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Theorem rnttrcl 33544
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 33530 . . . . 5 t++𝑅 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑛 ∈ (ω ∖ 1o)∃𝑓(𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑥 ∧ (𝑓𝑛) = 𝑦) ∧ ∀𝑎𝑛 (𝑓𝑎)𝑅(𝑓‘suc 𝑎))}
21rneqi 5821 . . . 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 5838 . . . 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 2766 . . 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 6736 . . . . . . . . . . . 12 (𝑎 = 𝑛 → (𝑓𝑎) = (𝑓 𝑛))
6 suceq 6296 . . . . . . . . . . . . 13 (𝑎 = 𝑛 → suc 𝑎 = suc 𝑛)
76fveq2d 6740 . . . . . . . . . . . 12 (𝑎 = 𝑛 → (𝑓‘suc 𝑎) = (𝑓‘suc 𝑛))
85, 7breq12d 5081 . . . . . . . . . . 11 (𝑎 = 𝑛 → ((𝑓𝑎)𝑅(𝑓‘suc 𝑎) ↔ (𝑓 𝑛)𝑅(𝑓‘suc 𝑛)))
9 simpr3 1198 . . . . . . . . . . 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 8223 . . . . . . . . . . . . . . . 16 1o = suc ∅
1110difeq2i 4049 . . . . . . . . . . . . . . 15 (ω ∖ 1o) = (ω ∖ suc ∅)
1211eleq2i 2830 . . . . . . . . . . . . . 14 (𝑛 ∈ (ω ∖ 1o) ↔ 𝑛 ∈ (ω ∖ suc ∅))
13 peano1 7686 . . . . . . . . . . . . . . 15 ∅ ∈ ω
14 eldifsucnn 33433 . . . . . . . . . . . . . . 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 4302 . . . . . . . . . . . . . . 15 (ω ∖ ∅) = ω
1716rexeqi 3337 . . . . . . . . . . . . . 14 (∃𝑥 ∈ (ω ∖ ∅)𝑛 = suc 𝑥 ↔ ∃𝑥 ∈ ω 𝑛 = suc 𝑥)
1812, 15, 173bitri 300 . . . . . . . . . . . . 13 (𝑛 ∈ (ω ∖ 1o) ↔ ∃𝑥 ∈ ω 𝑛 = suc 𝑥)
19 nnord 7671 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ ω → Ord 𝑥)
20 ordunisuc 7630 . . . . . . . . . . . . . . . . 17 (Ord 𝑥 suc 𝑥 = 𝑥)
2119, 20syl 17 . . . . . . . . . . . . . . . 16 (𝑥 ∈ ω → suc 𝑥 = 𝑥)
22 vex 3425 . . . . . . . . . . . . . . . . 17 𝑥 ∈ V
2322sucid 6310 . . . . . . . . . . . . . . . 16 𝑥 ∈ suc 𝑥
2421, 23eqeltrdi 2847 . . . . . . . . . . . . . . 15 (𝑥 ∈ ω → suc 𝑥 ∈ suc 𝑥)
25 unieq 4845 . . . . . . . . . . . . . . . 16 (𝑛 = suc 𝑥 𝑛 = suc 𝑥)
26 id 22 . . . . . . . . . . . . . . . 16 (𝑛 = suc 𝑥𝑛 = suc 𝑥)
2725, 26eleq12d 2833 . . . . . . . . . . . . . . 15 (𝑛 = suc 𝑥 → ( 𝑛𝑛 suc 𝑥 ∈ suc 𝑥))
2824, 27syl5ibrcom 250 . . . . . . . . . . . . . 14 (𝑥 ∈ ω → (𝑛 = suc 𝑥 𝑛𝑛))
2928rexlimiv 3207 . . . . . . . . . . . . 13 (∃𝑥 ∈ ω 𝑛 = suc 𝑥 𝑛𝑛)
3018, 29sylbi 220 . . . . . . . . . . . 12 (𝑛 ∈ (ω ∖ 1o) → 𝑛𝑛)
