
2		opexg 4344	. . . . 5 ⊢ ((∅ ∈ V ∧ (𝐺‘∅) ∈ 𝑉) → ⟨∅, (𝐺‘∅)⟩ ∈ V)
3	1, 2	mpan 424	. . . 4 ⊢ ((𝐺‘∅) ∈ 𝑉 → ⟨∅, (𝐺‘∅)⟩ ∈ V)
4		snidg 3718	. . . 4 ⊢ (⟨∅, (𝐺‘∅)⟩ ∈ V → ⟨∅, (𝐺‘∅)⟩ ∈ {⟨∅, (𝐺‘∅)⟩})
5	3, 4	syl 14	. . 3 ⊢ ((𝐺‘∅) ∈ 𝑉 → ⟨∅, (𝐺‘∅)⟩ ∈ {⟨∅, (𝐺‘∅)⟩})
6		fnsng 5403	. . . . 5 ⊢ ((∅ ∈ V ∧ (𝐺‘∅) ∈ 𝑉) → {⟨∅, (𝐺‘∅)⟩} Fn {∅})
7	1, 6	mpan 424	. . . 4 ⊢ ((𝐺‘∅) ∈ 𝑉 → {⟨∅, (𝐺‘∅)⟩} Fn {∅})
8		fvsng 5880	. . . . . . 7 ⊢ ((∅ ∈ V ∧ (𝐺‘∅) ∈ 𝑉) → ({⟨∅, (𝐺‘∅)⟩}‘∅) = (𝐺‘∅))
9	1, 8	mpan 424	. . . . . 6 ⊢ ((𝐺‘∅) ∈ 𝑉 → ({⟨∅, (𝐺‘∅)⟩}‘∅) = (𝐺‘∅))
10		res0 5042	. . . . . . 7 ⊢ ({⟨∅, (𝐺‘∅)⟩} ↾ ∅) = ∅
11	10	fveq2i 5673	. . . . . 6 ⊢ (𝐺‘({⟨∅, (𝐺‘∅)⟩} ↾ ∅)) = (𝐺‘∅)
12	9, 11	eqtr4di 2283	. . . . 5 ⊢ ((𝐺‘∅) ∈ 𝑉 → ({⟨∅, (𝐺‘∅)⟩}‘∅) = (𝐺‘({⟨∅, (𝐺‘∅)⟩} ↾ ∅)))
13		fveq2 5670	. . . . . . 7 ⊢ (𝑦 = ∅ → ({⟨∅, (𝐺‘∅)⟩}‘𝑦) = ({⟨∅, (𝐺‘∅)⟩}‘∅))
14		reseq2 5033	. . . . . . . 8 ⊢ (𝑦 = ∅ → ({⟨∅, (𝐺‘∅)⟩} ↾ 𝑦) = ({⟨∅, (𝐺‘∅)⟩} ↾ ∅))
15	14	fveq2d 5674	. . . . . . 7 ⊢ (𝑦 = ∅ → (𝐺‘({⟨∅, (𝐺‘∅)⟩} ↾ 𝑦)) = (𝐺‘({⟨∅, (𝐺‘∅)⟩} ↾ ∅)))
16	13, 15	eqeq12d 2247	. . . . . 6 ⊢ (𝑦 = ∅ → (({⟨∅, (𝐺‘∅)⟩}‘𝑦) = (𝐺‘({⟨∅, (𝐺‘∅)⟩} ↾ 𝑦)) ↔ ({⟨∅, (𝐺‘∅)⟩}‘∅) = (𝐺‘({⟨∅, (𝐺‘∅)⟩} ↾ ∅))))
17	1, 16	ralsn 3732	. . . . 5 ⊢ (∀𝑦 ∈ {∅} ({⟨∅, (𝐺‘∅)⟩}‘𝑦) = (𝐺‘({⟨∅, (𝐺‘∅)⟩} ↾ 𝑦)) ↔ ({⟨∅, (𝐺‘∅)⟩}‘∅) = (𝐺‘({⟨∅, (𝐺‘∅)⟩} ↾ ∅)))
18	12, 17	sylibr 134	. . . 4 ⊢ ((𝐺‘∅) ∈ 𝑉 → ∀𝑦 ∈ {∅} ({⟨∅, (𝐺‘∅)⟩}‘𝑦) = (𝐺‘({⟨∅, (𝐺‘∅)⟩} ↾ 𝑦)))
19		suc0 4532	. . . . . 6 ⊢ suc ∅ = {∅}
20		0elon 4513	. . . . . . 7 ⊢ ∅ ∈ On
21	20	onsuci 4638	. . . . . 6 ⊢ suc ∅ ∈ On
22	19, 21	eqeltrri 2306	. . . . 5 ⊢ {∅} ∈ On
23		fneq2 5445	. . . . . . 7 ⊢ (𝑥 = {∅} → ({⟨∅, (𝐺‘∅)⟩} Fn 𝑥 ↔ {⟨∅, (𝐺‘∅)⟩} Fn {∅}))
