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Theorem dfrtrclrec2 15097
Description: If two elements are connected by a reflexive, transitive closure, then they are connected via 𝑛 instances the relation, for some 𝑛. (Contributed by Drahflow, 12-Nov-2015.) (Revised by AV, 13-Jul-2024.)
Hypothesis
Ref Expression
dfrtrclrec2.1 (𝜑 → Rel 𝑅)
Assertion
Ref Expression
dfrtrclrec2 (𝜑 → (𝐴(t*rec‘𝑅)𝐵 ↔ ∃𝑛 ∈ ℕ0 𝐴(𝑅𝑟𝑛)𝐵))
Distinct variable groups:   𝑅,𝑛   𝐴,𝑛   𝐵,𝑛
Allowed substitution hint:   𝜑(𝑛)

Proof of Theorem dfrtrclrec2
Dummy variable 𝑟 is distinct from all other variables.
StepHypRef Expression
1 simpr 484 . . . . . 6 ((𝜑𝑅 ∈ V) → 𝑅 ∈ V)
2 nn0ex 12532 . . . . . . 7 0 ∈ V
3 ovex 7464 . . . . . . 7 (𝑅𝑟𝑛) ∈ V
42, 3iunex 7993 . . . . . 6 𝑛 ∈ ℕ0 (𝑅𝑟𝑛) ∈ V
5 oveq1 7438 . . . . . . . 8 (𝑟 = 𝑅 → (𝑟𝑟𝑛) = (𝑅𝑟𝑛))
65iuneq2d 5022 . . . . . . 7 (𝑟 = 𝑅 𝑛 ∈ ℕ0 (𝑟𝑟𝑛) = 𝑛 ∈ ℕ0 (𝑅𝑟𝑛))
7 eqid 2737 . . . . . . 7 (𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛)) = (𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))
86, 7fvmptg 7014 . . . . . 6 ((𝑅 ∈ V ∧ 𝑛 ∈ ℕ0 (𝑅𝑟𝑛) ∈ V) → ((𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))‘𝑅) = 𝑛 ∈ ℕ0 (𝑅𝑟𝑛))
91, 4, 8sylancl 586 . . . . 5 ((𝜑𝑅 ∈ V) → ((𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))‘𝑅) = 𝑛 ∈ ℕ0 (𝑅𝑟𝑛))
109ex 412 . . . 4 (𝜑 → (𝑅 ∈ V → ((𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))‘𝑅) = 𝑛 ∈ ℕ0 (𝑅𝑟𝑛)))
11 iun0 5062 . . . . . 6 𝑛 ∈ ℕ0 ∅ = ∅
1211a1i 11 . . . . 5 𝑅 ∈ V → 𝑛 ∈ ℕ0 ∅ = ∅)
13 reldmrelexp 15060 . . . . . . 7 Rel dom ↑𝑟
1413ovprc1 7470 . . . . . 6 𝑅 ∈ V → (𝑅𝑟𝑛) = ∅)
1514iuneq2d 5022 . . . . 5 𝑅 ∈ V → 𝑛 ∈ ℕ0 (𝑅𝑟𝑛) = 𝑛 ∈ ℕ0 ∅)
16 fvprc 6898 . . . . 5 𝑅 ∈ V → ((𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))‘𝑅) = ∅)
1712, 15, 163eqtr4rd 2788 . . . 4 𝑅 ∈ V → ((𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))‘𝑅) = 𝑛 ∈ ℕ0 (𝑅𝑟𝑛))
1810, 17pm2.61d1 180 . . 3 (𝜑 → ((𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))‘𝑅) = 𝑛 ∈ ℕ0 (𝑅𝑟𝑛))
19 breq 5145 . . . 4 (((𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))‘𝑅) = 𝑛 ∈ ℕ0 (𝑅𝑟𝑛) → (𝐴((𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))‘𝑅)𝐵𝐴 𝑛 ∈ ℕ0 (𝑅𝑟𝑛)𝐵))
20 eliun 4995 . . . . . 6 (⟨𝐴, 𝐵⟩ ∈ 𝑛 ∈ ℕ0 (𝑅𝑟𝑛) ↔ ∃𝑛 ∈ ℕ0𝐴, 𝐵⟩ ∈ (𝑅𝑟𝑛))
2120a1i 11 . . . . 5 (𝜑 → (⟨𝐴, 𝐵⟩ ∈ 𝑛 ∈ ℕ0 (𝑅𝑟𝑛) ↔ ∃𝑛 ∈ ℕ0𝐴, 𝐵⟩ ∈ (𝑅𝑟𝑛)))
