Theorem dmttrcl 9632

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 9619	. . . . 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 5853	. . . 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 5864	. . . 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 2759	. . 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 6834	. . . . . . . . . . . 12 ⊢ (𝑎 = ∅ → (𝑓‘𝑎) = (𝑓‘∅))
7		suceq 6385	. . . . . . . . . . . . . 14 ⊢ (𝑎 = ∅ → suc 𝑎 = suc ∅)
8		df-1o 8397	. . . . . . . . . . . . . 14 ⊢ 1o = suc ∅
9	7, 8	eqtr4di 2789	. . . . . . . . . . . . 13 ⊢ (𝑎 = ∅ → suc 𝑎 = 1o)
10	9	fveq2d 6838	. . . . . . . . . . . 12 ⊢ (𝑎 = ∅ → (𝑓‘suc 𝑎) = (𝑓‘1o))
11	6, 10	breq12d 5111	. . . . . . . . . . 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 3911	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ (ω ∖ 1o) ↔ (𝑛 ∈ ω ∧ ¬ 𝑛 ∈ 1o))
14		0ex 5252	. . . . . . . . . . . . . . . 16 ⊢ ∅ ∈ V
15		nnord 7816	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ ω → Ord 𝑛)
16		ordelsuc 7762	. . . . . . . . . . . . . . . 16 ⊢ ((∅ ∈ V ∧ Ord 𝑛) → (∅ ∈ 𝑛 ↔ suc ∅ ⊆ 𝑛))
17	14, 15, 16	sylancr 587	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ ω → (∅ ∈ 𝑛 ↔ suc ∅ ⊆ 𝑛))
18	8	sseq1i 3962	. . . . . . . . . . . . . . . 16 ⊢ (1o ⊆ 𝑛 ↔ suc ∅ ⊆ 𝑛)
19		1on 8409	. . . . . . . . . . . . . . . . . 18 ⊢ 1o ∈ On
20	19	onordi 6430	. . . . . . . . . . . . . . . . 17 ⊢ Ord 1o
21		ordtri1 6350	. . . . . . . . . . . . . . . . 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 3577	. . . . . . . . . 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 5122	. . . . . . . . 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 3444	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
31		fvex 6847	. . . . . . . . . 10 ⊢ (𝑓‘1o) ∈ V
32	30, 31	breldm 5857	. . . . . . . . 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 1934	. . . . . 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 3130	. . . . 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 1931	. . . 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 4020	. . 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 3980	. 2 ⊢ dom t++𝑅 ⊆ dom 𝑅
40		dmresv 6158	. . 3 ⊢ dom (𝑅 ↾ V) = dom 𝑅
41		relres 5964	. . . . . 6 ⊢ Rel (𝑅 ↾ V)
42		ssttrcl 9626	. . . . . 6 ⊢ (Rel (𝑅 ↾ V) → (𝑅 ↾ V) ⊆ t++(𝑅 ↾ V))
43	41, 42	ax-mp 5	. . . . 5 ⊢ (𝑅 ↾ V) ⊆ t++(𝑅 ↾ V)
44		ttrclresv 9628	. . . . 5 ⊢ t++(𝑅 ↾ V) = t++𝑅
45	43, 44	sseqtri 3982	. . . 4 ⊢ (𝑅 ↾ V) ⊆ t++𝑅
46		dmss 5851	. . . 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 3981	. 2 ⊢ dom 𝑅 ⊆ dom t++𝑅
49	39, 48	eqssi 3950	1 ⊢ dom t++𝑅 = dom 𝑅

Syntax hints:  ¬ wn 3   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541  ∃wex 1780   ∈ wcel 2113  {cab 2714  ∀wral 3051  ∃wrex 3060  Vcvv 3440   ∖ cdif 3898   ⊆ wss 3901  ∅c0 4285   class class class wbr 5098  {copab 5160  dom cdm 5624   ↾ cres 5626  Rel wrel 5629  Ord word 6316  suc csuc 6319   Fn wfn 6487  ‘cfv 6492  ωcom 7808  1_oc1o 8390  t++cttrcl 9618

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-tr 5206  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-om 7809  df-1o 8397  df-ttrcl 9619

This theorem is referenced by:  ttrclexg  9634  ttrclse  9638