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Theorem dfrtrclrec2 15093
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 12529 . . . . . . 7 0 ∈ V
3 ovex 7463 . . . . . . 7 (𝑅𝑟𝑛) ∈ V
42, 3iunex 7991 . . . . . 6 𝑛 ∈ ℕ0 (𝑅𝑟𝑛) ∈ V
5 oveq1 7437 . . . . . . . 8 (𝑟 = 𝑅 → (𝑟𝑟𝑛) = (𝑅𝑟𝑛))
65iuneq2d 5026 . . . . . . 7 (𝑟 = 𝑅 𝑛 ∈ ℕ0 (𝑟𝑟𝑛) = 𝑛 ∈ ℕ0 (𝑅𝑟𝑛))
7 eqid 2734 . . . . . . 7 (𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛)) = (𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))
86, 7fvmptg 7013 . . . . . 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 5066 . . . . . 6 𝑛 ∈ ℕ0 ∅ = ∅
1211a1i 11 . . . . 5 𝑅 ∈ V → 𝑛 ∈ ℕ0 ∅ = ∅)
13 reldmrelexp 15056 . . . . . . 7 Rel dom ↑𝑟
1413ovprc1 7469 . . . . . 6 𝑅 ∈ V → (𝑅𝑟𝑛) = ∅)
1514iuneq2d 5026 . . . . 5 𝑅 ∈ V → 𝑛 ∈ ℕ0 (𝑅𝑟𝑛) = 𝑛 ∈ ℕ0 ∅)
16 fvprc 6898 . . . . 5 𝑅 ∈ V → ((𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))‘𝑅) = ∅)
1712, 15, 163eqtr4rd 2785 . . . 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 5149 . . . 4 (((𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))‘𝑅) = 𝑛 ∈ ℕ0 (𝑅𝑟𝑛) → (𝐴((𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))‘𝑅)𝐵𝐴 𝑛 ∈ ℕ0 (𝑅𝑟𝑛)𝐵))
20 eliun 4999 . . . . . 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 5148 . . . . 5 (𝐴 𝑛 ∈ ℕ0 (𝑅𝑟𝑛)𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑛 ∈ ℕ0 (𝑅𝑟𝑛))
23 df-br 5148 . . . . . 6 (𝐴(𝑅𝑟𝑛)𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ (𝑅𝑟𝑛))
2423rexbii 3091 . . . . 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 15091 . . 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 5158 . . . . 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 1536  wcel 2105  wrex 3067  Vcvv 3477  c0 4338  cop 4636   ciun 4995   class class class wbr 5147  cmpt 5230  Rel wrel 5693  cfv 6562  (class class class)co 7430  0cn0 12523  𝑟crelexp 15054  t*reccrtrcl 15090
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pr 5437  ax-un 7753  ax-cnex 11208  ax-1cn 11210  ax-addcl 11212
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-nn 12264  df-n0 12524  df-relexp 15055  df-rtrclrec 15091
This theorem is referenced by:  rtrclreclem3  15095  rtrclind  15100
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