Theorem rnttrcl 9679

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 9665	. . . . 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 5918	. . . 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 5935	. . . 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 2788	. . 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 6871	. . . . . . . . . . . 12 ⊢ (𝑎 = ∪ 𝑛 → (𝑓‘𝑎) = (𝑓‘∪ 𝑛))
6		suceq 6418	. . . . . . . . . . . . 13 ⊢ (𝑎 = ∪ 𝑛 → suc 𝑎 = suc ∪ 𝑛)
7	6	fveq2d 6875	. . . . . . . . . . . 12 ⊢ (𝑎 = ∪ 𝑛 → (𝑓‘suc 𝑎) = (𝑓‘suc ∪ 𝑛))
8	5, 7	breq12d 5118	. . . . . . . . . . 11 ⊢ (𝑎 = ∪ 𝑛 → ((𝑓‘𝑎)𝑅(𝑓‘suc 𝑎) ↔ (𝑓‘∪ 𝑛)𝑅(𝑓‘suc ∪ 𝑛)))
9		simpr3 1213	. . . . . . . . . . 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 8441	. . . . . . . . . . . . . . . 16 ⊢ 1o = suc ∅
11	10	difeq2i 4080	. . . . . . . . . . . . . . 15 ⊢ (ω ∖ 1o) = (ω ∖ suc ∅)
12	11	eleq2i 2857	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ (ω ∖ 1o) ↔ 𝑛 ∈ (ω ∖ suc ∅))
13		peano1 7873	. . . . . . . . . . . . . . 15 ⊢ ∅ ∈ ω
14		eldifsucnn 8638	. . . . . . . . . . . . . . 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 4334	. . . . . . . . . . . . . . 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 7858	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ ω → Ord 𝑥)
20		ordunisuc 7816	. . . . . . . . . . . . . . . . 17 ⊢ (Ord 𝑥 → ∪ suc 𝑥 = 𝑥)
21	19, 20	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ω → ∪ suc 𝑥 = 𝑥)
22		vex 3461	. . . . . . . . . . . . . . . . 17 ⊢ 𝑥 ∈ V
23	22	sucid 6434	. . . . . . . . . . . . . . . 16 ⊢ 𝑥 ∈ suc 𝑥
24	21, 23	eqeltrdi 2873	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ω → ∪ suc 𝑥 ∈ suc 𝑥)
25		unieq 4879	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = suc 𝑥 → ∪ 𝑛 = ∪ suc 𝑥)
26		id 23	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = suc 𝑥 → 𝑛 = suc 𝑥)
27	25, 26	eleq12d 2859	. . . . . . . . . . . . . . 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 3159	. . . . . . . . . . . . 13 ⊢ (∃𝑥 ∈ ω 𝑛 = suc 𝑥 → ∪ 𝑛 ∈ 𝑛)
30	18, 29	sylbi 220	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ (ω ∖ 1o) → ∪ 𝑛 ∈ 𝑛)
31	30	adantr 485	. . . . . . . . . . 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 3585	. . . . . . . . . 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 6418	. . . . . . . . . . . . . . . . 17 ⊢ (∪ suc 𝑥 = 𝑥 → suc ∪ suc 𝑥 = suc 𝑥)
34	21, 33	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ω → suc ∪ suc 𝑥 = suc 𝑥)
35		suceq 6418	. . . . . . . . . . . . . . . . . 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 2781	. . . . . . . . . . . . . . . 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 3159	. . . . . . . . . . . . . 14 ⊢ (∃𝑥 ∈ ω 𝑛 = suc 𝑥 → suc ∪ 𝑛 = 𝑛)
40	18, 39	sylbi 220	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ (ω ∖ 1o) → suc ∪ 𝑛 = 𝑛)
41	40	fveq2d 6875	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ (ω ∖ 1o) → (𝑓‘suc ∪ 𝑛) = (𝑓‘𝑛))
42	41	adantr 485	. . . . . . . . . . 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 1250	. . . . . . . . . . 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 2800	. . . . . . . . . 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 5131	. . . . . . . . 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 6884	. . . . . . . . . 10 ⊢ (𝑓‘∪ 𝑛) ∈ V
47		vex 3461	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
48	46, 47	brelrn 5923	. . . . . . . . 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 417	. . . . . . 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 1956	. . . . . 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 3159	. . . . 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 1953	. . . 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 4024	. . 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 3985	. 2 ⊢ ran t++𝑅 ⊆ ran 𝑅
56		rnresv 6192	. . 3 ⊢ ran (𝑅 ↾ V) = ran 𝑅
57		relres 5995	. . . . . 6 ⊢ Rel (𝑅 ↾ V)
58		ssttrcl 9672	. . . . . 6 ⊢ (Rel (𝑅 ↾ V) → (𝑅 ↾ V) ⊆ t++(𝑅 ↾ V))
59	57, 58	ax-mp 5	. . . . 5 ⊢ (𝑅 ↾ V) ⊆ t++(𝑅 ↾ V)
60		ttrclresv 9674	. . . . 5 ⊢ t++(𝑅 ↾ V) = t++𝑅
61	59, 60	sseqtri 3987	. . . 4 ⊢ (𝑅 ↾ V) ⊆ t++𝑅
62	61	rnssi 5921	. . 3 ⊢ ran (𝑅 ↾ V) ⊆ ran t++𝑅
63	56, 62	eqsstrri 3986	. 2 ⊢ ran 𝑅 ⊆ ran t++𝑅
64	55, 63	eqssi 3955	1 ⊢ ran t++𝑅 = ran 𝑅

Syntax hints:   ↔ wb 209   ∧ wa 400   ∧ w3a 1101   = wceq 1563  ∃wex 1802   ∈ wcel 2145  {cab 2743  ∀wral 3079  ∃wrex 3089  Vcvv 3457   ∖ cdif 3904   ⊆ wss 3907  ∅c0 4288  ∪ cuni 4868   class class class wbr 5105  {copab 5167  ran crn 5653   ↾ cres 5654  Rel wrel 5657  Ord word 6349  suc csuc 6352   Fn wfn 6520  ‘cfv 6525  ωcom 7850  1_oc1o 8434  t++cttrcl 9664

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5232  ax-sep 5251  ax-nul 5261  ax-pr 5395  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-pss 3927  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-int 4909  df-iun 4954  df-br 5106  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5547  df-eprel 5552  df-po 5560  df-so 5561  df-fr 5605  df-we 5607  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-pred 6292  df-ord 6353  df-on 6354  df-lim 6355  df-suc 6356  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-ov 7403  df-oprab 7404  df-mpo 7405  df-om 7851  df-2nd 7975  df-frecs 8266  df-wrecs 8297  df-recs 8346  df-rdg 8385  df-1o 8441  df-oadd 8445  df-ttrcl 9665

This theorem is referenced by:  ttrclexg  9680