Theorem tfr0dm 6317

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 4127	. . . . 5 ⊢ ∅ ∈ V
2		opexg 4225	. . . . 5 ⊢ ((∅ ∈ V ∧ (𝐺‘∅) ∈ 𝑉) → ⟨∅, (𝐺‘∅)⟩ ∈ V)
3	1, 2	mpan 424	. . . 4 ⊢ ((𝐺‘∅) ∈ 𝑉 → ⟨∅, (𝐺‘∅)⟩ ∈ V)
4		snidg 3620	. . . 4 ⊢ (⟨∅, (𝐺‘∅)⟩ ∈ V → ⟨∅, (𝐺‘∅)⟩ ∈ {⟨∅, (𝐺‘∅)⟩})
5	3, 4	syl 14	. . 3 ⊢ ((𝐺‘∅) ∈ 𝑉 → ⟨∅, (𝐺‘∅)⟩ ∈ {⟨∅, (𝐺‘∅)⟩})
6		fnsng 5259	. . . . 5 ⊢ ((∅ ∈ V ∧ (𝐺‘∅) ∈ 𝑉) → {⟨∅, (𝐺‘∅)⟩} Fn {∅})
7	1, 6	mpan 424	. . . 4 ⊢ ((𝐺‘∅) ∈ 𝑉 → {⟨∅, (𝐺‘∅)⟩} Fn {∅})
8		fvsng 5708	. . . . . . 7 ⊢ ((∅ ∈ V ∧ (𝐺‘∅) ∈ 𝑉) → ({⟨∅, (𝐺‘∅)⟩}‘∅) = (𝐺‘∅))
9	1, 8	mpan 424	. . . . . 6 ⊢ ((𝐺‘∅) ∈ 𝑉 → ({⟨∅, (𝐺‘∅)⟩}‘∅) = (𝐺‘∅))
10		res0 4907	. . . . . . 7 ⊢ ({⟨∅, (𝐺‘∅)⟩} ↾ ∅) = ∅
11	10	fveq2i 5514	. . . . . 6 ⊢ (𝐺‘({⟨∅, (𝐺‘∅)⟩} ↾ ∅)) = (𝐺‘∅)
12	9, 11	eqtr4di 2228	. . . . 5 ⊢ ((𝐺‘∅) ∈ 𝑉 → ({⟨∅, (𝐺‘∅)⟩}‘∅) = (𝐺‘({⟨∅, (𝐺‘∅)⟩} ↾ ∅)))
13		fveq2 5511	. . . . . . 7 ⊢ (𝑦 = ∅ → ({⟨∅, (𝐺‘∅)⟩}‘𝑦) = ({⟨∅, (𝐺‘∅)⟩}‘∅))
14		reseq2 4898	. . . . . . . 8 ⊢ (𝑦 = ∅ → ({⟨∅, (𝐺‘∅)⟩} ↾ 𝑦) = ({⟨∅, (𝐺‘∅)⟩} ↾ ∅))
15	14	fveq2d 5515	. . . . . . 7 ⊢ (𝑦 = ∅ → (𝐺‘({⟨∅, (𝐺‘∅)⟩} ↾ 𝑦)) = (𝐺‘({⟨∅, (𝐺‘∅)⟩} ↾ ∅)))
16	13, 15	eqeq12d 2192	. . . . . 6 ⊢ (𝑦 = ∅ → (({⟨∅, (𝐺‘∅)⟩}‘𝑦) = (𝐺‘({⟨∅, (𝐺‘∅)⟩} ↾ 𝑦)) ↔ ({⟨∅, (𝐺‘∅)⟩}‘∅) = (𝐺‘({⟨∅, (𝐺‘∅)⟩} ↾ ∅))))
17	1, 16	ralsn 3634	. . . . 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 4408	. . . . . 6 ⊢ suc ∅ = {∅}
20		0elon 4389	. . . . . . 7 ⊢ ∅ ∈ On
21	20	onsuci 4512	. . . . . 6 ⊢ suc ∅ ∈ On
22	19, 21	eqeltrri 2251	. . . . 5 ⊢ {∅} ∈ On
23		fneq2 5301	. . . . . . 7 ⊢ (𝑥 = {∅} → ({⟨∅, (𝐺‘∅)⟩} Fn 𝑥 ↔ {⟨∅, (𝐺‘∅)⟩} Fn {∅}))
24		raleq 2672	. . . . . . 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 2841	. . . . 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 4181	. . . . 5 ⊢ (⟨∅, (𝐺‘∅)⟩ ∈ V → {⟨∅, (𝐺‘∅)⟩} ∈ V)
30		eleq2 2241	. . . . . . 7 ⊢ (𝑓 = {⟨∅, (𝐺‘∅)⟩} → (⟨∅, (𝐺‘∅)⟩ ∈ 𝑓 ↔ ⟨∅, (𝐺‘∅)⟩ ∈ {⟨∅, (𝐺‘∅)⟩}))
