Theorem tfr0dm 6583

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 4255	. . . . 5 ⊢ ∅ ∈ V
2		opexg 4363	. . . . 5 ⊢ ((∅ ∈ V ∧ (𝐺‘∅) ∈ 𝑉) → ⟨∅, (𝐺‘∅)⟩ ∈ V)
3	1, 2	mpan 428	. . . 4 ⊢ ((𝐺‘∅) ∈ 𝑉 → ⟨∅, (𝐺‘∅)⟩ ∈ V)
4		snidg 3734	. . . 4 ⊢ (⟨∅, (𝐺‘∅)⟩ ∈ V → ⟨∅, (𝐺‘∅)⟩ ∈ {⟨∅, (𝐺‘∅)⟩})
5	3, 4	syl 14	. . 3 ⊢ ((𝐺‘∅) ∈ 𝑉 → ⟨∅, (𝐺‘∅)⟩ ∈ {⟨∅, (𝐺‘∅)⟩})
6		fnsng 5423	. . . . 5 ⊢ ((∅ ∈ V ∧ (𝐺‘∅) ∈ 𝑉) → {⟨∅, (𝐺‘∅)⟩} Fn {∅})
7	1, 6	mpan 428	. . . 4 ⊢ ((𝐺‘∅) ∈ 𝑉 → {⟨∅, (𝐺‘∅)⟩} Fn {∅})
8		fvsng 5902	. . . . . . 7 ⊢ ((∅ ∈ V ∧ (𝐺‘∅) ∈ 𝑉) → ({⟨∅, (𝐺‘∅)⟩}‘∅) = (𝐺‘∅))
9	1, 8	mpan 428	. . . . . 6 ⊢ ((𝐺‘∅) ∈ 𝑉 → ({⟨∅, (𝐺‘∅)⟩}‘∅) = (𝐺‘∅))
10		res0 5062	. . . . . . 7 ⊢ ({⟨∅, (𝐺‘∅)⟩} ↾ ∅) = ∅
11	10	fveq2i 5693	. . . . . 6 ⊢ (𝐺‘({⟨∅, (𝐺‘∅)⟩} ↾ ∅)) = (𝐺‘∅)
12	9, 11	eqtr4di 2289	. . . . 5 ⊢ ((𝐺‘∅) ∈ 𝑉 → ({⟨∅, (𝐺‘∅)⟩}‘∅) = (𝐺‘({⟨∅, (𝐺‘∅)⟩} ↾ ∅)))
13		fveq2 5690	. . . . . . 7 ⊢ (𝑦 = ∅ → ({⟨∅, (𝐺‘∅)⟩}‘𝑦) = ({⟨∅, (𝐺‘∅)⟩}‘∅))
14		reseq2 5053	. . . . . . . 8 ⊢ (𝑦 = ∅ → ({⟨∅, (𝐺‘∅)⟩} ↾ 𝑦) = ({⟨∅, (𝐺‘∅)⟩} ↾ ∅))
15	14	fveq2d 5694	. . . . . . 7 ⊢ (𝑦 = ∅ → (𝐺‘({⟨∅, (𝐺‘∅)⟩} ↾ 𝑦)) = (𝐺‘({⟨∅, (𝐺‘∅)⟩} ↾ ∅)))
16	13, 15	eqeq12d 2253	. . . . . 6 ⊢ (𝑦 = ∅ → (({⟨∅, (𝐺‘∅)⟩}‘𝑦) = (𝐺‘({⟨∅, (𝐺‘∅)⟩} ↾ 𝑦)) ↔ ({⟨∅, (𝐺‘∅)⟩}‘∅) = (𝐺‘({⟨∅, (𝐺‘∅)⟩} ↾ ∅))))
17	1, 16	ralsn 3748	. . . . 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 4551	. . . . . 6 ⊢ suc ∅ = {∅}
20		0elon 4532	. . . . . . 7 ⊢ ∅ ∈ On
21	20	onsuci 4658	. . . . . 6 ⊢ suc ∅ ∈ On
22	19, 21	eqeltrri 2312	. . . . 5 ⊢ {∅} ∈ On
23		fneq2 5465	. . . . . . 7 ⊢ (𝑥 = {∅} → ({⟨∅, (𝐺‘∅)⟩} Fn 𝑥 ↔ {⟨∅, (𝐺‘∅)⟩} Fn {∅}))
24		raleq 2749	. . . . . . 7 ⊢ (𝑥 = {∅} → (∀𝑦 ∈ 𝑥 ({⟨∅, (𝐺‘∅)⟩}‘𝑦) = (𝐺‘({⟨∅, (𝐺‘∅)⟩} ↾ 𝑦)) ↔ ∀𝑦 ∈ {∅} ({⟨∅, (𝐺‘∅)⟩}‘𝑦) = (𝐺‘({⟨∅, (𝐺‘∅)⟩} ↾ 𝑦))))
25	23, 24	anbi12d 477	. . . . . 6 ⊢ (𝑥 = {∅} → (({⟨∅, (𝐺‘∅)⟩} Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 ({⟨∅, (𝐺‘∅)⟩}‘𝑦) = (𝐺‘({⟨∅, (𝐺‘∅)⟩} ↾ 𝑦))) ↔ ({⟨∅, (𝐺‘∅)⟩} Fn {∅} ∧ ∀𝑦 ∈ {∅} ({⟨∅, (𝐺‘∅)⟩}‘𝑦) = (𝐺‘({⟨∅, (𝐺‘∅)⟩} ↾ 𝑦)))))
26	25	rspcev 2929	. . . . 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 428	. . . 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 415	. . 3 ⊢ ((𝐺‘∅) ∈ 𝑉 → ∃𝑥 ∈ On ({⟨∅, (𝐺‘∅)⟩} Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 ({⟨∅, (𝐺‘∅)⟩}‘𝑦) = (𝐺‘({⟨∅, (𝐺‘∅)⟩} ↾ 𝑦))))
29		snexg 4316	. . . . 5 ⊢ (⟨∅, (𝐺‘∅)⟩ ∈ V → {⟨∅, (𝐺‘∅)⟩} ∈ V)