24		raleq 2741	. . . . . . 7 ⊢ (𝑥 = {∅} → (∀𝑦 ∈ 𝑥 ({⟨∅, (𝐺‘∅)⟩}‘𝑦) = (𝐺‘({⟨∅, (𝐺‘∅)⟩} ↾ 𝑦)) ↔ ∀𝑦 ∈ {∅} ({⟨∅, (𝐺‘∅)⟩}‘𝑦) = (𝐺‘({⟨∅, (𝐺‘∅)⟩} ↾ 𝑦))))
25	23, 24	anbi12d 473	. . . . . 6 ⊢ (𝑥 = {∅} → (({⟨∅, (𝐺‘∅)⟩} Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 ({⟨∅, (𝐺‘∅)⟩}‘𝑦) = (𝐺‘({⟨∅, (𝐺‘∅)⟩} ↾ 𝑦))) ↔ ({⟨∅, (𝐺‘∅)⟩} Fn {∅} ∧ ∀𝑦 ∈ {∅} ({⟨∅, (𝐺‘∅)⟩}‘𝑦) = (𝐺‘({⟨∅, (𝐺‘∅)⟩} ↾ 𝑦)))))
26	25	rspcev 2921	. . . . 5 ⊢ (({∅} ∈ On ∧ ({⟨∅, (𝐺‘∅)⟩} Fn {∅} ∧ ∀𝑦 ∈ {∅} ({⟨∅, (𝐺‘∅)⟩}‘𝑦) = (𝐺‘({⟨∅, (𝐺‘∅)⟩} ↾ 𝑦)))) → ∃𝑥 ∈ On ({⟨∅, (𝐺‘∅)⟩} Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 ({⟨∅, (𝐺‘∅)⟩}‘𝑦) = (𝐺‘({⟨∅, (𝐺‘∅)⟩} ↾ 𝑦))))
27	22, 26	mpan 424	. . . 4 ⊢ (({⟨∅, (𝐺‘∅)⟩} Fn {∅} ∧ ∀𝑦 ∈ {∅} ({⟨∅, (𝐺‘∅)⟩}‘𝑦) = (𝐺‘({⟨∅, (𝐺‘∅)⟩} ↾ 𝑦))) → ∃𝑥 ∈ On ({⟨∅, (𝐺‘∅)⟩} Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 ({⟨∅, (𝐺‘∅)⟩}‘𝑦) = (𝐺‘({⟨∅, (𝐺‘∅)⟩} ↾ 𝑦))))
28	7, 18, 27	syl2anc 411	. . 3 ⊢ ((𝐺‘∅) ∈ 𝑉 → ∃𝑥 ∈ On ({⟨∅, (𝐺‘∅)⟩} Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 ({⟨∅, (𝐺‘∅)⟩}‘𝑦) = (𝐺‘({⟨∅, (𝐺‘∅)⟩} ↾ 𝑦))))
29		snexg 4297	. . . . 5 ⊢ (⟨∅, (𝐺‘∅)⟩ ∈ V → {⟨∅, (𝐺‘∅)⟩} ∈ V)
30		eleq2 2296	. . . . . . 7 ⊢ (𝑓 = {⟨∅, (𝐺‘∅)⟩} → (⟨∅, (𝐺‘∅)⟩ ∈ 𝑓 ↔ ⟨∅, (𝐺‘∅)⟩ ∈ {⟨∅, (𝐺‘∅)⟩}))
31		fneq1 5444	. . . . . . . . 9 ⊢ (𝑓 = {⟨∅, (𝐺‘∅)⟩} → (𝑓 Fn 𝑥 ↔ {⟨∅, (𝐺‘∅)⟩} Fn 𝑥))
32		fveq1 5669	. . . . . . . . . . 11 ⊢ (𝑓 = {⟨∅, (𝐺‘∅)⟩} → (𝑓‘𝑦) = ({⟨∅, (𝐺‘∅)⟩}‘𝑦))
33		reseq1 5032	. . . . . . . . . . . 12 ⊢ (𝑓 = {⟨∅, (𝐺‘∅)⟩} → (𝑓 ↾ 𝑦) = ({⟨∅, (𝐺‘∅)⟩} ↾ 𝑦))
34	33	fveq2d 5674	. . . . . . . . . . 11 ⊢ (𝑓 = {⟨∅, (𝐺‘∅)⟩} → (𝐺‘(𝑓 ↾ 𝑦)) = (𝐺‘({⟨∅, (𝐺‘∅)⟩} ↾ 𝑦)))
35	32, 34	eqeq12d 2247	. . . . . . . . . 10 ⊢ (𝑓 = {⟨∅, (𝐺‘∅)⟩} → ((𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)) ↔ ({⟨∅, (𝐺‘∅)⟩}‘𝑦) = (𝐺‘({⟨∅, (𝐺‘∅)⟩} ↾ 𝑦))))
