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Theorem rnttrcl 9763
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.
StepHypRef Expression
1 df-ttrcl 9749 . . . . 5 t++𝑅 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑛 ∈ (ω ∖ 1o)∃𝑓(𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑥 ∧ (𝑓𝑛) = 𝑦) ∧ ∀𝑎𝑛 (𝑓𝑎)𝑅(𝑓‘suc 𝑎))}
21rneqi 5947 . . . 4 ran t++𝑅 = ran {⟨𝑥, 𝑦⟩ ∣ ∃𝑛 ∈ (ω ∖ 1o)∃𝑓(𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑥 ∧ (𝑓𝑛) = 𝑦) ∧ ∀𝑎𝑛 (𝑓𝑎)𝑅(𝑓‘suc 𝑎))}
3 rnopab 5964 . . . 4 ran {⟨𝑥, 𝑦⟩ ∣ ∃𝑛 ∈ (ω ∖ 1o)∃𝑓(𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑥 ∧ (𝑓𝑛) = 𝑦) ∧ ∀𝑎𝑛 (𝑓𝑎)𝑅(𝑓‘suc 𝑎))} = {𝑦 ∣ ∃𝑥𝑛 ∈ (ω ∖ 1o)∃𝑓(𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑥 ∧ (𝑓𝑛) = 𝑦) ∧ ∀𝑎𝑛 (𝑓𝑎)𝑅(𝑓‘suc 𝑎))}
42, 3eqtri 2764 . . 3 ran t++𝑅 = {𝑦 ∣ ∃𝑥𝑛 ∈ (ω ∖ 1o)∃𝑓(𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑥 ∧ (𝑓𝑛) = 𝑦) ∧ ∀𝑎𝑛 (𝑓𝑎)𝑅(𝑓‘suc 𝑎))}
5 fveq2 6905 . . . . . . . . . . . 12 (𝑎 = 𝑛 → (𝑓𝑎) = (𝑓 𝑛))
6 suceq 6449 . . . . . . . . . . . . 13 (𝑎 = 𝑛 → suc 𝑎 = suc 𝑛)
76fveq2d 6909 . . . . . . . . . . . 12 (𝑎 = 𝑛 → (𝑓‘suc 𝑎) = (𝑓‘suc 𝑛))
85, 7breq12d 5155 . . . . . . . . . . 11 (𝑎 = 𝑛 → ((𝑓𝑎)𝑅(𝑓‘suc 𝑎) ↔ (𝑓 𝑛)𝑅(𝑓‘suc 𝑛)))
9 simpr3 1196 . . . . . . . . . . 11 ((𝑛 ∈ (ω ∖ 1o) ∧ (𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑥 ∧ (𝑓𝑛) = 𝑦) ∧ ∀𝑎𝑛 (𝑓𝑎)𝑅(𝑓‘suc 𝑎))) → ∀𝑎𝑛 (𝑓𝑎)𝑅(𝑓‘suc 𝑎))
10 df-1o 8507 . . . . . . . . . . . . . . . 16 1o = suc ∅
1110difeq2i 4122 . . . . . . . . . . . . . . 15 (ω ∖ 1o) = (ω ∖ suc ∅)
1211eleq2i 2832 . . . . . . . . . . . . . 14 (𝑛 ∈ (ω ∖ 1o) ↔ 𝑛 ∈ (ω ∖ suc ∅))
13 peano1 7911 . . . . . . . . . . . . . . 15 ∅ ∈ ω
14 eldifsucnn 8703 . . . . . . . . . . . . . . 15 (∅ ∈ ω → (𝑛 ∈ (ω ∖ suc ∅) ↔ ∃𝑥 ∈ (ω ∖ ∅)𝑛 = suc 𝑥))
1513, 14ax-mp 5 . . . . . . . . . . . . . 14 (𝑛 ∈ (ω ∖ suc ∅) ↔ ∃𝑥 ∈ (ω ∖ ∅)𝑛 = suc 𝑥)
16 dif0 4377 . . . . . . . . . . . . . . 15 (ω ∖ ∅) = ω
1716rexeqi 3324 . . . . . . . . . . . . . 14 (∃𝑥 ∈ (ω ∖ ∅)𝑛 = suc 𝑥 ↔ ∃𝑥 ∈ ω 𝑛 = suc 𝑥)
1812, 15, 173bitri 297 . . . . . . . . . . . . 13 (𝑛 ∈ (ω ∖ 1o) ↔ ∃𝑥 ∈ ω 𝑛 = suc 𝑥)
19 nnord 7896 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ ω → Ord 𝑥)
20 ordunisuc 7853 . . . . . . . . . . . . . . . . 17 (Ord 𝑥 suc 𝑥 = 𝑥)
2119, 20syl 17 . . . . . . . . . . . . . . . 16 (𝑥 ∈ ω → suc 𝑥 = 𝑥)
