Theorem dmttrcl 9716

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 9703	. . . . 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 5905	. . . 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 5916	. . . 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 2761	. . 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 1233	. . . . . . . . . 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 6892	. . . . . . . . . . . 12 ⊢ (𝑎 = ∅ → (𝑓‘𝑎) = (𝑓‘∅))
7		suceq 6431	. . . . . . . . . . . . . 14 ⊢ (𝑎 = ∅ → suc 𝑎 = suc ∅)
8		df-1o 8466	. . . . . . . . . . . . . 14 ⊢ 1o = suc ∅
9	7, 8	eqtr4di 2791	. . . . . . . . . . . . 13 ⊢ (𝑎 = ∅ → suc 𝑎 = 1o)
10	9	fveq2d 6896	. . . . . . . . . . . 12 ⊢ (𝑎 = ∅ → (𝑓‘suc 𝑎) = (𝑓‘1o))
11	6, 10	breq12d 5162	. . . . . . . . . . 11 ⊢ (𝑎 = ∅ → ((𝑓‘𝑎)𝑅(𝑓‘suc 𝑎) ↔ (𝑓‘∅)𝑅(𝑓‘1o)))
12		simpr3 1197	. . . . . . . . . . 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 3959	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ (ω ∖ 1o) ↔ (𝑛 ∈ ω ∧ ¬ 𝑛 ∈ 1o))
14		0ex 5308	. . . . . . . . . . . . . . . 16 ⊢ ∅ ∈ V
15		nnord 7863	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ ω → Ord 𝑛)
16		ordelsuc 7808	. . . . . . . . . . . . . . . 16 ⊢ ((∅ ∈ V ∧ Ord 𝑛) → (∅ ∈ 𝑛 ↔ suc ∅ ⊆ 𝑛))
17	14, 15, 16	sylancr 588	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ω → (∅ ∈ 𝑛 ↔ suc ∅ ⊆ 𝑛))
18	8	sseq1i 4011	. . . . . . . . . . . . . . . 16 ⊢ (1o ⊆ 𝑛 ↔ suc ∅ ⊆ 𝑛)
19		1on 8478	. . . . . . . . . . . . . . . . . 18 ⊢ 1o ∈ On
20	19	onordi 6476	. . . . . . . . . . . . . . . . 17 ⊢ Ord 1o
21		ordtri1 6398	. . . . . . . . . . . . . . . . 17 ⊢ ((Ord 1o ∧ Ord 𝑛) → (1o ⊆ 𝑛 ↔ ¬ 𝑛 ∈ 1o))
22	20, 15, 21	sylancr 588	. . . . . . . . . . . . . . . 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 478	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ ω ∧ ¬ 𝑛 ∈ 1o) → ∅ ∈ 𝑛)
26	13, 25	sylbi 216	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ (ω ∖ 1o) → ∅ ∈ 𝑛)
27	26	adantr 482	. . . . . . . . . . 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 3614	. . . . . . . . . 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 5173	. . . . . . . . 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 3479	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
31		fvex 6905	. . . . . . . . . 10 ⊢ (𝑓‘1o) ∈ V
32	30, 31	breldm 5909	. . . . . . . . 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 414	. . . . . . 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 1937	. . . . . 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 3149	. . . . 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 1934	. . . 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 4068	. . 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 4017	. 2 ⊢ dom t++𝑅 ⊆ dom 𝑅
40		dmresv 6200	. . 3 ⊢ dom (𝑅 ↾ V) = dom 𝑅
41		relres 6011	. . . . . 6 ⊢ Rel (𝑅 ↾ V)
42		ssttrcl 9710	. . . . . 6 ⊢ (Rel (𝑅 ↾ V) → (𝑅 ↾ V) ⊆ t++(𝑅 ↾ V))
43	41, 42	ax-mp 5	. . . . 5 ⊢ (𝑅 ↾ V) ⊆ t++(𝑅 ↾ V)
44		ttrclresv 9712	. . . . 5 ⊢ t++(𝑅 ↾ V) = t++𝑅
45	43, 44	sseqtri 4019	. . . 4 ⊢ (𝑅 ↾ V) ⊆ t++𝑅
46		dmss 5903	. . . 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 4018	. 2 ⊢ dom 𝑅 ⊆ dom t++𝑅
49	39, 48	eqssi 3999	1 ⊢ dom t++𝑅 = dom 𝑅

Syntax hints:  ¬ wn 3   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542  ∃wex 1782   ∈ wcel 2107  {cab 2710  ∀wral 3062  ∃wrex 3071  Vcvv 3475   ∖ cdif 3946   ⊆ wss 3949  ∅c0 4323   class class class wbr 5149  {copab 5211  dom cdm 5677   ↾ cres 5679  Rel wrel 5682  Ord word 6364  suc csuc 6367   Fn wfn 6539  ‘cfv 6544  ωcom 7855  1_oc1o 8459  t++cttrcl 9702

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7725

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-om 7856  df-1o 8466  df-ttrcl 9703

This theorem is referenced by:  ttrclexg  9718  ttrclse  9722