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Theorem umgr3v3e3cycl 26923
Description: If and only if there is a 3-cycle in a multigraph, there are three (different) vertices in the graph which are mutually connected by edges. (Contributed by Alexander van der Vekens, 14-Nov-2017.) (Revised by AV, 12-Feb-2021.)
Hypotheses
Ref Expression
uhgr3cyclex.v 𝑉 = (Vtx‘𝐺)
uhgr3cyclex.e 𝐸 = (Edg‘𝐺)
Assertion
Ref Expression
umgr3v3e3cycl (𝐺 ∈ UMGraph → (∃𝑓𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ (#‘𝑓) = 3) ↔ ∃𝑎𝑉𝑏𝑉𝑐𝑉 ({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸 ∧ {𝑐, 𝑎} ∈ 𝐸)))
Distinct variable groups:   𝑓,𝑝,𝐺   𝐸,𝑎,𝑏,𝑐,𝑓,𝑝   𝐺,𝑎,𝑏,𝑐   𝑉,𝑎,𝑏,𝑐,𝑓,𝑝

Proof of Theorem umgr3v3e3cycl
StepHypRef Expression
1 umgrupgr 25906 . . . . . 6 (𝐺 ∈ UMGraph → 𝐺 ∈ UPGraph )
21adantr 481 . . . . 5 ((𝐺 ∈ UMGraph ∧ (𝑓(Cycles‘𝐺)𝑝 ∧ (#‘𝑓) = 3)) → 𝐺 ∈ UPGraph )
3 simpl 473 . . . . . 6 ((𝑓(Cycles‘𝐺)𝑝 ∧ (#‘𝑓) = 3) → 𝑓(Cycles‘𝐺)𝑝)
43adantl 482 . . . . 5 ((𝐺 ∈ UMGraph ∧ (𝑓(Cycles‘𝐺)𝑝 ∧ (#‘𝑓) = 3)) → 𝑓(Cycles‘𝐺)𝑝)
5 simpr 477 . . . . . 6 ((𝑓(Cycles‘𝐺)𝑝 ∧ (#‘𝑓) = 3) → (#‘𝑓) = 3)
65adantl 482 . . . . 5 ((𝐺 ∈ UMGraph ∧ (𝑓(Cycles‘𝐺)𝑝 ∧ (#‘𝑓) = 3)) → (#‘𝑓) = 3)
7 uhgr3cyclex.e . . . . . . 7 𝐸 = (Edg‘𝐺)
8 uhgr3cyclex.v . . . . . . 7 𝑉 = (Vtx‘𝐺)
97, 8upgr3v3e3cycl 26919 . . . . . 6 ((𝐺 ∈ UPGraph ∧ 𝑓(Cycles‘𝐺)𝑝 ∧ (#‘𝑓) = 3) → ∃𝑎𝑉𝑏𝑉𝑐𝑉 (({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸 ∧ {𝑐, 𝑎} ∈ 𝐸) ∧ (𝑎𝑏𝑏𝑐𝑐𝑎)))
10 simpl 473 . . . . . . . . 9 ((({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸 ∧ {𝑐, 𝑎} ∈ 𝐸) ∧ (𝑎𝑏𝑏𝑐𝑐𝑎)) → ({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸 ∧ {𝑐, 𝑎} ∈ 𝐸))
1110reximi 3006 . . . . . . . 8 (∃𝑐𝑉 (({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸 ∧ {𝑐, 𝑎} ∈ 𝐸) ∧ (𝑎𝑏𝑏𝑐𝑐𝑎)) → ∃𝑐𝑉 ({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸 ∧ {𝑐, 𝑎} ∈ 𝐸))
1211reximi 3006 . . . . . . 7 (∃𝑏𝑉𝑐𝑉 (({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸 ∧ {𝑐, 𝑎} ∈ 𝐸) ∧ (𝑎𝑏𝑏𝑐𝑐𝑎)) → ∃𝑏𝑉𝑐𝑉 ({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸 ∧ {𝑐, 𝑎} ∈ 𝐸))
1312reximi 3006 . . . . . 6 (∃𝑎𝑉𝑏𝑉𝑐𝑉 (({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸 ∧ {𝑐, 𝑎} ∈ 𝐸) ∧ (𝑎𝑏𝑏𝑐𝑐𝑎)) → ∃𝑎𝑉𝑏𝑉𝑐𝑉 ({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸 ∧ {𝑐, 𝑎} ∈ 𝐸))
