Theorem rnttrcl 9719

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

Assertion

Ref	Expression
rnttrcl	⊢ ran t++𝑅 = ran 𝑅

Proof of Theorem rnttrcl

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	rneqi 5936	. . . 4 ⊢ ran t++𝑅 = ran {⟨𝑥, 𝑦⟩ ∣ ∃𝑛 ∈ (ω ∖ 1o)∃𝑓(𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑥 ∧ (𝑓‘𝑛) = 𝑦) ∧ ∀𝑎 ∈ 𝑛 (𝑓‘𝑎)𝑅(𝑓‘suc 𝑎))}
3		rnopab 5953	. . . 4 ⊢ ran {⟨𝑥, 𝑦⟩ ∣ ∃𝑛 ∈ (ω ∖ 1o)∃𝑓(𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑥 ∧ (𝑓‘𝑛) = 𝑦) ∧ ∀𝑎 ∈ 𝑛 (𝑓‘𝑎)𝑅(𝑓‘suc 𝑎))} = {𝑦 ∣ ∃𝑥∃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 ⊢ ran t++𝑅 = {𝑦 ∣ ∃𝑥∃𝑛 ∈ (ω ∖ 1o)∃𝑓(𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑥 ∧ (𝑓‘𝑛) = 𝑦) ∧ ∀𝑎 ∈ 𝑛 (𝑓‘𝑎)𝑅(𝑓‘suc 𝑎))}
5		fveq2 6891	. . . . . . . . . . . 12 ⊢ (𝑎 = ∪ 𝑛 → (𝑓‘𝑎) = (𝑓‘∪ 𝑛))
6		suceq 6430	. . . . . . . . . . . . 13 ⊢ (𝑎 = ∪ 𝑛 → suc 𝑎 = suc ∪ 𝑛)
7	6	fveq2d 6895	. . . . . . . . . . . 12 ⊢ (𝑎 = ∪ 𝑛 → (𝑓‘suc 𝑎) = (𝑓‘suc ∪ 𝑛))
8	5, 7	breq12d 5161	. . . . . . . . . . 11 ⊢ (𝑎 = ∪ 𝑛 → ((𝑓‘𝑎)𝑅(𝑓‘suc 𝑎) ↔ (𝑓‘∪ 𝑛)𝑅(𝑓‘suc ∪ 𝑛)))
9		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))
10		df-1o 8468	. . . . . . . . . . . . . . . 16 ⊢ 1o = suc ∅
11	10	difeq2i 4119	. . . . . . . . . . . . . . 15 ⊢ (ω ∖ 1o) = (ω ∖ suc ∅)
12	11	eleq2i 2825	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ (ω ∖ 1o) ↔ 𝑛 ∈ (ω ∖ suc ∅))
13		peano1 7881	. . . . . . . . . . . . . . 15 ⊢ ∅ ∈ ω
14		eldifsucnn 8665	. . . . . . . . . . . . . . 15 ⊢ (∅ ∈ ω → (𝑛 ∈ (ω ∖ suc ∅) ↔ ∃𝑥 ∈ (ω ∖ ∅)𝑛 = suc 𝑥))
15	13, 14	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ (ω ∖ suc ∅) ↔ ∃𝑥 ∈ (ω ∖ ∅)𝑛 = suc 𝑥)
16		dif0 4372	. . . . . . . . . . . . . . 15 ⊢ (ω ∖ ∅) = ω
17	16	rexeqi 3324	. . . . . . . . . . . . . 14 ⊢ (∃𝑥 ∈ (ω ∖ ∅)𝑛 = suc 𝑥 ↔ ∃𝑥 ∈ ω 𝑛 = suc 𝑥)
18	12, 15, 17	3bitri 296	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ (ω ∖ 1o) ↔ ∃𝑥 ∈ ω 𝑛 = suc 𝑥)
19		nnord 7865	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ ω → Ord 𝑥)
20		ordunisuc 7822	. . . . . . . . . . . . . . . . 17 ⊢ (Ord 𝑥 → ∪ suc 𝑥 = 𝑥)
21	19, 20	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ω → ∪ suc 𝑥 = 𝑥)
22		vex 3478	. . . . . . . . . . . . . . . . 17 ⊢ 𝑥 ∈ V
23	22	sucid 6446	. . . . . . . . . . . . . . . 16 ⊢ 𝑥 ∈ suc 𝑥
24	21, 23	eqeltrdi 2841	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ω → ∪ suc 𝑥 ∈ suc 𝑥)
