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Theorem dfrtrclrec2 15007
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 485 . . . . . 6 ((πœ‘ ∧ 𝑅 ∈ V) β†’ 𝑅 ∈ V)
2 nn0ex 12480 . . . . . . 7 β„•0 ∈ V
3 ovex 7444 . . . . . . 7 (π‘…β†‘π‘Ÿπ‘›) ∈ V
42, 3iunex 7957 . . . . . 6 βˆͺ 𝑛 ∈ β„•0 (π‘…β†‘π‘Ÿπ‘›) ∈ V
5 oveq1 7418 . . . . . . . 8 (π‘Ÿ = 𝑅 β†’ (π‘Ÿβ†‘π‘Ÿπ‘›) = (π‘…β†‘π‘Ÿπ‘›))
65iuneq2d 5026 . . . . . . 7 (π‘Ÿ = 𝑅 β†’ βˆͺ 𝑛 ∈ β„•0 (π‘Ÿβ†‘π‘Ÿπ‘›) = βˆͺ 𝑛 ∈ β„•0 (π‘…β†‘π‘Ÿπ‘›))
7 eqid 2732 . . . . . . 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 6996 . . . . . 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 586 . . . . 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 413 . . . 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 5065 . . . . . 6 βˆͺ 𝑛 ∈ β„•0 βˆ… = βˆ…
1211a1i 11 . . . . 5 (Β¬ 𝑅 ∈ V β†’ βˆͺ 𝑛 ∈ β„•0 βˆ… = βˆ…)
13 reldmrelexp 14970 . . . . . . 7 Rel dom β†‘π‘Ÿ
1413ovprc1 7450 . . . . . 6 (Β¬ 𝑅 ∈ V β†’ (π‘…β†‘π‘Ÿπ‘›) = βˆ…)
1514iuneq2d 5026 . . . . 5 (Β¬ 𝑅 ∈ V β†’ βˆͺ 𝑛 ∈ β„•0 (π‘…β†‘π‘Ÿπ‘›) = βˆͺ 𝑛 ∈ β„•0 βˆ…)
16 fvprc 6883 . . . . 5 (Β¬ 𝑅 ∈ V β†’ ((π‘Ÿ ∈ V ↦ βˆͺ 𝑛 ∈ β„•0 (π‘Ÿβ†‘π‘Ÿπ‘›))β€˜π‘…) = βˆ…)
1712, 15, 163eqtr4rd 2783 . . . 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 5150 . . . 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 5001 . . . . . 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 5149 . . . . 5 (𝐴βˆͺ 𝑛 ∈ β„•0 (π‘…β†‘π‘Ÿπ‘›)𝐡 ↔ ⟨𝐴, 𝐡⟩ ∈ βˆͺ 𝑛 ∈ β„•0 (π‘…β†‘π‘Ÿπ‘›))
23 df-br 5149 . . . . . 6 (𝐴(π‘…β†‘π‘Ÿπ‘›)𝐡 ↔ ⟨𝐴, 𝐡⟩ ∈ (π‘…β†‘π‘Ÿπ‘›))
2423rexbii 3094 . . . . 5 (βˆƒπ‘› ∈ β„•0 𝐴(π‘…β†‘π‘Ÿπ‘›)𝐡 ↔ βˆƒπ‘› ∈ β„•0 ⟨𝐴, 𝐡⟩ ∈ (π‘…β†‘π‘Ÿπ‘›))
2521, 22, 243bitr4g 313 . . . 4 (πœ‘ β†’ (𝐴βˆͺ 𝑛 ∈ β„•0 (π‘…β†‘π‘Ÿπ‘›)𝐡 ↔ βˆƒπ‘› ∈ β„•0 𝐴(π‘…β†‘π‘Ÿπ‘›)𝐡))
2619, 25sylan9bb 510 . . 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 686 . 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 15005 . . 3 t*rec = (π‘Ÿ ∈ V ↦ βˆͺ 𝑛 ∈ β„•0 (π‘Ÿβ†‘π‘Ÿπ‘›))
29 fveq1 6890 . . . . . 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 5159 . . . . 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 343 . . . 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 340 . . 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 396   = wceq 1541   ∈ wcel 2106  βˆƒwrex 3070  Vcvv 3474  βˆ…c0 4322  βŸ¨cop 4634  βˆͺ ciun 4997   class class class wbr 5148   ↦ cmpt 5231  Rel wrel 5681  β€˜cfv 6543  (class class class)co 7411  β„•0cn0 12474  β†‘π‘Ÿcrelexp 14968  t*reccrtrcl 15004
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7727  ax-cnex 11168  ax-1cn 11170  ax-addcl 11172
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-nn 12215  df-n0 12475  df-relexp 14969  df-rtrclrec 15005
This theorem is referenced by:  rtrclreclem3  15009  rtrclind  15014
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