36	35	ralbidv 2542	. . . . . . . . 9 ⊢ (𝑓 = {⟨∅, (𝐺‘∅)⟩} → (∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)) ↔ ∀𝑦 ∈ 𝑥 ({⟨∅, (𝐺‘∅)⟩}‘𝑦) = (𝐺‘({⟨∅, (𝐺‘∅)⟩} ↾ 𝑦))))
37	31, 36	anbi12d 473	. . . . . . . 8 ⊢ (𝑓 = {⟨∅, (𝐺‘∅)⟩} → ((𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦))) ↔ ({⟨∅, (𝐺‘∅)⟩} Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 ({⟨∅, (𝐺‘∅)⟩}‘𝑦) = (𝐺‘({⟨∅, (𝐺‘∅)⟩} ↾ 𝑦)))))
38	37	rexbidv 2543	. . . . . . 7 ⊢ (𝑓 = {⟨∅, (𝐺‘∅)⟩} → (∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦))) ↔ ∃𝑥 ∈ On ({⟨∅, (𝐺‘∅)⟩} Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 ({⟨∅, (𝐺‘∅)⟩}‘𝑦) = (𝐺‘({⟨∅, (𝐺‘∅)⟩} ↾ 𝑦)))))
39	30, 38	anbi12d 473	. . . . . 6 ⊢ (𝑓 = {⟨∅, (𝐺‘∅)⟩} → ((⟨∅, (𝐺‘∅)⟩ ∈ 𝑓 ∧ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)))) ↔ (⟨∅, (𝐺‘∅)⟩ ∈ {⟨∅, (𝐺‘∅)⟩} ∧ ∃𝑥 ∈ On ({⟨∅, (𝐺‘∅)⟩} Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 ({⟨∅, (𝐺‘∅)⟩}‘𝑦) = (𝐺‘({⟨∅, (𝐺‘∅)⟩} ↾ 𝑦))))))
40	39	spcegv 2905	. . . . 5 ⊢ ({⟨∅, (𝐺‘∅)⟩} ∈ V → ((⟨∅, (𝐺‘∅)⟩ ∈ {⟨∅, (𝐺‘∅)⟩} ∧ ∃𝑥 ∈ On ({⟨∅, (𝐺‘∅)⟩} Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 ({⟨∅, (𝐺‘∅)⟩}‘𝑦) = (𝐺‘({⟨∅, (𝐺‘∅)⟩} ↾ 𝑦)))) → ∃𝑓(⟨∅, (𝐺‘∅)⟩ ∈ 𝑓 ∧ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦))))))
41	3, 29, 40	3syl 17	. . . 4 ⊢ ((𝐺‘∅) ∈ 𝑉 → ((⟨∅, (𝐺‘∅)⟩ ∈ {⟨∅, (𝐺‘∅)⟩} ∧ ∃𝑥 ∈ On ({⟨∅, (𝐺‘∅)⟩} Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 ({⟨∅, (𝐺‘∅)⟩}‘𝑦) = (𝐺‘({⟨∅, (𝐺‘∅)⟩} ↾ 𝑦)))) → ∃𝑓(⟨∅, (𝐺‘∅)⟩ ∈ 𝑓 ∧ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦))))))
42		tfr.1	. . . . . 6 ⊢ 𝐹 = recs(𝐺)
43	42	eleq2i 2299	. . . . 5 ⊢ (⟨∅, (𝐺‘∅)⟩ ∈ 𝐹 ↔ ⟨∅, (𝐺‘∅)⟩ ∈ recs(𝐺))