149, 13syl 17 . . . . 5 ((𝐺 ∈ UPGraph ∧ 𝑓(Cycles‘𝐺)𝑝 ∧ (#‘𝑓) = 3) → ∃𝑎𝑉𝑏𝑉𝑐𝑉 ({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸 ∧ {𝑐, 𝑎} ∈ 𝐸))
152, 4, 6, 14syl3anc 1323 . . . 4 ((𝐺 ∈ UMGraph ∧ (𝑓(Cycles‘𝐺)𝑝 ∧ (#‘𝑓) = 3)) → ∃𝑎𝑉𝑏𝑉𝑐𝑉 ({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸 ∧ {𝑐, 𝑎} ∈ 𝐸))
1615ex 450 . . 3 (𝐺 ∈ UMGraph → ((𝑓(Cycles‘𝐺)𝑝 ∧ (#‘𝑓) = 3) → ∃𝑎𝑉𝑏𝑉𝑐𝑉 ({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸 ∧ {𝑐, 𝑎} ∈ 𝐸)))
1716exlimdvv 1859 . 2 (𝐺 ∈ UMGraph → (∃𝑓𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ (#‘𝑓) = 3) → ∃𝑎𝑉𝑏𝑉𝑐𝑉 ({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸 ∧ {𝑐, 𝑎} ∈ 𝐸)))
18 simplll 797 . . . . . 6 ((((𝐺 ∈ UMGraph ∧ (𝑎𝑉𝑏𝑉)) ∧ 𝑐𝑉) ∧ ({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸 ∧ {𝑐, 𝑎} ∈ 𝐸)) → 𝐺 ∈ UMGraph )
19 df-3an 1038 . . . . . . . 8 ((𝑎𝑉𝑏𝑉𝑐𝑉) ↔ ((𝑎𝑉𝑏𝑉) ∧ 𝑐𝑉))
2019biimpri 218 . . . . . . 7 (((𝑎𝑉𝑏𝑉) ∧ 𝑐𝑉) → (𝑎𝑉𝑏𝑉𝑐𝑉))
2120ad4ant23 1294 . . . . . 6 ((((𝐺 ∈ UMGraph ∧ (𝑎𝑉𝑏𝑉)) ∧ 𝑐𝑉) ∧ ({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸 ∧ {𝑐, 𝑎} ∈ 𝐸)) → (𝑎𝑉𝑏𝑉𝑐𝑉))
22 simpr 477 . . . . . 6 ((((𝐺 ∈ UMGraph ∧ (𝑎𝑉𝑏𝑉)) ∧ 𝑐𝑉) ∧ ({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸 ∧ {𝑐, 𝑎} ∈ 𝐸)) → ({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸 ∧ {𝑐, 𝑎} ∈ 𝐸))
238, 7umgr3cyclex 26922 . . . . . . 7 ((𝐺 ∈ UMGraph ∧ (𝑎𝑉𝑏𝑉𝑐𝑉) ∧ ({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸 ∧ {𝑐, 𝑎} ∈ 𝐸)) → ∃𝑓𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ (#‘𝑓) = 3 ∧ (𝑝‘0) = 𝑎))
24 3simpa 1056 . . . . . . . 8 ((𝑓(Cycles‘𝐺)𝑝 ∧ (#‘𝑓) = 3 ∧ (𝑝‘0) = 𝑎) → (𝑓(Cycles‘𝐺)𝑝 ∧ (#‘𝑓) = 3))
25242eximi 1760 . . . . . . 7 (∃𝑓𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ (#‘𝑓) = 3 ∧ (𝑝‘0) = 𝑎) → ∃𝑓𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ (#‘𝑓) = 3))
2623, 25syl 17 . . . . . 6 ((𝐺 ∈ UMGraph ∧ (𝑎𝑉𝑏𝑉𝑐𝑉) ∧ ({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸 ∧ {𝑐, 𝑎} ∈ 𝐸)) → ∃𝑓𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ (#‘𝑓) = 3))
2718, 21, 22, 26syl3anc 1323 . . . . 5 ((((𝐺 ∈ UMGraph ∧ (𝑎𝑉𝑏𝑉)) ∧ 𝑐𝑉) ∧ ({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸 ∧ {𝑐, 𝑎} ∈ 𝐸)) → ∃𝑓𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ (#‘𝑓) = 3))
2827ex 450 . . . 4 (((𝐺 ∈ UMGraph ∧ (𝑎𝑉𝑏𝑉)) ∧ 𝑐𝑉) → (({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸 ∧ {𝑐, 𝑎} ∈ 𝐸) → ∃𝑓𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ (#‘𝑓) = 3)))
