Theorem rnttrcl 9612

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 9598	. . . . 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 5877	. . . 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 5894	. . . 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 2754	. . 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 6822	. . . . . . . . . . . 12 ⊢ (𝑎 = ∪ 𝑛 → (𝑓‘𝑎) = (𝑓‘∪ 𝑛))
6		suceq 6374	. . . . . . . . . . . . 13 ⊢ (𝑎 = ∪ 𝑛 → suc 𝑎 = suc ∪ 𝑛)
7	6	fveq2d 6826	. . . . . . . . . . . 12 ⊢ (𝑎 = ∪ 𝑛 → (𝑓‘suc 𝑎) = (𝑓‘suc ∪ 𝑛))
8	5, 7	breq12d 5104	. . . . . . . . . . 11 ⊢ (𝑎 = ∪ 𝑛 → ((𝑓‘𝑎)𝑅(𝑓‘suc 𝑎) ↔ (𝑓‘∪ 𝑛)𝑅(𝑓‘suc ∪ 𝑛)))
9		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))
10		df-1o 8385	. . . . . . . . . . . . . . . 16 ⊢ 1o = suc ∅
11	10	difeq2i 4073	. . . . . . . . . . . . . . 15 ⊢ (ω ∖ 1o) = (ω ∖ suc ∅)
12	11	eleq2i 2823	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ (ω ∖ 1o) ↔ 𝑛 ∈ (ω ∖ suc ∅))
13		peano1 7819	. . . . . . . . . . . . . . 15 ⊢ ∅ ∈ ω
14		eldifsucnn 8579	. . . . . . . . . . . . . . 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 4328	. . . . . . . . . . . . . . 15 ⊢ (ω ∖ ∅) = ω
17	16	rexeqi 3291	. . . . . . . . . . . . . 14 ⊢ (∃𝑥 ∈ (ω ∖ ∅)𝑛 = suc 𝑥 ↔ ∃𝑥 ∈ ω 𝑛 = suc 𝑥)
18	12, 15, 17	3bitri 297	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ (ω ∖ 1o) ↔ ∃𝑥 ∈ ω 𝑛 = suc 𝑥)
19		nnord 7804	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ ω → Ord 𝑥)
20		ordunisuc 7762	. . . . . . . . . . . . . . . . 17 ⊢ (Ord 𝑥 → ∪ suc 𝑥 = 𝑥)
21	19, 20	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ω → ∪ suc 𝑥 = 𝑥)
22		vex 3440	. . . . . . . . . . . . . . . . 17 ⊢ 𝑥 ∈ V
23	22	sucid 6390	. . . . . . . . . . . . . . . 16 ⊢ 𝑥 ∈ suc 𝑥
24	21, 23	eqeltrdi 2839	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ω → ∪ suc 𝑥 ∈ suc 𝑥)
25		unieq 4870	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = suc 𝑥 → ∪ 𝑛 = ∪ suc 𝑥)
26		id 22	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = suc 𝑥 → 𝑛 = suc 𝑥)
27	25, 26	eleq12d 2825	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = suc 𝑥 → (∪ 𝑛 ∈ 𝑛 ↔ ∪ suc 𝑥 ∈ suc 𝑥))
28	24, 27	syl5ibrcom 247	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ω → (𝑛 = suc 𝑥 → ∪ 𝑛 ∈ 𝑛))
29	28	rexlimiv 3126	. . . . . . . . . . . . 13 ⊢ (∃𝑥 ∈ ω 𝑛 = suc 𝑥 → ∪ 𝑛 ∈ 𝑛)
30	18, 29	sylbi 217	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ (ω ∖ 1o) → ∪ 𝑛 ∈ 𝑛)
31	30	adantr 480	. . . . . . . . . . 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 3578	. . . . . . . . . 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 6374	. . . . . . . . . . . . . . . . 17 ⊢ (∪ suc 𝑥 = 𝑥 → suc ∪ suc 𝑥 = suc 𝑥)
34	21, 33	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ω → suc ∪ suc 𝑥 = suc 𝑥)
35		suceq 6374	. . . . . . . . . . . . . . . . . 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 2747	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = suc 𝑥 → (suc ∪ 𝑛 = 𝑛 ↔ suc ∪ suc 𝑥 = suc 𝑥))
38	34, 37	syl5ibrcom 247	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ω → (𝑛 = suc 𝑥 → suc ∪ 𝑛 = 𝑛))
39	38	rexlimiv 3126	. . . . . . . . . . . . . 14 ⊢ (∃𝑥 ∈ ω 𝑛 = suc 𝑥 → suc ∪ 𝑛 = 𝑛)
40	18, 39	sylbi 217	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ (ω ∖ 1o) → suc ∪ 𝑛 = 𝑛)
41	40	fveq2d 6826	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ (ω ∖ 1o) → (𝑓‘suc ∪ 𝑛) = (𝑓‘𝑛))
42	41	adantr 480	. . . . . . . . . . 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 1234	. . . . . . . . . . 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 2766	. . . . . . . . . 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 5117	. . . . . . . . 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 6835	. . . . . . . . . 10 ⊢ (𝑓‘∪ 𝑛) ∈ V
47		vex 3440	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
48	46, 47	brelrn 5882	. . . . . . . . 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 412	. . . . . . 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 1934	. . . . . 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 3126	. . . . 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 1931	. . . 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 4020	. . 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 3981	. 2 ⊢ ran t++𝑅 ⊆ ran 𝑅
56		rnresv 6148	. . 3 ⊢ ran (𝑅 ↾ V) = ran 𝑅
57		relres 5954	. . . . . 6 ⊢ Rel (𝑅 ↾ V)
58		ssttrcl 9605	. . . . . 6 ⊢ (Rel (𝑅 ↾ V) → (𝑅 ↾ V) ⊆ t++(𝑅 ↾ V))
59	57, 58	ax-mp 5	. . . . 5 ⊢ (𝑅 ↾ V) ⊆ t++(𝑅 ↾ V)
60		ttrclresv 9607	. . . . 5 ⊢ t++(𝑅 ↾ V) = t++𝑅
61	59, 60	sseqtri 3983	. . . 4 ⊢ (𝑅 ↾ V) ⊆ t++𝑅
62	61	rnssi 5880	. . 3 ⊢ ran (𝑅 ↾ V) ⊆ ran t++𝑅
63	56, 62	eqsstrri 3982	. 2 ⊢ ran 𝑅 ⊆ ran t++𝑅
64	55, 63	eqssi 3951	1 ⊢ ran t++𝑅 = ran 𝑅

Syntax hints:   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541  ∃wex 1780   ∈ wcel 2111  {cab 2709  ∀wral 3047  ∃wrex 3056  Vcvv 3436   ∖ cdif 3899   ⊆ wss 3902  ∅c0 4283  ∪ cuni 4859   class class class wbr 5091  {copab 5153  ran crn 5617   ↾ cres 5618  Rel wrel 5621  Ord word 6305  suc csuc 6308   Fn wfn 6476  ‘cfv 6481  ωcom 7796  1_oc1o 8378  t++cttrcl 9597

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 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5217  ax-sep 5234  ax-nul 5244  ax-pr 5370  ax-un 7668

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 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-int 4898  df-iun 4943  df-br 5092  df-opab 5154  df-mpt 5173  df-tr 5199  df-id 5511  df-eprel 5516  df-po 5524  df-so 5525  df-fr 5569  df-we 5571  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-ov 7349  df-oprab 7350  df-mpo 7351  df-om 7797  df-2nd 7922  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329  df-1o 8385  df-oadd 8389  df-ttrcl 9598

This theorem is referenced by:  ttrclexg  9613