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Theorem dfrtrclrec2 15005
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 (πœ‘ β†’ Rel 𝑅)
Assertion
Ref Expression
dfrtrclrec2 (πœ‘ β†’ (𝐴(t*recβ€˜π‘…)𝐡 ↔ βˆƒπ‘› ∈ β„•0 𝐴(π‘…β†‘π‘Ÿπ‘›)𝐡))
Distinct variable groups:   𝑅,𝑛   𝐴,𝑛   𝐡,𝑛
Allowed substitution hint:   πœ‘(𝑛)
Proof of Theorem dfrtrclrec2
Dummy variable π‘Ÿ is distinct from all other variables.
StepHypRef Expression
1 simpr 486 . . . . . 6 ((πœ‘ ∧ 𝑅 ∈ V) β†’ 𝑅 ∈ V)
2 nn0ex 12478 . . . . . . 7 β„•0 ∈ V
3 ovex 7442 . . . . . . 7 (π‘…β†‘π‘Ÿπ‘›) ∈ V
42, 3iunex 7955 . . . . . 6 βˆͺ 𝑛 ∈ β„•0 (π‘…β†‘π‘Ÿπ‘›) ∈ V
5 oveq1 7416 . . . . . . . 8 (π‘Ÿ = 𝑅 β†’ (π‘Ÿβ†‘π‘Ÿπ‘›) = (π‘…β†‘π‘Ÿπ‘›))
65iuneq2d 5027 . . . . . . 7 (π‘Ÿ = 𝑅 β†’ βˆͺ 𝑛 ∈ β„•0 (π‘Ÿβ†‘π‘Ÿπ‘›) = βˆͺ 𝑛 ∈ β„•0 (π‘…β†‘π‘Ÿπ‘›))
7 eqid 2733 . . . . . . 7 (π‘Ÿ ∈ V ↦ βˆͺ 𝑛 ∈ β„•0 (π‘Ÿβ†‘π‘Ÿπ‘›)) = (π‘Ÿ ∈ V ↦ βˆͺ 𝑛 ∈ β„•0 (π‘Ÿβ†‘π‘Ÿπ‘›))
86, 7fvmptg 6997 . . . . . 6 ((𝑅 ∈ V ∧ βˆͺ 𝑛 ∈ β„•0 (π‘…β†‘π‘Ÿπ‘›) ∈ V) β†’ ((π‘Ÿ ∈ V ↦ βˆͺ 𝑛 ∈ β„•0 (π‘Ÿβ†‘π‘Ÿπ‘›))β€˜π‘…) = βˆͺ 𝑛 ∈ β„•0 (π‘…β†‘π‘Ÿπ‘›))
91, 4, 8sylancl 587 . . . . 5 ((πœ‘ ∧ 𝑅 ∈ V) β†’ ((π‘Ÿ ∈ V ↦ βˆͺ 𝑛 ∈ β„•0 (π‘Ÿβ†‘π‘Ÿπ‘›))β€˜π‘…) = βˆͺ 𝑛 ∈ β„•0 (π‘…β†‘π‘Ÿπ‘›))
109ex 414 . . . 4 (πœ‘ β†’ (𝑅 ∈ V β†’ ((π‘Ÿ ∈ V ↦ βˆͺ 𝑛 ∈ β„•0 (π‘Ÿβ†‘π‘Ÿπ‘›))β€˜π‘…) = βˆͺ 𝑛 ∈ β„•0 (π‘…β†‘π‘Ÿπ‘›)))
11 iun0 5066 . . . . . 6 βˆͺ 𝑛 ∈ β„•0 βˆ… = βˆ…
1211a1i 11 . . . . 5 (Β¬ 𝑅 ∈ V β†’ βˆͺ 𝑛 ∈ β„•0 βˆ… = βˆ…)
13 reldmrelexp 14968 . . . . . . 7 Rel dom β†‘π‘Ÿ
1413ovprc1 7448 . . . . . 6 (Β¬ 𝑅 ∈ V β†’ (π‘…β†‘π‘Ÿπ‘›) = βˆ…)