44		df-recs 6536	. . . . . 6 ⊢ recs(𝐺) = ∪ {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)))}
45	44	eleq2i 2299	. . . . 5 ⊢ (⟨∅, (𝐺‘∅)⟩ ∈ recs(𝐺) ↔ ⟨∅, (𝐺‘∅)⟩ ∈ ∪ {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)))})
46		eluniab 3926	. . . . 5 ⊢ (⟨∅, (𝐺‘∅)⟩ ∈ ∪ {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)))} ↔ ∃𝑓(⟨∅, (𝐺‘∅)⟩ ∈ 𝑓 ∧ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)))))
47	43, 45, 46	3bitri 206	. . . 4 ⊢ (⟨∅, (𝐺‘∅)⟩ ∈ 𝐹 ↔ ∃𝑓(⟨∅, (𝐺‘∅)⟩ ∈ 𝑓 ∧ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)))))
48	41, 47	imbitrrdi 162	. . 3 ⊢ ((𝐺‘∅) ∈ 𝑉 → ((⟨∅, (𝐺‘∅)⟩ ∈ {⟨∅, (𝐺‘∅)⟩} ∧ ∃𝑥 ∈ On ({⟨∅, (𝐺‘∅)⟩} Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 ({⟨∅, (𝐺‘∅)⟩}‘𝑦) = (𝐺‘({⟨∅, (𝐺‘∅)⟩} ↾ 𝑦)))) → ⟨∅, (𝐺‘∅)⟩ ∈ 𝐹))
49	5, 28, 48	mp2and 433	. 2 ⊢ ((𝐺‘∅) ∈ 𝑉 → ⟨∅, (𝐺‘∅)⟩ ∈ 𝐹)
50		opeldmg 4961	. . 3 ⊢ ((∅ ∈ V ∧ (𝐺‘∅) ∈ 𝑉) → (⟨∅, (𝐺‘∅)⟩ ∈ 𝐹 → ∅ ∈ dom 𝐹))
51	1, 50	mpan 424	. 2 ⊢ ((𝐺‘∅) ∈ 𝑉 → (⟨∅, (𝐺‘∅)⟩ ∈ 𝐹 → ∅ ∈ dom 𝐹))
52	49, 51	mpd 13	1 ⊢ ((𝐺‘∅) ∈ 𝑉 → ∅ ∈ dom 𝐹)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1398  ∃wex 1541   ∈ wcel 2203  {cab 2218  ∀wral 2520  ∃wrex 2521  Vcvv 2813  ∅c0 3508  {csn 3689  ⟨cop 3692  ∪ cuni 3914  Oncon0 4484  suc csuc 4486  dom cdm 4749   ↾ cres 4751   Fn wfn 5347  ‘cfv 5352  recscrecs 6535

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2815  df-sbc 3043  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-br 4110  df-opab 4172  df-tr 4209  df-id 4414  df-iord 4487  df-on 4489  df-suc 4492  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-res 4761  df-iota 5312  df-fun 5354  df-fn 5355  df-fv 5360  df-recs 6536

This theorem is referenced by:  tfr0  6554