2928rexlimdva 3025 . . 3 ((𝐺 ∈ UMGraph ∧ (𝑎𝑉𝑏𝑉)) → (∃𝑐𝑉 ({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸 ∧ {𝑐, 𝑎} ∈ 𝐸) → ∃𝑓𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ (#‘𝑓) = 3)))
3029rexlimdvva 3032 . 2 (𝐺 ∈ UMGraph → (∃𝑎𝑉𝑏𝑉𝑐𝑉 ({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸 ∧ {𝑐, 𝑎} ∈ 𝐸) → ∃𝑓𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ (#‘𝑓) = 3)))
3117, 30impbid 202 1 (𝐺 ∈ UMGraph → (∃𝑓𝑝(𝑓(Cycles‘𝐺)𝑝 ∧ (#‘𝑓) = 3) ↔ ∃𝑎𝑉𝑏𝑉𝑐𝑉 ({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸 ∧ {𝑐, 𝑎} ∈ 𝐸)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wex 1701  wcel 1987  wne 2790  wrex 2908  {cpr 4155   class class class wbr 4618  cfv 5852  0cc0 9887  3c3 11022  #chash 13064  Vtxcvtx 25787  Edgcedg 25852   UPGraph cupgr 25884   UMGraph cumgr 25885  Cyclesccycls 26562
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-cnex 9943  ax-resscn 9944  ax-1cn 9945  ax-icn 9946  ax-addcl 9947  ax-addrcl 9948  ax-mulcl 9949  ax-mulrcl 9950  ax-mulcom 9951  ax-addass 9952  ax-mulass 9953  ax-distr 9954  ax-i2m1 9955  ax-1ne0 9956  ax-1rid 9957  ax-rnegex 9958  ax-rrecex 9959  ax-cnre 9960  ax-pre-lttri 9961  ax-pre-lttrn 9962  ax-pre-ltadd 9963  ax-pre-mulgt0 9964
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ifp 1012  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-riota 6571  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-om 7020  df-1st 7120  df-2nd 7121  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-1o 7512  df-2o 7513  df-oadd 7516  df-er 7694  df-map 7811  df-pm 7812  df-en 7907  df-dom 7908  df-sdom 7909  df-fin 7910  df-card 8716  df-cda 8941  df-pnf 10027  df-mnf 10028  df-xr 10029  df-ltxr 10030  df-le 10031  df-sub 10219  df-neg 10220  df-nn 10972  df-2 11030  df-3 11031  df-4 11032  df-n0 11244  df-xnn0 11315  df-z 11329  df-uz 11639  df-fz 12276  df-fzo 12414  df-hash 13065  df-word 13245  df-concat 13247  df-s1 13248  df-s2 13537  df-s3 13538  df-s4 13539  df-edg 25853  df-uhgr 25862  df-upgr 25886  df-umgr 25887  df-wlks 26378  df-trls 26471  df-pths 26494  df-cycls 26564
This theorem is referenced by: (None)
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