22 vex 3483 . . . . . . . . . . . . . . . . 17 𝑥 ∈ V
2322sucid 6465 . . . . . . . . . . . . . . . 16 𝑥 ∈ suc 𝑥
2421, 23eqeltrdi 2848 . . . . . . . . . . . . . . 15 (𝑥 ∈ ω → suc 𝑥 ∈ suc 𝑥)
25 unieq 4917 . . . . . . . . . . . . . . . 16 (𝑛 = suc 𝑥 𝑛 = suc 𝑥)
26 id 22 . . . . . . . . . . . . . . . 16 (𝑛 = suc 𝑥𝑛 = suc 𝑥)
2725, 26eleq12d 2834 . . . . . . . . . . . . . . 15 (𝑛 = suc 𝑥 → ( 𝑛𝑛 suc 𝑥 ∈ suc 𝑥))
2824, 27syl5ibrcom 247 . . . . . . . . . . . . . 14 (𝑥 ∈ ω → (𝑛 = suc 𝑥 𝑛𝑛))
2928rexlimiv 3147 . . . . . . . . . . . . 13 (∃𝑥 ∈ ω 𝑛 = suc 𝑥 𝑛𝑛)
3018, 29sylbi 217 . . . . . . . . . . . 12 (𝑛 ∈ (ω ∖ 1o) → 𝑛𝑛)
3130adantr 480 . . . . . . . . . . 11 ((𝑛 ∈ (ω ∖ 1o) ∧ (𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑥 ∧ (𝑓𝑛) = 𝑦) ∧ ∀𝑎𝑛 (𝑓𝑎)𝑅(𝑓‘suc 𝑎))) → 𝑛𝑛)
328, 9, 31rspcdva 3622 . . . . . . . . . 10 ((𝑛 ∈ (ω ∖ 1o) ∧ (𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑥 ∧ (𝑓𝑛) = 𝑦) ∧ ∀𝑎𝑛 (𝑓𝑎)𝑅(𝑓‘suc 𝑎))) → (𝑓 𝑛)𝑅(𝑓‘suc 𝑛))
33 suceq 6449 . . . . . . . . . . . . . . . . 17 ( suc 𝑥 = 𝑥 → suc suc 𝑥 = suc 𝑥)
3421, 33syl 17 . . . . . . . . . . . . . . . 16 (𝑥 ∈ ω → suc suc 𝑥 = suc 𝑥)
35 suceq 6449 . . . . . . . . . . . . . . . . . 18 ( 𝑛 = suc 𝑥 → suc 𝑛 = suc suc 𝑥)
3625, 35syl 17 . . . . . . . . . . . . . . . . 17 (𝑛 = suc 𝑥 → suc 𝑛 = suc suc 𝑥)
3736, 26eqeq12d 2752 . . . . . . . . . . . . . . . 16 (𝑛 = suc 𝑥 → (suc 𝑛 = 𝑛 ↔ suc suc 𝑥 = suc 𝑥))
3834, 37syl5ibrcom 247 . . . . . . . . . . . . . . 15 (𝑥 ∈ ω → (𝑛 = suc 𝑥 → suc 𝑛 = 𝑛))
3938rexlimiv 3147 . . . . . . . . . . . . . 14 (∃𝑥 ∈ ω 𝑛 = suc 𝑥 → suc 𝑛 = 𝑛)
4018, 39sylbi 217 . . . . . . . . . . . . 13 (𝑛 ∈ (ω ∖ 1o) → suc 𝑛 = 𝑛)
4140fveq2d 6909 . . . . . . . . . . . 12 (𝑛 ∈ (ω ∖ 1o) → (𝑓‘suc 𝑛) = (𝑓𝑛))
4241adantr 480 . . . . . . . . . . 11 ((𝑛 ∈ (ω ∖ 1o) ∧ (𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑥 ∧ (𝑓𝑛) = 𝑦) ∧ ∀𝑎𝑛 (𝑓𝑎)𝑅(𝑓‘suc 𝑎))) → (𝑓‘suc 𝑛) = (𝑓𝑛))
43 simpr2r 1233 . . . . . . . . . . 11 ((𝑛 ∈ (ω ∖ 1o) ∧ (𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑥 ∧ (𝑓𝑛) = 𝑦) ∧ ∀𝑎𝑛 (𝑓𝑎)𝑅(𝑓‘suc 𝑎))) → (𝑓𝑛) = 𝑦)
4442, 43eqtrd 2776 . . . . . . . . . 10 ((𝑛 ∈ (ω ∖ 1o) ∧ (𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑥 ∧ (𝑓𝑛) = 𝑦) ∧ ∀𝑎𝑛 (𝑓𝑎)𝑅(𝑓‘suc 𝑎))) → (𝑓‘suc 𝑛) = 𝑦)
4532, 44breqtrd 5168 . . . . . . . . 9 ((𝑛 ∈ (ω ∖ 1o) ∧ (𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑥 ∧ (𝑓𝑛) = 𝑦) ∧ ∀𝑎𝑛 (𝑓𝑎)𝑅(𝑓‘suc 𝑎))) → (𝑓 𝑛)𝑅𝑦)