1514iuneq2d 5027 . . . . 5 (Β¬ 𝑅 ∈ V β†’ βˆͺ 𝑛 ∈ β„•0 (π‘…β†‘π‘Ÿπ‘›) = βˆͺ 𝑛 ∈ β„•0 βˆ…)
16 fvprc 6884 . . . . 5 (Β¬ 𝑅 ∈ V β†’ ((π‘Ÿ ∈ V ↦ βˆͺ 𝑛 ∈ β„•0 (π‘Ÿβ†‘π‘Ÿπ‘›))β€˜π‘…) = βˆ…)
1712, 15, 163eqtr4rd 2784 . . . 4 (Β¬ 𝑅 ∈ V β†’ ((π‘Ÿ ∈ V ↦ βˆͺ 𝑛 ∈ β„•0 (π‘Ÿβ†‘π‘Ÿπ‘›))β€˜π‘…) = βˆͺ 𝑛 ∈ β„•0 (π‘…β†‘π‘Ÿπ‘›))
1810, 17pm2.61d1 180 . . 3 (πœ‘ β†’ ((π‘Ÿ ∈ V ↦ βˆͺ 𝑛 ∈ β„•0 (π‘Ÿβ†‘π‘Ÿπ‘›))β€˜π‘…) = βˆͺ 𝑛 ∈ β„•0 (π‘…β†‘π‘Ÿπ‘›))
19 breq 5151 . . . 4 (((π‘Ÿ ∈ V ↦ βˆͺ 𝑛 ∈ β„•0 (π‘Ÿβ†‘π‘Ÿπ‘›))β€˜π‘…) = βˆͺ 𝑛 ∈ β„•0 (π‘…β†‘π‘Ÿπ‘›) β†’ (𝐴((π‘Ÿ ∈ V ↦ βˆͺ 𝑛 ∈ β„•0 (π‘Ÿβ†‘π‘Ÿπ‘›))β€˜π‘…)𝐡 ↔ 𝐴βˆͺ 𝑛 ∈ β„•0 (π‘…β†‘π‘Ÿπ‘›)𝐡))
20 eliun 5002 . . . . . 6 (⟨𝐴, 𝐡⟩ ∈ βˆͺ 𝑛 ∈ β„•0 (π‘…β†‘π‘Ÿπ‘›) ↔ βˆƒπ‘› ∈ β„•0 ⟨𝐴, 𝐡⟩ ∈ (π‘…β†‘π‘Ÿπ‘›))
2120a1i 11 . . . . 5 (πœ‘ β†’ (⟨𝐴, 𝐡⟩ ∈ βˆͺ 𝑛 ∈ β„•0 (π‘…β†‘π‘Ÿπ‘›) ↔ βˆƒπ‘› ∈ β„•0 ⟨𝐴, 𝐡⟩ ∈ (π‘…β†‘π‘Ÿπ‘›)))
22 df-br 5150 . . . . 5 (𝐴βˆͺ 𝑛 ∈ β„•0 (π‘…β†‘π‘Ÿπ‘›)𝐡 ↔ ⟨𝐴, 𝐡⟩ ∈ βˆͺ 𝑛 ∈ β„•0 (π‘…β†‘π‘Ÿπ‘›))
23 df-br 5150 . . . . . 6 (𝐴(π‘…β†‘π‘Ÿπ‘›)𝐡 ↔ ⟨𝐴, 𝐡⟩ ∈ (π‘…β†‘π‘Ÿπ‘›))
2423rexbii 3095 . . . . 5 (βˆƒπ‘› ∈ β„•0 𝐴(π‘…β†‘π‘Ÿπ‘›)𝐡 ↔ βˆƒπ‘› ∈ β„•0 ⟨𝐴, 𝐡⟩ ∈ (π‘…β†‘π‘Ÿπ‘›))
2521, 22, 243bitr4g 314 . . . 4 (πœ‘ β†’ (𝐴βˆͺ 𝑛 ∈ β„•0 (π‘…β†‘π‘Ÿπ‘›)𝐡 ↔ βˆƒπ‘› ∈ β„•0 𝐴(π‘…β†‘π‘Ÿπ‘›)𝐡))
2619, 25sylan9bb 511 . . 3 ((((π‘Ÿ ∈ V ↦ βˆͺ 𝑛 ∈ β„•0 (π‘Ÿβ†‘π‘Ÿπ‘›))β€˜π‘…) = βˆͺ 𝑛 ∈ β„•0 (π‘…β†‘π‘Ÿπ‘›) ∧ πœ‘) β†’ (𝐴((π‘Ÿ ∈ V ↦ βˆͺ 𝑛 ∈ β„•0 (π‘Ÿβ†‘π‘Ÿπ‘›))β€˜π‘…)𝐡 ↔ βˆƒπ‘› ∈ β„•0 𝐴(π‘…β†‘π‘Ÿπ‘›)𝐡))
2718, 26mpancom 687 . 2 (πœ‘ β†’ (𝐴((π‘Ÿ ∈ V ↦ βˆͺ 𝑛 ∈ β„•0 (π‘Ÿβ†‘π‘Ÿπ‘›))β€˜π‘…)𝐡 ↔ βˆƒπ‘› ∈ β„•0 𝐴(π‘…β†‘π‘Ÿπ‘›)𝐡))
28 df-rtrclrec 15003 . . 3 t*rec = (π‘Ÿ ∈ V ↦ βˆͺ 𝑛 ∈ β„•0 (π‘Ÿβ†‘π‘Ÿπ‘›))
