Theorem dmttrcl 9758

Description: The domain of a transitive closure is the same as the domain of the original class. (Contributed by Scott Fenton, 26-Oct-2024.)

Assertion

Ref	Expression
dmttrcl	⊢ dom t++𝑅 = dom 𝑅

Proof of Theorem dmttrcl

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

Step	Hyp	Ref	Expression
1		df-ttrcl 9745	. . . . 5 ⊢ t++𝑅 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑛 ∈ (ω ∖ 1o)∃𝑓(𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑥 ∧ (𝑓‘𝑛) = 𝑦) ∧ ∀𝑎 ∈ 𝑛 (𝑓‘𝑎)𝑅(𝑓‘suc 𝑎))}
2	1	dmeqi 5917	. . . 4 ⊢ dom t++𝑅 = dom {⟨𝑥, 𝑦⟩ ∣ ∃𝑛 ∈ (ω ∖ 1o)∃𝑓(𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑥 ∧ (𝑓‘𝑛) = 𝑦) ∧ ∀𝑎 ∈ 𝑛 (𝑓‘𝑎)𝑅(𝑓‘suc 𝑎))}
3		dmopab 5928	. . . 4 ⊢ dom {⟨𝑥, 𝑦⟩ ∣ ∃𝑛 ∈ (ω ∖ 1o)∃𝑓(𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑥 ∧ (𝑓‘𝑛) = 𝑦) ∧ ∀𝑎 ∈ 𝑛 (𝑓‘𝑎)𝑅(𝑓‘suc 𝑎))} = {𝑥 ∣ ∃𝑦∃𝑛 ∈ (ω ∖ 1o)∃𝑓(𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑥 ∧ (𝑓‘𝑛) = 𝑦) ∧ ∀𝑎 ∈ 𝑛 (𝑓‘𝑎)𝑅(𝑓‘suc 𝑎))}
4	2, 3	eqtri 2762	. . 3 ⊢ dom t++𝑅 = {𝑥 ∣ ∃𝑦∃𝑛 ∈ (ω ∖ 1o)∃𝑓(𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑥 ∧ (𝑓‘𝑛) = 𝑦) ∧ ∀𝑎 ∈ 𝑛 (𝑓‘𝑎)𝑅(𝑓‘suc 𝑎))}
5		simpr2l 1231	. . . . . . . . . 10 ⊢ ((𝑛 ∈ (ω ∖ 1o) ∧ (𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑥 ∧ (𝑓‘𝑛) = 𝑦) ∧ ∀𝑎 ∈ 𝑛 (𝑓‘𝑎)𝑅(𝑓‘suc 𝑎))) → (𝑓‘∅) = 𝑥)
6		fveq2 6906	. . . . . . . . . . . 12 ⊢ (𝑎 = ∅ → (𝑓‘𝑎) = (𝑓‘∅))
7		suceq 6451	. . . . . . . . . . . . . 14 ⊢ (𝑎 = ∅ → suc 𝑎 = suc ∅)
8		df-1o 8504	. . . . . . . . . . . . . 14 ⊢ 1o = suc ∅
9	7, 8	eqtr4di 2792	. . . . . . . . . . . . 13 ⊢ (𝑎 = ∅ → suc 𝑎 = 1o)
10	9	fveq2d 6910	. . . . . . . . . . . 12 ⊢ (𝑎 = ∅ → (𝑓‘suc 𝑎) = (𝑓‘1o))
11	6, 10	breq12d 5160	. . . . . . . . . . 11 ⊢ (𝑎 = ∅ → ((𝑓‘𝑎)𝑅(𝑓‘suc 𝑎) ↔ (𝑓‘∅)𝑅(𝑓‘1o)))
12		simpr3 1195	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ (ω ∖ 1o) ∧ (𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑥 ∧ (𝑓‘𝑛) = 𝑦) ∧ ∀𝑎 ∈ 𝑛 (𝑓‘𝑎)𝑅(𝑓‘suc 𝑎))) → ∀𝑎 ∈ 𝑛 (𝑓‘𝑎)𝑅(𝑓‘suc 𝑎))
13		eldif 3972	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ (ω ∖ 1o) ↔ (𝑛 ∈ ω ∧ ¬ 𝑛 ∈ 1o))
