Theorem rnttrcl 9753

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 9739	. . . . 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 5927	. . . 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 5944	. . . 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 2786	. . 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 6911	. . . . . . . . . . . 12 ⊢ (𝑎 = ∪ 𝑛 → (𝑓‘𝑎) = (𝑓‘∪ 𝑛))
6		suceq 6452	. . . . . . . . . . . . 13 ⊢ (𝑎 = ∪ 𝑛 → suc 𝑎 = suc ∪ 𝑛)
7	6	fveq2d 6915	. . . . . . . . . . . 12 ⊢ (𝑎 = ∪ 𝑛 → (𝑓‘suc 𝑎) = (𝑓‘suc ∪ 𝑛))
8	5, 7	breq12d 5119	. . . . . . . . . . 11 ⊢ (𝑎 = ∪ 𝑛 → ((𝑓‘𝑎)𝑅(𝑓‘suc 𝑎) ↔ (𝑓‘∪ 𝑛)𝑅(𝑓‘suc ∪ 𝑛)))
9		simpr3 1215	. . . . . . . . . . 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 8507	. . . . . . . . . . . . . . . 16 ⊢ 1o = suc ∅
11	10	difeq2i 4063	. . . . . . . . . . . . . . 15 ⊢ (ω ∖ 1o) = (ω ∖ suc ∅)
12	11	eleq2i 2856	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ (ω ∖ 1o) ↔ 𝑛 ∈ (ω ∖ suc ∅))
13		peano1 7927	. . . . . . . . . . . . . . 15 ⊢ ∅ ∈ ω
14		eldifsucnn 8704	. . . . . . . . . . . . . . 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 4330	. . . . . . . . . . . . . . 15 ⊢ (ω ∖ ∅) = ω
17	16	rexeqi 3322	. . . . . . . . . . . . . 14 ⊢ (∃𝑥 ∈ (ω ∖ ∅)𝑛 = suc 𝑥 ↔ ∃𝑥 ∈ ω 𝑛 = suc 𝑥)
18	12, 15, 17	3bitri 300	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ (ω ∖ 1o) ↔ ∃𝑥 ∈ ω 𝑛 = suc 𝑥)
19		nnord 7912	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ ω → Ord 𝑥)
20		ordunisuc 7870	. . . . . . . . . . . . . . . . 17 ⊢ (Ord 𝑥 → ∪ suc 𝑥 = 𝑥)
21	19, 20	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ω → ∪ suc 𝑥 = 𝑥)
22		vex 3458	. . . . . . . . . . . . . . . . 17 ⊢ 𝑥 ∈ V
23	22	sucid 6468	. . . . . . . . . . . . . . . 16 ⊢ 𝑥 ∈ suc 𝑥
24	21, 23	eqeltrdi 2872	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ω → ∪ suc 𝑥 ∈ suc 𝑥)
25		unieq 4881	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = suc 𝑥 → ∪ 𝑛 = ∪ suc 𝑥)
26		id 23	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = suc 𝑥 → 𝑛 = suc 𝑥)
27	25, 26	eleq12d 2858	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = suc 𝑥 → (∪ 𝑛 ∈ 𝑛 ↔ ∪ suc 𝑥 ∈ suc 𝑥))
28	24, 27	syl5ibrcom 250	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ω → (𝑛 = suc 𝑥 → ∪ 𝑛 ∈ 𝑛))
29	28	rexlimiv 3160	. . . . . . . . . . . . 13 ⊢ (∃𝑥 ∈ ω 𝑛 = suc 𝑥 → ∪ 𝑛 ∈ 𝑛)
30	18, 29	sylbi 220	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ (ω ∖ 1o) → ∪ 𝑛 ∈ 𝑛)
31	30	adantr 486	. . . . . . . . . . 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 3581	. . . . . . . . . 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 6452	. . . . . . . . . . . . . . . . 17 ⊢ (∪ suc 𝑥 = 𝑥 → suc ∪ suc 𝑥 = suc 𝑥)
34	21, 33	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ω → suc ∪ suc 𝑥 = suc 𝑥)
35		suceq 6452	. . . . . . . . . . . . . . . . . 18 ⊢ (∪ 𝑛 = ∪ suc 𝑥 → suc ∪ 𝑛 = suc ∪ suc 𝑥)
36	25, 35	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = suc 𝑥 → suc ∪ 𝑛 = suc ∪ suc 𝑥)