25		unieq 4919	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = suc 𝑥 → ∪ 𝑛 = ∪ suc 𝑥)
26		id 22	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = suc 𝑥 → 𝑛 = suc 𝑥)
27	25, 26	eleq12d 2827	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = suc 𝑥 → (∪ 𝑛 ∈ 𝑛 ↔ ∪ suc 𝑥 ∈ suc 𝑥))
28	24, 27	syl5ibrcom 246	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ω → (𝑛 = suc 𝑥 → ∪ 𝑛 ∈ 𝑛))
29	28	rexlimiv 3148	. . . . . . . . . . . . 13 ⊢ (∃𝑥 ∈ ω 𝑛 = suc 𝑥 → ∪ 𝑛 ∈ 𝑛)
30	18, 29	sylbi 216	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ (ω ∖ 1o) → ∪ 𝑛 ∈ 𝑛)
31	30	adantr 481	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ (ω ∖ 1o) ∧ (𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑥 ∧ (𝑓‘𝑛) = 𝑦) ∧ ∀𝑎 ∈ 𝑛 (𝑓‘𝑎)𝑅(𝑓‘suc 𝑎))) → ∪ 𝑛 ∈ 𝑛)
32	8, 9, 31	rspcdva 3613	. . . . . . . . . 10 ⊢ ((𝑛 ∈ (ω ∖ 1o) ∧ (𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑥 ∧ (𝑓‘𝑛) = 𝑦) ∧ ∀𝑎 ∈ 𝑛 (𝑓‘𝑎)𝑅(𝑓‘suc 𝑎))) → (𝑓‘∪ 𝑛)𝑅(𝑓‘suc ∪ 𝑛))
33		suceq 6430	. . . . . . . . . . . . . . . . 17 ⊢ (∪ suc 𝑥 = 𝑥 → suc ∪ suc 𝑥 = suc 𝑥)
34	21, 33	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ω → suc ∪ suc 𝑥 = suc 𝑥)
35		suceq 6430	. . . . . . . . . . . . . . . . . 18 ⊢ (∪ 𝑛 = ∪ suc 𝑥 → suc ∪ 𝑛 = suc ∪ suc 𝑥)
36	25, 35	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = suc 𝑥 → suc ∪ 𝑛 = suc ∪ suc 𝑥)
37	36, 26	eqeq12d 2748	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = suc 𝑥 → (suc ∪ 𝑛 = 𝑛 ↔ suc ∪ suc 𝑥 = suc 𝑥))
38	34, 37	syl5ibrcom 246	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ω → (𝑛 = suc 𝑥 → suc ∪ 𝑛 = 𝑛))
39	38	rexlimiv 3148	. . . . . . . . . . . . . 14 ⊢ (∃𝑥 ∈ ω 𝑛 = suc 𝑥 → suc ∪ 𝑛 = 𝑛)
40	18, 39	sylbi 216	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ (ω ∖ 1o) → suc ∪ 𝑛 = 𝑛)
41	40	fveq2d 6895	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ (ω ∖ 1o) → (𝑓‘suc ∪ 𝑛) = (𝑓‘𝑛))
42	41	adantr 481	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ (ω ∖ 1o) ∧ (𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑥 ∧ (𝑓‘𝑛) = 𝑦) ∧ ∀𝑎 ∈ 𝑛 (𝑓‘𝑎)𝑅(𝑓‘suc 𝑎))) → (𝑓‘suc ∪ 𝑛) = (𝑓‘𝑛))
43		simpr2r 1233	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ (ω ∖ 1o) ∧ (𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑥 ∧ (𝑓‘𝑛) = 𝑦) ∧ ∀𝑎 ∈ 𝑛 (𝑓‘𝑎)𝑅(𝑓‘suc 𝑎))) → (𝑓‘𝑛) = 𝑦)
44	42, 43	eqtrd 2772	. . . . . . . . . 10 ⊢ ((𝑛 ∈ (ω ∖ 1o) ∧ (𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑥 ∧ (𝑓‘𝑛) = 𝑦) ∧ ∀𝑎 ∈ 𝑛 (𝑓‘𝑎)𝑅(𝑓‘suc 𝑎))) → (𝑓‘suc ∪ 𝑛) = 𝑦)
