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Theorem rnttrcl 9760
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 9746 . . . . 5 t++𝑅 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑛 ∈ (ω ∖ 1o)∃𝑓(𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑥 ∧ (𝑓𝑛) = 𝑦) ∧ ∀𝑎𝑛 (𝑓𝑎)𝑅(𝑓‘suc 𝑎))}
21rneqi 5951 . . . 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 5968 . . . 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 2763 . . 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 6907 . . . . . . . . . . . 12 (𝑎 = 𝑛 → (𝑓𝑎) = (𝑓 𝑛))
6 suceq 6452 . . . . . . . . . . . . 13 (𝑎 = 𝑛 → suc 𝑎 = suc 𝑛)
76fveq2d 6911 . . . . . . . . . . . 12 (𝑎 = 𝑛 → (𝑓‘suc 𝑎) = (𝑓‘suc 𝑛))
85, 7breq12d 5161 . . . . . . . . . . 11 (𝑎 = 𝑛 → ((𝑓𝑎)𝑅(𝑓‘suc 𝑎) ↔ (𝑓 𝑛)𝑅(𝑓‘suc 𝑛)))
9 simpr3 1195 . . . . . . . . . . 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 8505 . . . . . . . . . . . . . . . 16 1o = suc ∅
1110difeq2i 4133 . . . . . . . . . . . . . . 15 (ω ∖ 1o) = (ω ∖ suc ∅)
1211eleq2i 2831 . . . . . . . . . . . . . 14 (𝑛 ∈ (ω ∖ 1o) ↔ 𝑛 ∈ (ω ∖ suc ∅))
13 peano1 7911 . . . . . . . . . . . . . . 15 ∅ ∈ ω
14 eldifsucnn 8701 . . . . . . . . . . . . . . 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 4384 . . . . . . . . . . . . . . 15 (ω ∖ ∅) = ω
1716rexeqi 3323 . . . . . . . . . . . . . 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 7895 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ ω → Ord 𝑥)
20 ordunisuc 7852 . . . . . . . . . . . . . . . . 17 (Ord 𝑥 suc 𝑥 = 𝑥)
2119, 20syl 17 . . . . . . . . . . . . . . . 16 (𝑥 ∈ ω → suc 𝑥 = 𝑥)
22 vex 3482 . . . . . . . . . . . . . . . . 17 𝑥 ∈ V
2322sucid 6468 . . . . . . . . . . . . . . . 16 𝑥 ∈ suc 𝑥
2421, 23eqeltrdi 2847 . . . . . . . . . . . . . . 15 (𝑥 ∈ ω → suc 𝑥 ∈ suc 𝑥)
25 unieq 4923 . . . . . . . . . . . . . . . 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 247 . . . . . . . . . . . . . 14 (𝑥 ∈ ω → (𝑛 = suc 𝑥 𝑛𝑛))
2928rexlimiv 3146 . . . . . . . . . . . . 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 3623 . . . . . . . . . 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 6452 . . . . . . . . . . . . . . . . 17 ( suc 𝑥 = 𝑥 → suc suc 𝑥 = suc 𝑥)
3421, 33syl 17 . . . . . . . . . . . . . . . 16 (𝑥 ∈ ω → suc suc 𝑥 = suc 𝑥)
35 suceq 6452 . . . . . . . . . . . . . . . . . 18 ( 𝑛 = suc 𝑥 → suc 𝑛 = suc suc 𝑥)
3625, 35syl 17 . . . . . . . . . . . . . . . . 17 (𝑛 = suc 𝑥 → suc 𝑛 = suc suc 𝑥)
