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Theorem rnttrcl 9524
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 9510 . . . . 5 t++𝑅 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑛 ∈ (ω ∖ 1o)∃𝑓(𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑥 ∧ (𝑓𝑛) = 𝑦) ∧ ∀𝑎𝑛 (𝑓𝑎)𝑅(𝑓‘suc 𝑎))}
21rneqi 5858 . . . 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 5875 . . . 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 6804 . . . . . . . . . . . 12 (𝑎 = 𝑛 → (𝑓𝑎) = (𝑓 𝑛))
6 suceq 6346 . . . . . . . . . . . . 13 (𝑎 = 𝑛 → suc 𝑎 = suc 𝑛)
76fveq2d 6808 . . . . . . . . . . . 12 (𝑎 = 𝑛 → (𝑓‘suc 𝑎) = (𝑓‘suc 𝑛))
85, 7breq12d 5094 . . . . . . . . . . 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 8328 . . . . . . . . . . . . . . . 16 1o = suc ∅
1110difeq2i 4060 . . . . . . . . . . . . . . 15 (ω ∖ 1o) = (ω ∖ suc ∅)
1211eleq2i 2828 . . . . . . . . . . . . . 14 (𝑛 ∈ (ω ∖ 1o) ↔ 𝑛 ∈ (ω ∖ suc ∅))
13 peano1 7767 . . . . . . . . . . . . . . 15 ∅ ∈ ω
14 eldifsucnn 8525 . . . . . . . . . . . . . . 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 4312 . . . . . . . . . . . . . . 15 (ω ∖ ∅) = ω
1716rexeqi 3359 . . . . . . . . . . . . . 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 7752 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ ω → Ord 𝑥)
20 ordunisuc 7711 . . . . . . . . . . . . . . . . 17 (Ord 𝑥 suc 𝑥 = 𝑥)
2119, 20syl 17 . . . . . . . . . . . . . . . 16 (𝑥 ∈ ω → suc 𝑥 = 𝑥)
22 vex 3441 . . . . . . . . . . . . . . . . 17 𝑥 ∈ V
2322sucid 6362 . . . . . . . . . . . . . . . 16 𝑥 ∈ suc 𝑥
2421, 23eqeltrdi 2845 . . . . . . . . . . . . . . 15 (𝑥 ∈ ω → suc 𝑥 ∈ suc 𝑥)
25 unieq 4855 . . . . . . . . . . . . . . . 16 (𝑛 = suc 𝑥 𝑛 = suc 𝑥)
26 id 22 . . . . . . . . . . . . . . . 16 (𝑛 = suc 𝑥𝑛 = suc 𝑥)
2725, 26eleq12d 2831 . . . . . . . . . . . . . . 15 (𝑛 = suc 𝑥 → ( 𝑛𝑛 suc 𝑥 ∈ suc 𝑥))
2824, 27syl5ibrcom 247 . . . . . . . . . . . . . 14 (𝑥 ∈ ω → (𝑛 = suc 𝑥 𝑛𝑛))
2928rexlimiv 3142 . . . . . . . . . . . . 13 (∃𝑥 ∈ ω 𝑛 = suc 𝑥 𝑛𝑛)
3018, 29sylbi 216 . . . . . . . . . . . 12 (𝑛 ∈ (ω ∖ 1o) → 𝑛𝑛)
3130adantr 482 . . . . . . . . . . 11 ((𝑛 ∈ (ω ∖ 1o) ∧ (𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑥 ∧ (𝑓𝑛) = 𝑦) ∧ ∀𝑎𝑛 (𝑓𝑎)𝑅(𝑓‘suc 𝑎))) → 𝑛𝑛)
328, 9, 31rspcdva 3567 . . . . . . . . . 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 6346 . . . . . . . . . . . . . . . . 17 ( suc 𝑥 = 𝑥 → suc suc 𝑥 = suc 𝑥)
3421, 33syl 17 . . . . . . . . . . . . . . . 16 (𝑥 ∈ ω → suc suc 𝑥 = suc 𝑥)
35 suceq 6346 . . . . . . . . . . . . . . . . . 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 3142 . . . . . . . . . . . . . 14 (∃𝑥 ∈ ω 𝑛 = suc 𝑥 → suc 𝑛 = 𝑛)
4018, 39sylbi 216 . . . . . . . . . . . . 13 (𝑛 ∈ (ω ∖ 1o) → suc 𝑛 = 𝑛)
4140fveq2d 6808 . . . . . . . . . . . 12 (𝑛 ∈ (ω ∖ 1o) → (𝑓‘suc 𝑛) = (𝑓𝑛))
4241adantr 482 . . . . . . . . . . 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 5107 . . . . . . . . 9 ((𝑛 ∈ (ω ∖ 1o) ∧ (𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑥 ∧ (𝑓𝑛) = 𝑦) ∧ ∀𝑎𝑛 (𝑓𝑎)𝑅(𝑓‘suc 𝑎))) → (𝑓 𝑛)𝑅𝑦)