3130adantr 484 . . . . . . . . . . 11 ((𝑛 ∈ (ω ∖ 1o) ∧ (𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑥 ∧ (𝑓𝑛) = 𝑦) ∧ ∀𝑎𝑛 (𝑓𝑎)𝑅(𝑓‘suc 𝑎))) → 𝑛𝑛)
328, 9, 31rspcdva 3552 . . . . . . . . . 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 6296 . . . . . . . . . . . . . . . . 17 ( suc 𝑥 = 𝑥 → suc suc 𝑥 = suc 𝑥)
3421, 33syl 17 . . . . . . . . . . . . . . . 16 (𝑥 ∈ ω → suc suc 𝑥 = suc 𝑥)
35 suceq 6296 . . . . . . . . . . . . . . . . . 18 ( 𝑛 = suc 𝑥 → suc 𝑛 = suc suc 𝑥)
3625, 35syl 17 . . . . . . . . . . . . . . . . 17 (𝑛 = suc 𝑥 → suc 𝑛 = suc suc 𝑥)
3736, 26eqeq12d 2754 . . . . . . . . . . . . . . . 16 (𝑛 = suc 𝑥 → (suc 𝑛 = 𝑛 ↔ suc suc 𝑥 = suc 𝑥))
3834, 37syl5ibrcom 250 . . . . . . . . . . . . . . 15 (𝑥 ∈ ω → (𝑛 = suc 𝑥 → suc 𝑛 = 𝑛))
3938rexlimiv 3207 . . . . . . . . . . . . . 14 (∃𝑥 ∈ ω 𝑛 = suc 𝑥 → suc 𝑛 = 𝑛)
4018, 39sylbi 220 . . . . . . . . . . . . 13 (𝑛 ∈ (ω ∖ 1o) → suc 𝑛 = 𝑛)
4140fveq2d 6740 . . . . . . . . . . . 12 (𝑛 ∈ (ω ∖ 1o) → (𝑓‘suc 𝑛) = (𝑓𝑛))
4241adantr 484 . . . . . . . . . . 11 ((𝑛 ∈ (ω ∖ 1o) ∧ (𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑥 ∧ (𝑓𝑛) = 𝑦) ∧ ∀𝑎𝑛 (𝑓𝑎)𝑅(𝑓‘suc 𝑎))) → (𝑓‘suc 𝑛) = (𝑓𝑛))
43 simpr2r 1235 . . . . . . . . . . 11 ((𝑛 ∈ (ω ∖ 1o) ∧ (𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑥 ∧ (𝑓𝑛) = 𝑦) ∧ ∀𝑎𝑛 (𝑓𝑎)𝑅(𝑓‘suc 𝑎))) → (𝑓𝑛) = 𝑦)
4442, 43eqtrd 2778 . . . . . . . . . 10 ((𝑛 ∈ (ω ∖ 1o) ∧ (𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑥 ∧ (𝑓𝑛) = 𝑦) ∧ ∀𝑎𝑛 (𝑓𝑎)𝑅(𝑓‘suc 𝑎))) → (𝑓‘suc 𝑛) = 𝑦)
4532, 44breqtrd 5094 . . . . . . . . 9 ((𝑛 ∈ (ω ∖ 1o) ∧ (𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑥 ∧ (𝑓𝑛) = 𝑦) ∧ ∀𝑎𝑛 (𝑓𝑎)𝑅(𝑓‘suc 𝑎))) → (𝑓 𝑛)𝑅𝑦)
46 fvex 6749 . . . . . . . . . 10 (𝑓 𝑛) ∈ V
47 vex 3425 . . . . . . . . . 10 𝑦 ∈ V
4846, 47brelrn 5826 . . . . . . . . 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 416 . . . . . . 7 (𝑛 ∈ (ω ∖ 1o) → ((𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑥 ∧ (𝑓𝑛) = 𝑦) ∧ ∀𝑎𝑛 (𝑓𝑎)𝑅(𝑓‘suc 𝑎)) → 𝑦 ∈ ran 𝑅))
5150exlimdv 1941 . . . . . 6 (𝑛 ∈ (ω ∖ 1o) → (∃𝑓(𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑥 ∧ (𝑓𝑛) = 𝑦) ∧ ∀𝑎𝑛 (𝑓𝑎)𝑅(𝑓‘suc 𝑎)) → 𝑦 ∈ ran 𝑅))
5251rexlimiv 3207 . . . . 5 (∃𝑛 ∈ (ω ∖ 1o)∃𝑓(𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑥 ∧ (𝑓𝑛) = 𝑦) ∧ ∀𝑎𝑛 (𝑓𝑎)𝑅(𝑓‘suc 𝑎)) → 𝑦 ∈ ran 𝑅)