45	32, 44	breqtrd 5174	. . . . . . . . 9 ⊢ ((𝑛 ∈ (ω ∖ 1o) ∧ (𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑥 ∧ (𝑓‘𝑛) = 𝑦) ∧ ∀𝑎 ∈ 𝑛 (𝑓‘𝑎)𝑅(𝑓‘suc 𝑎))) → (𝑓‘∪ 𝑛)𝑅𝑦)
46		fvex 6904	. . . . . . . . . 10 ⊢ (𝑓‘∪ 𝑛) ∈ V
47		vex 3478	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
48	46, 47	brelrn 5941	. . . . . . . . 9 ⊢ ((𝑓‘∪ 𝑛)𝑅𝑦 → 𝑦 ∈ ran 𝑅)
49	45, 48	syl 17	. . . . . . . 8 ⊢ ((𝑛 ∈ (ω ∖ 1o) ∧ (𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑥 ∧ (𝑓‘𝑛) = 𝑦) ∧ ∀𝑎 ∈ 𝑛 (𝑓‘𝑎)𝑅(𝑓‘suc 𝑎))) → 𝑦 ∈ ran 𝑅)
50	49	ex 413	. . . . . . 7 ⊢ (𝑛 ∈ (ω ∖ 1o) → ((𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑥 ∧ (𝑓‘𝑛) = 𝑦) ∧ ∀𝑎 ∈ 𝑛 (𝑓‘𝑎)𝑅(𝑓‘suc 𝑎)) → 𝑦 ∈ ran 𝑅))
51	50	exlimdv 1936	. . . . . 6 ⊢ (𝑛 ∈ (ω ∖ 1o) → (∃𝑓(𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑥 ∧ (𝑓‘𝑛) = 𝑦) ∧ ∀𝑎 ∈ 𝑛 (𝑓‘𝑎)𝑅(𝑓‘suc 𝑎)) → 𝑦 ∈ ran 𝑅))
52	51	rexlimiv 3148	. . . . 5 ⊢ (∃𝑛 ∈ (ω ∖ 1o)∃𝑓(𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑥 ∧ (𝑓‘𝑛) = 𝑦) ∧ ∀𝑎 ∈ 𝑛 (𝑓‘𝑎)𝑅(𝑓‘suc 𝑎)) → 𝑦 ∈ ran 𝑅)
53	52	exlimiv 1933	. . . 4 ⊢ (∃𝑥∃𝑛 ∈ (ω ∖ 1o)∃𝑓(𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑥 ∧ (𝑓‘𝑛) = 𝑦) ∧ ∀𝑎 ∈ 𝑛 (𝑓‘𝑎)𝑅(𝑓‘suc 𝑎)) → 𝑦 ∈ ran 𝑅)
54	53	abssi 4067	. . 3 ⊢ {𝑦 ∣ ∃𝑥∃𝑛 ∈ (ω ∖ 1o)∃𝑓(𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑥 ∧ (𝑓‘𝑛) = 𝑦) ∧ ∀𝑎 ∈ 𝑛 (𝑓‘𝑎)𝑅(𝑓‘suc 𝑎))} ⊆ ran 𝑅
55	4, 54	eqsstri 4016	. 2 ⊢ ran t++𝑅 ⊆ ran 𝑅
56		rnresv 6200	. . 3 ⊢ ran (𝑅 ↾ V) = ran 𝑅
57		relres 6010	. . . . . 6 ⊢ Rel (𝑅 ↾ V)
58		ssttrcl 9712	. . . . . 6 ⊢ (Rel (𝑅 ↾ V) → (𝑅 ↾ V) ⊆ t++(𝑅 ↾ V))
59	57, 58	ax-mp 5	. . . . 5 ⊢ (𝑅 ↾ V) ⊆ t++(𝑅 ↾ V)
60		ttrclresv 9714	. . . . 5 ⊢ t++(𝑅 ↾ V) = t++𝑅
61	59, 60	sseqtri 4018	. . . 4 ⊢ (𝑅 ↾ V) ⊆ t++𝑅
62	61	rnssi 5939	. . 3 ⊢ ran (𝑅 ↾ V) ⊆ ran t++𝑅
63	56, 62	eqsstrri 4017	. 2 ⊢ ran 𝑅 ⊆ ran t++𝑅
64	55, 63	eqssi 3998	1 ⊢ ran t++𝑅 = ran 𝑅

Syntax hints:   ↔ 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  ∪ cuni 4908   class class class wbr 5148  {copab 5210  ran crn 5677   ↾ 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-int 4951  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-pred 6300  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-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-1o 8468  df-oadd 8472  df-ttrcl 9705

This theorem is referenced by:  ttrclexg  9720