Theorem dmttrcl 9718

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 9705	. . . . 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 5904	. . . 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 5915	. . . 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 2760	. . 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 1232	. . . . . . . . . 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 6891	. . . . . . . . . . . 12 ⊢ (𝑎 = ∅ → (𝑓‘𝑎) = (𝑓‘∅))
7		suceq 6430	. . . . . . . . . . . . . 14 ⊢ (𝑎 = ∅ → suc 𝑎 = suc ∅)
8		df-1o 8468	. . . . . . . . . . . . . 14 ⊢ 1o = suc ∅
9	7, 8	eqtr4di 2790	. . . . . . . . . . . . 13 ⊢ (𝑎 = ∅ → suc 𝑎 = 1o)
10	9	fveq2d 6895	. . . . . . . . . . . 12 ⊢ (𝑎 = ∅ → (𝑓‘suc 𝑎) = (𝑓‘1o))
11	6, 10	breq12d 5161	. . . . . . . . . . 11 ⊢ (𝑎 = ∅ → ((𝑓‘𝑎)𝑅(𝑓‘suc 𝑎) ↔ (𝑓‘∅)𝑅(𝑓‘1o)))
12		simpr3 1196	. . . . . . . . . . 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 3958	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ (ω ∖ 1o) ↔ (𝑛 ∈ ω ∧ ¬ 𝑛 ∈ 1o))
14		0ex 5307	. . . . . . . . . . . . . . . 16 ⊢ ∅ ∈ V
15		nnord 7865	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ ω → Ord 𝑛)
16		ordelsuc 7810	. . . . . . . . . . . . . . . 16 ⊢ ((∅ ∈ V ∧ Ord 𝑛) → (∅ ∈ 𝑛 ↔ suc ∅ ⊆ 𝑛))
17	14, 15, 16	sylancr 587	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ω → (∅ ∈ 𝑛 ↔ suc ∅ ⊆ 𝑛))
18	8	sseq1i 4010	. . . . . . . . . . . . . . . 16 ⊢ (1o ⊆ 𝑛 ↔ suc ∅ ⊆ 𝑛)
19		1on 8480	. . . . . . . . . . . . . . . . . 18 ⊢ 1o ∈ On
20	19	onordi 6475	. . . . . . . . . . . . . . . . 17 ⊢ Ord 1o
21		ordtri1 6397	. . . . . . . . . . . . . . . . 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 284	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ω → (suc ∅ ⊆ 𝑛 ↔ ¬ 𝑛 ∈ 1o))
24	17, 23	bitr2d 279	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ω → (¬ 𝑛 ∈ 1o ↔ ∅ ∈ 𝑛))
25	24	biimpa 477	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ ω ∧ ¬ 𝑛 ∈ 1o) → ∅ ∈ 𝑛)
26	13, 25	sylbi 216	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ (ω ∖ 1o) → ∅ ∈ 𝑛)
27	26	adantr 481	. . . . . . . . . . 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 3613	. . . . . . . . . 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 5172	. . . . . . . . 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 3478	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
31		fvex 6904	. . . . . . . . . 10 ⊢ (𝑓‘1o) ∈ V
32	30, 31	breldm 5908	. . . . . . . . 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 413	. . . . . . 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 1936	. . . . . 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 3148	. . . . 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 1933	. . . 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 4067	. . 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 4016	. 2 ⊢ dom t++𝑅 ⊆ dom 𝑅
40		dmresv 6199	. . 3 ⊢ dom (𝑅 ↾ V) = dom 𝑅
41		relres 6010	. . . . . 6 ⊢ Rel (𝑅 ↾ V)
42		ssttrcl 9712	. . . . . 6 ⊢ (Rel (𝑅 ↾ V) → (𝑅 ↾ V) ⊆ t++(𝑅 ↾ V))
43	41, 42	ax-mp 5	. . . . 5 ⊢ (𝑅 ↾ V) ⊆ t++(𝑅 ↾ V)
44		ttrclresv 9714	. . . . 5 ⊢ t++(𝑅 ↾ V) = t++𝑅
45	43, 44	sseqtri 4018	. . . 4 ⊢ (𝑅 ↾ V) ⊆ t++𝑅
46		dmss 5902	. . . 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 4017	. 2 ⊢ dom 𝑅 ⊆ dom t++𝑅
49	39, 48	eqssi 3998	1 ⊢ dom t++𝑅 = dom 𝑅

Syntax hints:  ¬ wn 3   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541  ∃wex 1781   ∈ wcel 2106  {cab 2709  ∀wral 3061  ∃wrex 3070  Vcvv 3474   ∖ cdif 3945   ⊆ wss 3948  ∅c0 4322   class class class wbr 5148  {copab 5210  dom cdm 5676   ↾ cres 5678  Rel wrel 5681  Ord word 6363  suc csuc 6366   Fn wfn 6538  ‘cfv 6543  ωcom 7857  1_oc1o 8461  t++cttrcl 9704

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7727

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-om 7858  df-1o 8468  df-ttrcl 9705

This theorem is referenced by:  ttrclexg  9720  ttrclse  9724