3736, 26eqeq12d 2751 . . . . . . . . . . . . . . . 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 3146 . . . . . . . . . . . . . 14 (∃𝑥 ∈ ω 𝑛 = suc 𝑥 → suc 𝑛 = 𝑛)
4018, 39sylbi 217 . . . . . . . . . . . . 13 (𝑛 ∈ (ω ∖ 1o) → suc 𝑛 = 𝑛)
4140fveq2d 6911 . . . . . . . . . . . 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 1232 . . . . . . . . . . 11 ((𝑛 ∈ (ω ∖ 1o) ∧ (𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑥 ∧ (𝑓𝑛) = 𝑦) ∧ ∀𝑎𝑛 (𝑓𝑎)𝑅(𝑓‘suc 𝑎))) → (𝑓𝑛) = 𝑦)
4442, 43eqtrd 2775 . . . . . . . . . 10 ((𝑛 ∈ (ω ∖ 1o) ∧ (𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑥 ∧ (𝑓𝑛) = 𝑦) ∧ ∀𝑎𝑛 (𝑓𝑎)𝑅(𝑓‘suc 𝑎))) → (𝑓‘suc 𝑛) = 𝑦)
4532, 44breqtrd 5174 . . . . . . . . 9 ((𝑛 ∈ (ω ∖ 1o) ∧ (𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑥 ∧ (𝑓𝑛) = 𝑦) ∧ ∀𝑎𝑛 (𝑓𝑎)𝑅(𝑓‘suc 𝑎))) → (𝑓 𝑛)𝑅𝑦)
46 fvex 6920 . . . . . . . . . 10 (𝑓 𝑛) ∈ V
47 vex 3482 . . . . . . . . . 10 𝑦 ∈ V
4846, 47brelrn 5956 . . . . . . . . 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 1931 . . . . . 6 (𝑛 ∈ (ω ∖ 1o) → (∃𝑓(𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑥 ∧ (𝑓𝑛) = 𝑦) ∧ ∀𝑎𝑛 (𝑓𝑎)𝑅(𝑓‘suc 𝑎)) → 𝑦 ∈ ran 𝑅))
5251rexlimiv 3146 . . . . 5 (∃𝑛 ∈ (ω ∖ 1o)∃𝑓(𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑥 ∧ (𝑓𝑛) = 𝑦) ∧ ∀𝑎𝑛 (𝑓𝑎)𝑅(𝑓‘suc 𝑎)) → 𝑦 ∈ ran 𝑅)
5352exlimiv 1928 . . . 4 (∃𝑥𝑛 ∈ (ω ∖ 1o)∃𝑓(𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑥 ∧ (𝑓𝑛) = 𝑦) ∧ ∀𝑎𝑛 (𝑓𝑎)𝑅(𝑓‘suc 𝑎)) → 𝑦 ∈ ran 𝑅)
5453abssi 4080 . . 3 {𝑦 ∣ ∃𝑥𝑛 ∈ (ω ∖ 1o)∃𝑓(𝑓 Fn suc 𝑛 ∧ ((𝑓‘∅) = 𝑥 ∧ (𝑓𝑛) = 𝑦) ∧ ∀𝑎𝑛 (𝑓𝑎)𝑅(𝑓‘suc 𝑎))} ⊆ ran 𝑅
554, 54eqsstri 4030 . 2 ran t++𝑅 ⊆ ran 𝑅
56 rnresv 6223 . . 3 ran (𝑅 ↾ V) = ran 𝑅
57 relres 6026 . . . . . 6 Rel (𝑅 ↾ V)
58 ssttrcl 9753 . . . . . 6 (Rel (𝑅 ↾ V) → (𝑅 ↾ V) ⊆ t++(𝑅 ↾ V))
5957, 58ax-mp 5 . . . . 5 (𝑅 ↾ V) ⊆ t++(𝑅 ↾ V)
60 ttrclresv 9755 . . . . 5 t++(𝑅 ↾ V) = t++𝑅
6159, 60sseqtri 4032 . . . 4 (𝑅 ↾ V) ⊆ t++𝑅
6261rnssi 5954 . . 3 ran (𝑅 ↾ V) ⊆ ran t++𝑅
6356, 62eqsstrri 4031 . 2 ran 𝑅 ⊆ ran t++𝑅
6455, 63eqssi 4012 1 ran t++𝑅 = ran 𝑅
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395  w3a 1086   = wceq 1537  wex 1776  wcel 2106  {cab 2712  wral 3059  wrex 3068  Vcvv 3478  cdif 3960  wss 3963  c0 4339   cuni 4912   class class class wbr 5148  {copab 5210  ran crn 5690  cres 5691  Rel wrel 5694  Ord word 6385  suc csuc 6388   Fn wfn 6558  cfv 6563  ωcom 7887  1oc1o 8498  t++cttrcl 9745
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-oadd 8509  df-ttrcl 9746
This theorem is referenced by:  ttrclexg  9761
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