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Theorem dfrtrclrec2 13841
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 11336 . . . . 5 0 ∈ V
3 ovex 6718 . . . . 5 (𝑅𝑟𝑛) ∈ V
42, 3iunex 7189 . . . 4 𝑛 ∈ ℕ0 (𝑅𝑟𝑛) ∈ V
5 oveq1 6697 . . . . . 6 (𝑟 = 𝑅 → (𝑟𝑟𝑛) = (𝑅𝑟𝑛))
65iuneq2d 4579 . . . . 5 (𝑟 = 𝑅 𝑛 ∈ ℕ0 (𝑟𝑟𝑛) = 𝑛 ∈ ℕ0 (𝑅𝑟𝑛))
7 eqid 2651 . . . . 5 (𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛)) = (𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))
86, 7fvmptg 6319 . . . 4 ((𝑅 ∈ V ∧ 𝑛 ∈ ℕ0 (𝑅𝑟𝑛) ∈ V) → ((𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))‘𝑅) = 𝑛 ∈ ℕ0 (𝑅𝑟𝑛))
91, 4, 8sylancl 695 . . 3 (𝜑 → ((𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))‘𝑅) = 𝑛 ∈ ℕ0 (𝑅𝑟𝑛))
10 breq 4687 . . . 4 (((𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))‘𝑅) = 𝑛 ∈ ℕ0 (𝑅𝑟𝑛) → (𝐴((𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))‘𝑅)𝐵𝐴 𝑛 ∈ ℕ0 (𝑅𝑟𝑛)𝐵))
11 eliun 4556 . . . . . 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 4686 . . . . 5 (𝐴 𝑛 ∈ ℕ0 (𝑅𝑟𝑛)𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑛 ∈ ℕ0 (𝑅𝑟𝑛))
14 df-br 4686 . . . . . 6 (𝐴(𝑅𝑟𝑛)𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ (𝑅𝑟𝑛))
1514rexbii 3070 . . . . 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 736 . . 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 704 . 2 (𝜑 → (𝐴((𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))‘𝑅)𝐵 ↔ ∃𝑛 ∈ ℕ0 𝐴(𝑅𝑟𝑛)𝐵))
19 df-rtrclrec 13840 . . 3 t*rec = (𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))
20 fveq1 6228 . . . . . 6 (t*rec = (𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛)) → (t*rec‘𝑅) = ((𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))‘𝑅))
2120breqd 4696 . . . . 5 (t*rec = (𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛)) → (𝐴(t*rec‘𝑅)𝐵𝐴((𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))‘𝑅)𝐵))
2221bibi1d 332 . . . 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 329 . . 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 1523  wcel 2030  wrex 2942  Vcvv 3231  cop 4216   ciun 4552   class class class wbr 4685  cmpt 4762  Rel wrel 5148  cfv 5926  (class class class)co 6690  0cn0 11330  𝑟crelexp 13804  t*reccrtrcl 13839
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-i2m1 10042  ax-1ne0 10043  ax-rrecex 10046  ax-cnre 10047
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-om 7108  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-nn 11059  df-n0 11331  df-rtrclrec 13840
This theorem is referenced by:  rtrclreclem3  13844  rtrclind  13849
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