14		0ex 5312	. . . . . . . . . . . . . . . 16 ⊢ ∅ ∈ V
15		nnord 7894	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ ω → Ord 𝑛)
16		ordelsuc 7839	. . . . . . . . . . . . . . . 16 ⊢ ((∅ ∈ V ∧ Ord 𝑛) → (∅ ∈ 𝑛 ↔ suc ∅ ⊆ 𝑛))
17	14, 15, 16	sylancr 587	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ω → (∅ ∈ 𝑛 ↔ suc ∅ ⊆ 𝑛))
18	8	sseq1i 4023	. . . . . . . . . . . . . . . 16 ⊢ (1o ⊆ 𝑛 ↔ suc ∅ ⊆ 𝑛)
19		1on 8516	. . . . . . . . . . . . . . . . . 18 ⊢ 1o ∈ On
20	19	onordi 6496	. . . . . . . . . . . . . . . . 17 ⊢ Ord 1o
21		ordtri1 6418	. . . . . . . . . . . . . . . . 17 ⊢ ((Ord 1o ∧ Ord 𝑛) → (1o ⊆ 𝑛 ↔ ¬ 𝑛 ∈ 1o))
22	20, 15, 21	sylancr 587	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ ω → (1o ⊆ 𝑛 ↔ ¬ 𝑛 ∈ 1o))
23	18, 22	bitr3id 285	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ω → (suc ∅ ⊆ 𝑛 ↔ ¬ 𝑛 ∈ 1o))
24	17, 23	bitr2d 280	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ω → (¬ 𝑛 ∈ 1o ↔ ∅ ∈ 𝑛))
25	24	biimpa 476	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ ω ∧ ¬ 𝑛 ∈ 1o) → ∅ ∈ 𝑛)
26	13, 25	sylbi 217	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ (ω ∖ 1o) → ∅ ∈ 𝑛)
27	26	adantr 480	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ (ω ∖ 1o) ∧ (𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑥 ∧ (𝑓‘𝑛) = 𝑦) ∧ ∀𝑎 ∈ 𝑛 (𝑓‘𝑎)𝑅(𝑓‘suc 𝑎))) → ∅ ∈ 𝑛)
28	11, 12, 27	rspcdva 3622	. . . . . . . . . 10 ⊢ ((𝑛 ∈ (ω ∖ 1o) ∧ (𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑥 ∧ (𝑓‘𝑛) = 𝑦) ∧ ∀𝑎 ∈ 𝑛 (𝑓‘𝑎)𝑅(𝑓‘suc 𝑎))) → (𝑓‘∅)𝑅(𝑓‘1o))
29	5, 28	eqbrtrrd 5171	. . . . . . . . 9 ⊢ ((𝑛 ∈ (ω ∖ 1o) ∧ (𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑥 ∧ (𝑓‘𝑛) = 𝑦) ∧ ∀𝑎 ∈ 𝑛 (𝑓‘𝑎)𝑅(𝑓‘suc 𝑎))) → 𝑥𝑅(𝑓‘1o))
30		vex 3481	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
31		fvex 6919	. . . . . . . . . 10 ⊢ (𝑓‘1o) ∈ V
32	30, 31	breldm 5921	. . . . . . . . 9 ⊢ (𝑥𝑅(𝑓‘1o) → 𝑥 ∈ dom 𝑅)
33	29, 32	syl 17	. . . . . . . 8 ⊢ ((𝑛 ∈ (ω ∖ 1o) ∧ (𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑥 ∧ (𝑓‘𝑛) = 𝑦) ∧ ∀𝑎 ∈ 𝑛 (𝑓‘𝑎)𝑅(𝑓‘suc 𝑎))) → 𝑥 ∈ dom 𝑅)
