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Theorem dfrtrclrec2 14411
 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 488 . . . . . 6 ((𝜑𝑅 ∈ V) → 𝑅 ∈ V)
2 nn0ex 11893 . . . . . . 7 0 ∈ V
3 ovex 7168 . . . . . . 7 (𝑅𝑟𝑛) ∈ V
42, 3iunex 7653 . . . . . 6 𝑛 ∈ ℕ0 (𝑅𝑟𝑛) ∈ V
5 oveq1 7142 . . . . . . . 8 (𝑟 = 𝑅 → (𝑟𝑟𝑛) = (𝑅𝑟𝑛))
65iuneq2d 4910 . . . . . . 7 (𝑟 = 𝑅 𝑛 ∈ ℕ0 (𝑟𝑟𝑛) = 𝑛 ∈ ℕ0 (𝑅𝑟𝑛))
7 eqid 2798 . . . . . . 7 (𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛)) = (𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))
86, 7fvmptg 6743 . . . . . 6 ((𝑅 ∈ V ∧ 𝑛 ∈ ℕ0 (𝑅𝑟𝑛) ∈ V) → ((𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))‘𝑅) = 𝑛 ∈ ℕ0 (𝑅𝑟𝑛))
91, 4, 8sylancl 589 . . . . 5 ((𝜑𝑅 ∈ V) → ((𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))‘𝑅) = 𝑛 ∈ ℕ0 (𝑅𝑟𝑛))
109ex 416 . . . 4 (𝜑 → (𝑅 ∈ V → ((𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))‘𝑅) = 𝑛 ∈ ℕ0 (𝑅𝑟𝑛)))
11 iun0 4948 . . . . . 6 𝑛 ∈ ℕ0 ∅ = ∅
1211a1i 11 . . . . 5 𝑅 ∈ V → 𝑛 ∈ ℕ0 ∅ = ∅)
13 reldmrelexp 14374 . . . . . . 7 Rel dom ↑𝑟
1413ovprc1 7174 . . . . . 6 𝑅 ∈ V → (𝑅𝑟𝑛) = ∅)
1514iuneq2d 4910 . . . . 5 𝑅 ∈ V → 𝑛 ∈ ℕ0 (𝑅𝑟𝑛) = 𝑛 ∈ ℕ0 ∅)
16 fvprc 6638 . . . . 5 𝑅 ∈ V → ((𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))‘𝑅) = ∅)
1712, 15, 163eqtr4rd 2844 . . . 4 𝑅 ∈ V → ((𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))‘𝑅) = 𝑛 ∈ ℕ0 (𝑅𝑟𝑛))
1810, 17pm2.61d1 183 . . 3 (𝜑 → ((𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))‘𝑅) = 𝑛 ∈ ℕ0 (𝑅𝑟𝑛))
19 breq 5032 . . . 4 (((𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))‘𝑅) = 𝑛 ∈ ℕ0 (𝑅𝑟𝑛) → (𝐴((𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))‘𝑅)𝐵𝐴 𝑛 ∈ ℕ0 (𝑅𝑟𝑛)𝐵))
20 eliun 4885 . . . . . 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 5031 . . . . 5 (𝐴 𝑛 ∈ ℕ0 (𝑅𝑟𝑛)𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑛 ∈ ℕ0 (𝑅𝑟𝑛))
23 df-br 5031 . . . . . 6 (𝐴(𝑅𝑟𝑛)𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ (𝑅𝑟𝑛))
2423rexbii 3210 . . . . 5 (∃𝑛 ∈ ℕ0 𝐴(𝑅𝑟𝑛)𝐵 ↔ ∃𝑛 ∈ ℕ0𝐴, 𝐵⟩ ∈ (𝑅𝑟𝑛))
2521, 22, 243bitr4g 317 . . . 4 (𝜑 → (𝐴 𝑛 ∈ ℕ0 (𝑅𝑟𝑛)𝐵 ↔ ∃𝑛 ∈ ℕ0 𝐴(𝑅𝑟𝑛)𝐵))
2619, 25sylan9bb 513 . . 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 687 . 2 (𝜑 → (𝐴((𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))‘𝑅)𝐵 ↔ ∃𝑛 ∈ ℕ0 𝐴(𝑅𝑟𝑛)𝐵))
28 df-rtrclrec 14409 . . 3 t*rec = (𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))
29 fveq1 6644 . . . . . 6 (t*rec = (𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛)) → (t*rec‘𝑅) = ((𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))‘𝑅))
3029breqd 5041 . . . . 5 (t*rec = (𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛)) → (𝐴(t*rec‘𝑅)𝐵𝐴((𝑟 ∈ V ↦ 𝑛 ∈ ℕ0 (𝑟𝑟𝑛))‘𝑅)𝐵))
3130bibi1d 347 . . . 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 344 . . 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 234 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 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∃wrex 3107  Vcvv 3441  ∅c0 4243  ⟨cop 4531  ∪ ciun 4881   class class class wbr 5030   ↦ cmpt 5110  Rel wrel 5524  ‘cfv 6324  (class class class)co 7135  ℕ0cn0 11887  ↑𝑟crelexp 14372  t*reccrtrcl 14408 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7443  ax-cnex 10584  ax-1cn 10586  ax-addcl 10588 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7563  df-wrecs 7932  df-recs 7993  df-rdg 8031  df-nn 11628  df-n0 11888  df-relexp 14373  df-rtrclrec 14409 This theorem is referenced by:  rtrclreclem3  14413  rtrclind  14418
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