5352exlimiv 1938 . . . 4 (∃𝑥𝑛 ∈ (ω ∖ 1o)∃𝑓(𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑥 ∧ (𝑓𝑛) = 𝑦) ∧ ∀𝑎𝑛 (𝑓𝑎)𝑅(𝑓‘suc 𝑎)) → 𝑦 ∈ ran 𝑅)
5453abssi 3998 . . 3 {𝑦 ∣ ∃𝑥𝑛 ∈ (ω ∖ 1o)∃𝑓(𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑥 ∧ (𝑓𝑛) = 𝑦) ∧ ∀𝑎𝑛 (𝑓𝑎)𝑅(𝑓‘suc 𝑎))} ⊆ ran 𝑅
554, 54eqsstri 3950 . 2 ran t++𝑅 ⊆ ran 𝑅
56 rnresv 6079 . . 3 ran (𝑅 ↾ V) = ran 𝑅
57 relres 5895 . . . . . 6 Rel (𝑅 ↾ V)
58 ssttrcl 33537 . . . . . 6 (Rel (𝑅 ↾ V) → (𝑅 ↾ V) ⊆ t++(𝑅 ↾ V))
5957, 58ax-mp 5 . . . . 5 (𝑅 ↾ V) ⊆ t++(𝑅 ↾ V)
60 ttrclresv 33539 . . . . 5 t++(𝑅 ↾ V) = t++𝑅
6159, 60sseqtri 3952 . . . 4 (𝑅 ↾ V) ⊆ t++𝑅
6261rnssi 5824 . . 3 ran (𝑅 ↾ V) ⊆ ran t++𝑅
6356, 62eqsstrri 3951 . 2 ran 𝑅 ⊆ ran t++𝑅
6455, 63eqssi 3932 1 ran t++𝑅 = ran 𝑅
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399  w3a 1089   = wceq 1543  wex 1787  wcel 2111  {cab 2715  wral 3062  wrex 3063  Vcvv 3421  cdif 3878  wss 3881  c0 4252   cuni 4834   class class class wbr 5068  {copab 5130  ran crn 5567  cres 5568  Rel wrel 5571  Ord word 6230  suc csuc 6233   Fn wfn 6393  cfv 6398  ωcom 7663  1oc1o 8216  t++cttrcl 33529
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2159  ax-12 2176  ax-ext 2709  ax-rep 5194  ax-sep 5207  ax-nul 5214  ax-pr 5337  ax-un 7542
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2072  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2887  df-ne 2942  df-ral 3067  df-rex 3068  df-reu 3069  df-rab 3071  df-v 3423  df-sbc 3710  df-csb 3827  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4253  df-if 4455  df-pw 4530  df-sn 4557  df-pr 4559  df-tp 4561  df-op 4563  df-uni 4835  df-int 4875  df-iun 4921  df-br 5069  df-opab 5131  df-mpt 5151  df-tr 5177  df-id 5470  df-eprel 5475  df-po 5483  df-so 5484  df-fr 5524  df-we 5526  df-xp 5572  df-rel 5573  df-cnv 5574  df-co 5575  df-dm 5576  df-rn 5577  df-res 5578  df-ima 5579  df-pred 6176  df-ord 6234  df-on 6235  df-lim 6236  df-suc 6237  df-iota 6356  df-fun 6400  df-fn 6401  df-f 6402  df-f1 6403  df-fo 6404  df-f1o 6405  df-fv 6406  df-ov 7235  df-oprab 7236  df-mpo 7237  df-om 7664  df-wrecs 8068  df-recs 8129  df-rdg 8167  df-1o 8223  df-oadd 8227  df-ttrcl 33530
This theorem is referenced by:  ttrclexg  33545
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