46 fvex 6918 . . . . . . . . . 10 (𝑓 𝑛) ∈ V
47 vex 3483 . . . . . . . . . 10 𝑦 ∈ V
4846, 47brelrn 5952 . . . . . . . . 9 ((𝑓 𝑛)𝑅𝑦𝑦 ∈ ran 𝑅)
4945, 48syl 17 . . . . . . . 8 ((𝑛 ∈ (ω ∖ 1o) ∧ (𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑥 ∧ (𝑓𝑛) = 𝑦) ∧ ∀𝑎𝑛 (𝑓𝑎)𝑅(𝑓‘suc 𝑎))) → 𝑦 ∈ ran 𝑅)
5049ex 412 . . . . . . 7 (𝑛 ∈ (ω ∖ 1o) → ((𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑥 ∧ (𝑓𝑛) = 𝑦) ∧ ∀𝑎𝑛 (𝑓𝑎)𝑅(𝑓‘suc 𝑎)) → 𝑦 ∈ ran 𝑅))
5150exlimdv 1932 . . . . . 6 (𝑛 ∈ (ω ∖ 1o) → (∃𝑓(𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑥 ∧ (𝑓𝑛) = 𝑦) ∧ ∀𝑎𝑛 (𝑓𝑎)𝑅(𝑓‘suc 𝑎)) → 𝑦 ∈ ran 𝑅))
5251rexlimiv 3147 . . . . 5 (∃𝑛 ∈ (ω ∖ 1o)∃𝑓(𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑥 ∧ (𝑓𝑛) = 𝑦) ∧ ∀𝑎𝑛 (𝑓𝑎)𝑅(𝑓‘suc 𝑎)) → 𝑦 ∈ ran 𝑅)
5352exlimiv 1929 . . . 4 (∃𝑥𝑛 ∈ (ω ∖ 1o)∃𝑓(𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑥 ∧ (𝑓𝑛) = 𝑦) ∧ ∀𝑎𝑛 (𝑓𝑎)𝑅(𝑓‘suc 𝑎)) → 𝑦 ∈ ran 𝑅)
5453abssi 4069 . . 3 {𝑦 ∣ ∃𝑥𝑛 ∈ (ω ∖ 1o)∃𝑓(𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑥 ∧ (𝑓𝑛) = 𝑦) ∧ ∀𝑎𝑛 (𝑓𝑎)𝑅(𝑓‘suc 𝑎))} ⊆ ran 𝑅
554, 54eqsstri 4029 . 2 ran t++𝑅 ⊆ ran 𝑅
56 rnresv 6220 . . 3 ran (𝑅 ↾ V) = ran 𝑅
57 relres 6022 . . . . . 6 Rel (𝑅 ↾ V)
58 ssttrcl 9756 . . . . . 6 (Rel (𝑅 ↾ V) → (𝑅 ↾ V) ⊆ t++(𝑅 ↾ V))
5957, 58ax-mp 5 . . . . 5 (𝑅 ↾ V) ⊆ t++(𝑅 ↾ V)
60 ttrclresv 9758 . . . . 5 t++(𝑅 ↾ V) = t++𝑅
6159, 60sseqtri 4031 . . . 4 (𝑅 ↾ V) ⊆ t++𝑅
6261rnssi 5950 . . 3 ran (𝑅 ↾ V) ⊆ ran t++𝑅
6356, 62eqsstrri 4030 . 2 ran 𝑅 ⊆ ran t++𝑅
6455, 63eqssi 3999 1 ran t++𝑅 = ran 𝑅
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395  w3a 1086   = wceq 1539  wex 1778  wcel 2107  {cab 2713  wral 3060  wrex 3069  Vcvv 3479  cdif 3947  wss 3950  c0 4332   cuni 4906   class class class wbr 5142  {copab 5204  ran crn 5685  cres 5686  Rel wrel 5689  Ord word 6382  suc csuc 6385   Fn wfn 6555  cfv 6560  ωcom 7888  1oc1o 8500  t++cttrcl 9748
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-int 4946  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-ov 7435  df-oprab 7436  df-mpo 7437  df-om 7889  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-1o 8507  df-oadd 8511  df-ttrcl 9749
This theorem is referenced by:  ttrclexg  9764
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