22 df-br 5144 . . . . 5 (𝐴 𝑛 ∈ ℕ0 (𝑅𝑟𝑛)𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑛 ∈ ℕ0 (𝑅𝑟𝑛))
23 df-br 5144 . . . . . 6 (𝐴(𝑅𝑟𝑛)𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ (𝑅𝑟𝑛))
2423rexbii 3094 . . . . 5 (∃𝑛 ∈ ℕ0 𝐴(𝑅𝑟𝑛)𝐵 ↔ ∃𝑛 ∈ ℕ0𝐴, 𝐵⟩ ∈ (𝑅𝑟𝑛))
2521, 22, 243bitr4g 314 . . . 4 (𝜑 → (𝐴 𝑛 ∈ ℕ0 (𝑅𝑟𝑛)𝐵 ↔ ∃𝑛 ∈ ℕ0 𝐴(𝑅𝑟𝑛)𝐵))
2619, 25sylan9bb 509 . . 3 ((((𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))‘𝑅) = 𝑛 ∈ ℕ0 (𝑅𝑟𝑛) ∧ 𝜑) → (𝐴((𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))‘𝑅)𝐵 ↔ ∃𝑛 ∈ ℕ0 𝐴(𝑅𝑟𝑛)𝐵))
2718, 26mpancom 688 . 2 (𝜑 → (𝐴((𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))‘𝑅)𝐵 ↔ ∃𝑛 ∈ ℕ0 𝐴(𝑅𝑟𝑛)𝐵))
28 df-rtrclrec 15095 . . 3 t*rec = (𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))
29 fveq1 6905 . . . . . 6 (t*rec = (𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛)) → (t*rec‘𝑅) = ((𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))‘𝑅))
3029breqd 5154 . . . . 5 (t*rec = (𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛)) → (𝐴(t*rec‘𝑅)𝐵𝐴((𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))‘𝑅)𝐵))
3130bibi1d 343 . . . 4 (t*rec = (𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛)) → ((𝐴(t*rec‘𝑅)𝐵 ↔ ∃𝑛 ∈ ℕ0 𝐴(𝑅𝑟𝑛)𝐵) ↔ (𝐴((𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))‘𝑅)𝐵 ↔ ∃𝑛 ∈ ℕ0 𝐴(𝑅𝑟𝑛)𝐵)))
3231imbi2d 340 . . 3 (t*rec = (𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛)) → ((𝜑 → (𝐴(t*rec‘𝑅)𝐵 ↔ ∃𝑛 ∈ ℕ0 𝐴(𝑅𝑟𝑛)𝐵)) ↔ (𝜑 → (𝐴((𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))‘𝑅)𝐵 ↔ ∃𝑛 ∈ ℕ0 𝐴(𝑅𝑟𝑛)𝐵))))
3328, 32ax-mp 5 . 2 ((𝜑 → (𝐴(t*rec‘𝑅)𝐵 ↔ ∃𝑛 ∈ ℕ0 𝐴(𝑅𝑟𝑛)𝐵)) ↔ (𝜑 → (𝐴((𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))‘𝑅)𝐵 ↔ ∃𝑛 ∈ ℕ0 𝐴(𝑅𝑟𝑛)𝐵)))
3427, 33mpbir 231 1 (𝜑 → (𝐴(t*rec‘𝑅)𝐵 ↔ ∃𝑛 ∈ ℕ0 𝐴(𝑅𝑟𝑛)𝐵))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  wrex 3070  Vcvv 3480  c0 4333  cop 4632   ciun 4991   class class class wbr 5143  cmpt 5225  Rel wrel 5690  cfv 6561  (class class class)co 7431  0cn0 12526  𝑟crelexp 15058  t*reccrtrcl 15094
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 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-1cn 11213  ax-addcl 11215
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-nn 12267  df-n0 12527  df-relexp 15059  df-rtrclrec 15095
This theorem is referenced by:  rtrclreclem3  15099  rtrclind  15104
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