46 fvex 6817 . . . . . . . . . 10 (𝑓 𝑛) ∈ V
47 vex 3441 . . . . . . . . . 10 𝑦 ∈ V
4846, 47brelrn 5863 . . . . . . . . 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 414 . . . . . . 7 (𝑛 ∈ (ω ∖ 1o) → ((𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑥 ∧ (𝑓𝑛) = 𝑦) ∧ ∀𝑎𝑛 (𝑓𝑎)𝑅(𝑓‘suc 𝑎)) → 𝑦 ∈ ran 𝑅))
5150exlimdv 1934 . . . . . 6 (𝑛 ∈ (ω ∖ 1o) → (∃𝑓(𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑥 ∧ (𝑓𝑛) = 𝑦) ∧ ∀𝑎𝑛 (𝑓𝑎)𝑅(𝑓‘suc 𝑎)) → 𝑦 ∈ ran 𝑅))
5251rexlimiv 3142 . . . . 5 (∃𝑛 ∈ (ω ∖ 1o)∃𝑓(𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑥 ∧ (𝑓𝑛) = 𝑦) ∧ ∀𝑎𝑛 (𝑓𝑎)𝑅(𝑓‘suc 𝑎)) → 𝑦 ∈ ran 𝑅)
5352exlimiv 1931 . . . 4 (∃𝑥𝑛 ∈ (ω ∖ 1o)∃𝑓(𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑥 ∧ (𝑓𝑛) = 𝑦) ∧ ∀𝑎𝑛 (𝑓𝑎)𝑅(𝑓‘suc 𝑎)) → 𝑦 ∈ ran 𝑅)
5453abssi 4009 . . 3 {𝑦 ∣ ∃𝑥𝑛 ∈ (ω ∖ 1o)∃𝑓(𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑥 ∧ (𝑓𝑛) = 𝑦) ∧ ∀𝑎𝑛 (𝑓𝑎)𝑅(𝑓‘suc 𝑎))} ⊆ ran 𝑅
554, 54eqsstri 3960 . 2 ran t++𝑅 ⊆ ran 𝑅
56 rnresv 6119 . . 3 ran (𝑅 ↾ V) = ran 𝑅
57 relres 5932 . . . . . 6 Rel (𝑅 ↾ V)
58 ssttrcl 9517 . . . . . 6 (Rel (𝑅 ↾ V) → (𝑅 ↾ V) ⊆ t++(𝑅 ↾ V))
5957, 58ax-mp 5 . . . . 5 (𝑅 ↾ V) ⊆ t++(𝑅 ↾ V)
60 ttrclresv 9519 . . . . 5 t++(𝑅 ↾ V) = t++𝑅
6159, 60sseqtri 3962 . . . 4 (𝑅 ↾ V) ⊆ t++𝑅
6261rnssi 5861 . . 3 ran (𝑅 ↾ V) ⊆ ran t++𝑅
6356, 62eqsstrri 3961 . 2 ran 𝑅 ⊆ ran t++𝑅
6455, 63eqssi 3942 1 ran t++𝑅 = ran 𝑅
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 397  w3a 1087   = wceq 1539  wex 1779  wcel 2104  {cab 2713  wral 3062  wrex 3071  Vcvv 3437  cdif 3889  wss 3892  c0 4262   cuni 4844   class class class wbr 5081  {copab 5143  ran crn 5601  cres 5602  Rel wrel 5605  Ord word 6280  suc csuc 6283   Fn wfn 6453  cfv 6458  ωcom 7744  1oc1o 8321  t++cttrcl 9509
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2707  ax-rep 5218  ax-sep 5232  ax-nul 5239  ax-pr 5361  ax-un 7620
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3or 1088  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3286  df-rab 3287  df-v 3439  df-sbc 3722  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-pss 3911  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4566  df-pr 4568  df-op 4572  df-uni 4845  df-int 4887  df-iun 4933  df-br 5082  df-opab 5144  df-mpt 5165  df-tr 5199  df-id 5500  df-eprel 5506  df-po 5514  df-so 5515  df-fr 5555  df-we 5557  df-xp 5606  df-rel 5607  df-cnv 5608  df-co 5609  df-dm 5610  df-rn 5611  df-res 5612  df-ima 5613  df-pred 6217  df-ord 6284  df-on 6285  df-lim 6286  df-suc 6287  df-iota 6410  df-fun 6460  df-fn 6461  df-f 6462  df-f1 6463  df-fo 6464  df-f1o 6465  df-fv 6466  df-ov 7310  df-oprab 7311  df-mpo 7312  df-om 7745  df-2nd 7864  df-frecs 8128  df-wrecs 8159  df-recs 8233  df-rdg 8272  df-1o 8328  df-oadd 8332  df-ttrcl 9510
This theorem is referenced by:  ttrclexg  9525
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