37	36, 26	eqeq12d 2779	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = suc 𝑥 → (suc ∪ 𝑛 = 𝑛 ↔ suc ∪ suc 𝑥 = suc 𝑥))
38	34, 37	syl5ibrcom 250	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ω → (𝑛 = suc 𝑥 → suc ∪ 𝑛 = 𝑛))
39	38	rexlimiv 3160	. . . . . . . . . . . . . 14 ⊢ (∃𝑥 ∈ ω 𝑛 = suc 𝑥 → suc ∪ 𝑛 = 𝑛)
40	18, 39	sylbi 220	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ (ω ∖ 1o) → suc ∪ 𝑛 = 𝑛)
41	40	fveq2d 6915	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ (ω ∖ 1o) → (𝑓‘suc ∪ 𝑛) = (𝑓‘𝑛))
42	41	adantr 486	. . . . . . . . . . 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 1252	. . . . . . . . . . 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 2798	. . . . . . . . . 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 5135	. . . . . . . . 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 6924	. . . . . . . . . 10 ⊢ (𝑓‘∪ 𝑛) ∈ V
47		vex 3458	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
48	46, 47	brelrn 5932	. . . . . . . . 9 ⊢ ((𝑓‘∪ 𝑛)𝑅𝑦 → 𝑦 ∈ ran 𝑅)
49	45, 48	syl 18	. . . . . . . 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 418	. . . . . . 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 1968	. . . . . 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 3160	. . . . 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 1965	. . . 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 3996	. . 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 3959	. 2 ⊢ ran t++𝑅 ⊆ ran 𝑅
56		rnresv 6209	. . 3 ⊢ ran (𝑅 ↾ V) = ran 𝑅
57		relres 6008	. . . . . 6 ⊢ Rel (𝑅 ↾ V)
58		ssttrcl 9746	. . . . . 6 ⊢ (Rel (𝑅 ↾ V) → (𝑅 ↾ V) ⊆ t++(𝑅 ↾ V))
59	57, 58	ax-mp 5	. . . . 5 ⊢ (𝑅 ↾ V) ⊆ t++(𝑅 ↾ V)
60		ttrclresv 9748	. . . . 5 ⊢ t++(𝑅 ↾ V) = t++𝑅
61	59, 60	sseqtri 3961	. . . 4 ⊢ (𝑅 ↾ V) ⊆ t++𝑅
62	61	rnssi 5930	. . 3 ⊢ ran (𝑅 ↾ V) ⊆ ran t++𝑅
63	56, 62	eqsstrri 3960	. 2 ⊢ ran 𝑅 ⊆ ran t++𝑅
64	55, 63	eqssi 3929	1 ⊢ ran t++𝑅 = ran 𝑅

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570  ∃wex 1812   ∈ wcel 2147  {cab 2741  ∀wral 3080  ∃wrex 3090  Vcvv 3454   ⊆ wss 3900   ∖ cdif 4048  ∅c0 4281  ∪ cuni 4870   class class class wbr 5106  {copab 5171  ran crn 5657   ↾ cres 5658  Rel wrel 5661  Ord word 6382  suc csuc 6385   Fn wfn 6556  ‘cfv 6561  ωcom 7904  1_oc1o 8500  t++cttrcl 9738

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5236  ax-sep 5253  ax-nul 5264  ax-pr 5396  ax-un 7778

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5551  df-eprel 5556  df-po 5564  df-so 5565  df-fr 5609  df-we 5611  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-pred 6325  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6515  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7449  df-oprab 7450  df-mpo 7451  df-om 7905  df-2nd 8029  df-frecs 8330  df-wrecs 8361  df-recs 8410  df-rdg 8449  df-1o 8507  df-oadd 8511  df-ttrcl 9739

This theorem is used by:  ttrclexg  9754