29 fveq1 6891 . . . . . 6 (t*rec = (π‘Ÿ ∈ V ↦ βˆͺ 𝑛 ∈ β„•0 (π‘Ÿβ†‘π‘Ÿπ‘›)) β†’ (t*recβ€˜π‘…) = ((π‘Ÿ ∈ V ↦ βˆͺ 𝑛 ∈ β„•0 (π‘Ÿβ†‘π‘Ÿπ‘›))β€˜π‘…))
3029breqd 5160 . . . . 5 (t*rec = (π‘Ÿ ∈ V ↦ βˆͺ 𝑛 ∈ β„•0 (π‘Ÿβ†‘π‘Ÿπ‘›)) β†’ (𝐴(t*recβ€˜π‘…)𝐡 ↔ 𝐴((π‘Ÿ ∈ V ↦ βˆͺ 𝑛 ∈ β„•0 (π‘Ÿβ†‘π‘Ÿπ‘›))β€˜π‘…)𝐡))
3130bibi1d 344 . . . 4 (t*rec = (π‘Ÿ ∈ V ↦ βˆͺ 𝑛 ∈ β„•0 (π‘Ÿβ†‘π‘Ÿπ‘›)) β†’ ((𝐴(t*recβ€˜π‘…)𝐡 ↔ βˆƒπ‘› ∈ β„•0 𝐴(π‘…β†‘π‘Ÿπ‘›)𝐡) ↔ (𝐴((π‘Ÿ ∈ V ↦ βˆͺ 𝑛 ∈ β„•0 (π‘Ÿβ†‘π‘Ÿπ‘›))β€˜π‘…)𝐡 ↔ βˆƒπ‘› ∈ β„•0 𝐴(π‘…β†‘π‘Ÿπ‘›)𝐡)))
3231imbi2d 341 . . 3 (t*rec = (π‘Ÿ ∈ V ↦ βˆͺ 𝑛 ∈ β„•0 (π‘Ÿβ†‘π‘Ÿπ‘›)) β†’ ((πœ‘ β†’ (𝐴(t*recβ€˜π‘…)𝐡 ↔ βˆƒπ‘› ∈ β„•0 𝐴(π‘…β†‘π‘Ÿπ‘›)𝐡)) ↔ (πœ‘ β†’ (𝐴((π‘Ÿ ∈ V ↦ βˆͺ 𝑛 ∈ β„•0 (π‘Ÿβ†‘π‘Ÿπ‘›))β€˜π‘…)𝐡 ↔ βˆƒπ‘› ∈ β„•0 𝐴(π‘…β†‘π‘Ÿπ‘›)𝐡))))
3328, 32ax-mp 5 . 2 ((πœ‘ β†’ (𝐴(t*recβ€˜π‘…)𝐡 ↔ βˆƒπ‘› ∈ β„•0 𝐴(π‘…β†‘π‘Ÿπ‘›)𝐡)) ↔ (πœ‘ β†’ (𝐴((π‘Ÿ ∈ V ↦ βˆͺ 𝑛 ∈ β„•0 (π‘Ÿβ†‘π‘Ÿπ‘›))β€˜π‘…)𝐡 ↔ βˆƒπ‘› ∈ β„•0 𝐴(π‘…β†‘π‘Ÿπ‘›)𝐡)))
3427, 33mpbir 230 1 (πœ‘ β†’ (𝐴(t*recβ€˜π‘…)𝐡 ↔ βˆƒπ‘› ∈ β„•0 𝐴(π‘…β†‘π‘Ÿπ‘›)𝐡))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107  βˆƒwrex 3071  Vcvv 3475  βˆ…c0 4323  βŸ¨cop 4635  βˆͺ ciun 4998   class class class wbr 5149   ↦ cmpt 5232  Rel wrel 5682  β€˜cfv 6544  (class class class)co 7409  β„•0cn0 12472  β†‘π‘Ÿcrelexp 14966  t*reccrtrcl 15002
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-1cn 11168  ax-addcl 11170
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-nn 12213  df-n0 12473  df-relexp 14967  df-rtrclrec 15003
This theorem is referenced by:  rtrclreclem3  15007  rtrclind  15012
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