31		fneq1 5300	. . . . . . . . 9 ⊢ (𝑓 = {⟨∅, (𝐺‘∅)⟩} → (𝑓 Fn 𝑥 ↔ {⟨∅, (𝐺‘∅)⟩} Fn 𝑥))
32		fveq1 5510	. . . . . . . . . . 11 ⊢ (𝑓 = {⟨∅, (𝐺‘∅)⟩} → (𝑓‘𝑦) = ({⟨∅, (𝐺‘∅)⟩}‘𝑦))
33		reseq1 4897	. . . . . . . . . . . 12 ⊢ (𝑓 = {⟨∅, (𝐺‘∅)⟩} → (𝑓 ↾ 𝑦) = ({⟨∅, (𝐺‘∅)⟩} ↾ 𝑦))
34	33	fveq2d 5515	. . . . . . . . . . 11 ⊢ (𝑓 = {⟨∅, (𝐺‘∅)⟩} → (𝐺‘(𝑓 ↾ 𝑦)) = (𝐺‘({⟨∅, (𝐺‘∅)⟩} ↾ 𝑦)))
35	32, 34	eqeq12d 2192	. . . . . . . . . 10 ⊢ (𝑓 = {⟨∅, (𝐺‘∅)⟩} → ((𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)) ↔ ({⟨∅, (𝐺‘∅)⟩}‘𝑦) = (𝐺‘({⟨∅, (𝐺‘∅)⟩} ↾ 𝑦))))
36	35	ralbidv 2477	. . . . . . . . 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 2478	. . . . . . 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 2825	. . . . 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 2244	. . . . 5 ⊢ (⟨∅, (𝐺‘∅)⟩ ∈ 𝐹 ↔ ⟨∅, (𝐺‘∅)⟩ ∈ recs(𝐺))
44		df-recs 6300	. . . . . 6 ⊢ recs(𝐺) = ∪ {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)))}
45	44	eleq2i 2244	. . . . 5 ⊢ (⟨∅, (𝐺‘∅)⟩ ∈ recs(𝐺) ↔ ⟨∅, (𝐺‘∅)⟩ ∈ ∪ {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)))})
46		eluniab 3819	. . . . 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	syl6ibr 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 4828	. . 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 1353  ∃wex 1492   ∈ wcel 2148  {cab 2163  ∀wral 2455  ∃wrex 2456  Vcvv 2737  ∅c0 3422  {csn 3591  ⟨cop 3594  ∪ cuni 3807  Oncon0 4360  suc csuc 4362  dom cdm 4623   ↾ cres 4625   Fn wfn 5207  ‘cfv 5212  recscrecs 6299

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-br 4001  df-opab 4062  df-tr 4099  df-id 4290  df-iord 4363  df-on 4365  df-suc 4368  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-res 4635  df-iota 5174  df-fun 5214  df-fn 5215  df-fv 5220  df-recs 6300

This theorem is referenced by:  tfr0  6318