34	33	ex 412	. . . . . . 7 ⊢ (𝑛 ∈ (ω ∖ 1o) → ((𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑥 ∧ (𝑓‘𝑛) = 𝑦) ∧ ∀𝑎 ∈ 𝑛 (𝑓‘𝑎)𝑅(𝑓‘suc 𝑎)) → 𝑥 ∈ dom 𝑅))
35	34	exlimdv 1930	. . . . . 6 ⊢ (𝑛 ∈ (ω ∖ 1o) → (∃𝑓(𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑥 ∧ (𝑓‘𝑛) = 𝑦) ∧ ∀𝑎 ∈ 𝑛 (𝑓‘𝑎)𝑅(𝑓‘suc 𝑎)) → 𝑥 ∈ dom 𝑅))
36	35	rexlimiv 3145	. . . . 5 ⊢ (∃𝑛 ∈ (ω ∖ 1o)∃𝑓(𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑥 ∧ (𝑓‘𝑛) = 𝑦) ∧ ∀𝑎 ∈ 𝑛 (𝑓‘𝑎)𝑅(𝑓‘suc 𝑎)) → 𝑥 ∈ dom 𝑅)
37	36	exlimiv 1927	. . . 4 ⊢ (∃𝑦∃𝑛 ∈ (ω ∖ 1o)∃𝑓(𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑥 ∧ (𝑓‘𝑛) = 𝑦) ∧ ∀𝑎 ∈ 𝑛 (𝑓‘𝑎)𝑅(𝑓‘suc 𝑎)) → 𝑥 ∈ dom 𝑅)
38	37	abssi 4079	. . 3 ⊢ {𝑥 ∣ ∃𝑦∃𝑛 ∈ (ω ∖ 1o)∃𝑓(𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑥 ∧ (𝑓‘𝑛) = 𝑦) ∧ ∀𝑎 ∈ 𝑛 (𝑓‘𝑎)𝑅(𝑓‘suc 𝑎))} ⊆ dom 𝑅
39	4, 38	eqsstri 4029	. 2 ⊢ dom t++𝑅 ⊆ dom 𝑅
40		dmresv 6221	. . 3 ⊢ dom (𝑅 ↾ V) = dom 𝑅
41		relres 6025	. . . . . 6 ⊢ Rel (𝑅 ↾ V)
42		ssttrcl 9752	. . . . . 6 ⊢ (Rel (𝑅 ↾ V) → (𝑅 ↾ V) ⊆ t++(𝑅 ↾ V))
43	41, 42	ax-mp 5	. . . . 5 ⊢ (𝑅 ↾ V) ⊆ t++(𝑅 ↾ V)
44		ttrclresv 9754	. . . . 5 ⊢ t++(𝑅 ↾ V) = t++𝑅
45	43, 44	sseqtri 4031	. . . 4 ⊢ (𝑅 ↾ V) ⊆ t++𝑅
46		dmss 5915	. . . 4 ⊢ ((𝑅 ↾ V) ⊆ t++𝑅 → dom (𝑅 ↾ V) ⊆ dom t++𝑅)
47	45, 46	ax-mp 5	. . 3 ⊢ dom (𝑅 ↾ V) ⊆ dom t++𝑅
48	40, 47	eqsstrri 4030	. 2 ⊢ dom 𝑅 ⊆ dom t++𝑅
49	39, 48	eqssi 4011	1 ⊢ dom t++𝑅 = dom 𝑅

Syntax hints:  ¬ wn 3   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1536  ∃wex 1775   ∈ wcel 2105  {cab 2711  ∀wral 3058  ∃wrex 3067  Vcvv 3477   ∖ cdif 3959   ⊆ wss 3962  ∅c0 4338   class class class wbr 5147  {copab 5209  dom cdm 5688   ↾ cres 5690  Rel wrel 5693  Ord word 6384  suc csuc 6387   Fn wfn 6557  ‘cfv 6562  ωcom 7886  1_oc1o 8497  t++cttrcl 9744

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pr 5437  ax-un 7753

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-om 7887  df-1o 8504  df-ttrcl 9745

This theorem is referenced by:  ttrclexg  9760  ttrclse  9764