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Theorem dfrtrclrec2 13726
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.)
Hypotheses
Ref Expression
rtrclreclem.1 (𝜑 → Rel 𝑅)
rtrclreclem.2 (𝜑𝑅 ∈ V)
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 rtrclreclem.2 . . . 4 (𝜑𝑅 ∈ V)
2 nn0ex 11243 . . . . 5 0 ∈ V
3 ovex 6633 . . . . 5 (𝑅𝑟𝑛) ∈ V
42, 3iunex 7096 . . . 4 𝑛 ∈ ℕ0 (𝑅𝑟𝑛) ∈ V
5 oveq1 6612 . . . . . 6 (𝑟 = 𝑅 → (𝑟𝑟𝑛) = (𝑅𝑟𝑛))
65iuneq2d 4518 . . . . 5 (𝑟 = 𝑅 𝑛 ∈ ℕ0 (𝑟𝑟𝑛) = 𝑛 ∈ ℕ0 (𝑅𝑟𝑛))
7 eqid 2626 . . . . 5 (𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛)) = (𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))
86, 7fvmptg 6238 . . . 4 ((𝑅 ∈ V ∧ 𝑛 ∈ ℕ0 (𝑅𝑟𝑛) ∈ V) → ((𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))‘𝑅) = 𝑛 ∈ ℕ0 (𝑅𝑟𝑛))
91, 4, 8sylancl 693 . . 3 (𝜑 → ((𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))‘𝑅) = 𝑛 ∈ ℕ0 (𝑅𝑟𝑛))
10 breq 4620 . . . 4 (((𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))‘𝑅) = 𝑛 ∈ ℕ0 (𝑅𝑟𝑛) → (𝐴((𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))‘𝑅)𝐵𝐴 𝑛 ∈ ℕ0 (𝑅𝑟𝑛)𝐵))
11 eliun 4495 . . . . . 6 (⟨𝐴, 𝐵⟩ ∈ 𝑛 ∈ ℕ0 (𝑅𝑟𝑛) ↔ ∃𝑛 ∈ ℕ0𝐴, 𝐵⟩ ∈ (𝑅𝑟𝑛))
1211a1i 11 . . . . 5 (𝜑 → (⟨𝐴, 𝐵⟩ ∈ 𝑛 ∈ ℕ0 (𝑅𝑟𝑛) ↔ ∃𝑛 ∈ ℕ0𝐴, 𝐵⟩ ∈ (𝑅𝑟𝑛)))
13 df-br 4619 . . . . 5 (𝐴 𝑛 ∈ ℕ0 (𝑅𝑟𝑛)𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑛 ∈ ℕ0 (𝑅𝑟𝑛))
14 df-br 4619 . . . . . 6 (𝐴(𝑅𝑟𝑛)𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ (𝑅𝑟𝑛))
1514rexbii 3039 . . . . 5 (∃𝑛 ∈ ℕ0 𝐴(𝑅𝑟𝑛)𝐵 ↔ ∃𝑛 ∈ ℕ0𝐴, 𝐵⟩ ∈ (𝑅𝑟𝑛))
1612, 13, 153bitr4g 303 . . . 4 (𝜑 → (𝐴 𝑛 ∈ ℕ0 (𝑅𝑟𝑛)𝐵 ↔ ∃𝑛 ∈ ℕ0 𝐴(𝑅𝑟𝑛)𝐵))
1710, 16sylan9bb 735 . . 3 ((((𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))‘𝑅) = 𝑛 ∈ ℕ0 (𝑅𝑟𝑛) ∧ 𝜑) → (𝐴((𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))‘𝑅)𝐵 ↔ ∃𝑛 ∈ ℕ0 𝐴(𝑅𝑟𝑛)𝐵))
189, 17mpancom 702 . 2 (𝜑 → (𝐴((𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))‘𝑅)𝐵 ↔ ∃𝑛 ∈ ℕ0 𝐴(𝑅𝑟𝑛)𝐵))
19 df-rtrclrec 13725 . . 3 t*rec = (𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))
20 fveq1 6149 . . . . . 6 (t*rec = (𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛)) → (t*rec‘𝑅) = ((𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))‘𝑅))
2120breqd 4629 . . . . 5 (t*rec = (𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛)) → (𝐴(t*rec‘𝑅)𝐵𝐴((𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))‘𝑅)𝐵))
2221bibi1d 333 . . . 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)))
2322imbi2d 330 . . 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))))
2419, 23ax-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)))
2518, 24mpbir 221 1 (𝜑 → (𝐴(t*rec‘𝑅)𝐵 ↔ ∃𝑛 ∈ ℕ0 𝐴(𝑅𝑟𝑛)𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1480  wcel 1992  wrex 2913  Vcvv 3191  cop 4159   ciun 4490   class class class wbr 4618  cmpt 4678  Rel wrel 5084  cfv 5850  (class class class)co 6605  0cn0 11237  𝑟crelexp 13689  t*reccrtrcl 13724
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903  ax-cnex 9937  ax-resscn 9938  ax-1cn 9939  ax-icn 9940  ax-addcl 9941  ax-addrcl 9942  ax-mulcl 9943  ax-mulrcl 9944  ax-i2m1 9949  ax-1ne0 9950  ax-rrecex 9953  ax-cnre 9954
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5642  df-ord 5688  df-on 5689  df-lim 5690  df-suc 5691  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-ov 6608  df-om 7014  df-wrecs 7353  df-recs 7414  df-rdg 7452  df-nn 10966  df-n0 11238  df-rtrclrec 13725
This theorem is referenced by:  rtrclreclem3  13729  rtrclind  13734
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