30		eleq2 2302	. . . . . . 7 ⊢ (𝑓 = {⟨∅, (𝐺‘∅)⟩} → (⟨∅, (𝐺‘∅)⟩ ∈ 𝑓 ↔ ⟨∅, (𝐺‘∅)⟩ ∈ {⟨∅, (𝐺‘∅)⟩}))
31		fneq1 5464	. . . . . . . . 9 ⊢ (𝑓 = {⟨∅, (𝐺‘∅)⟩} → (𝑓 Fn 𝑥 ↔ {⟨∅, (𝐺‘∅)⟩} Fn 𝑥))
32		fveq1 5689	. . . . . . . . . . 11 ⊢ (𝑓 = {⟨∅, (𝐺‘∅)⟩} → (𝑓‘𝑦) = ({⟨∅, (𝐺‘∅)⟩}‘𝑦))
33		reseq1 5052	. . . . . . . . . . . 12 ⊢ (𝑓 = {⟨∅, (𝐺‘∅)⟩} → (𝑓 ↾ 𝑦) = ({⟨∅, (𝐺‘∅)⟩} ↾ 𝑦))
34	33	fveq2d 5694	. . . . . . . . . . 11 ⊢ (𝑓 = {⟨∅, (𝐺‘∅)⟩} → (𝐺‘(𝑓 ↾ 𝑦)) = (𝐺‘({⟨∅, (𝐺‘∅)⟩} ↾ 𝑦)))
35	32, 34	eqeq12d 2253	. . . . . . . . . 10 ⊢ (𝑓 = {⟨∅, (𝐺‘∅)⟩} → ((𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)) ↔ ({⟨∅, (𝐺‘∅)⟩}‘𝑦) = (𝐺‘({⟨∅, (𝐺‘∅)⟩} ↾ 𝑦))))
36	35	ralbidv 2550	. . . . . . . . 9 ⊢ (𝑓 = {⟨∅, (𝐺‘∅)⟩} → (∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)) ↔ ∀𝑦 ∈ 𝑥 ({⟨∅, (𝐺‘∅)⟩}‘𝑦) = (𝐺‘({⟨∅, (𝐺‘∅)⟩} ↾ 𝑦))))
37	31, 36	anbi12d 477	. . . . . . . 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 2551	. . . . . . 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 477	. . . . . 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 2913	. . . . 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 2305	. . . . 5 ⊢ (⟨∅, (𝐺‘∅)⟩ ∈ 𝐹 ↔ ⟨∅, (𝐺‘∅)⟩ ∈ recs(𝐺))
44		df-recs 6566	. . . . . 6 ⊢ recs(𝐺) = ∪ {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)))}
45	44	eleq2i 2305	. . . . 5 ⊢ (⟨∅, (𝐺‘∅)⟩ ∈ recs(𝐺) ↔ ⟨∅, (𝐺‘∅)⟩ ∈ ∪ {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)))})
46		eluniab 3942	. . . . 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 437	. 2 ⊢ ((𝐺‘∅) ∈ 𝑉 → ⟨∅, (𝐺‘∅)⟩ ∈ 𝐹)
50		opeldmg 4981	. . 3 ⊢ ((∅ ∈ V ∧ (𝐺‘∅) ∈ 𝑉) → (⟨∅, (𝐺‘∅)⟩ ∈ 𝐹 → ∅ ∈ dom 𝐹))
51	1, 50	mpan 428	. 2 ⊢ ((𝐺‘∅) ∈ 𝑉 → (⟨∅, (𝐺‘∅)⟩ ∈ 𝐹 → ∅ ∈ dom 𝐹))
52	49, 51	mpd 13	1 ⊢ ((𝐺‘∅) ∈ 𝑉 → ∅ ∈ dom 𝐹)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1402  ∃wex 1545   ∈ wcel 2209  {cab 2224  ∀wral 2528  ∃wrex 2529  Vcvv 2821  ∅c0 3520  {csn 3705  ⟨cop 3708  ∪ cuni 3930  Oncon0 4503  suc csuc 4505  dom cdm 4769   ↾ cres 4771   Fn wfn 5367  ‘cfv 5372  recscrecs 6565

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-opab 4188  df-tr 4225  df-id 4433  df-iord 4506  df-on 4508  df-suc 4511  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-res 4781  df-iota 5332  df-fun 5374  df-fn 5375  df-fv 5380  df-recs 6566

This theorem is referenced by:  tfr0  6584