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Theorem umgr3v3e3cycl 41351
 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 25769 . . . . . 6 (𝐺 ∈ UMGraph → 𝐺 ∈ UPGraph )
21adantr 480 . . . . 5 ((𝐺 ∈ UMGraph ∧ (𝑓(CycleS‘𝐺)𝑝 ∧ (#‘𝑓) = 3)) → 𝐺 ∈ UPGraph )
3 simpl 472 . . . . . 6 ((𝑓(CycleS‘𝐺)𝑝 ∧ (#‘𝑓) = 3) → 𝑓(CycleS‘𝐺)𝑝)
43adantl 481 . . . . 5 ((𝐺 ∈ UMGraph ∧ (𝑓(CycleS‘𝐺)𝑝 ∧ (#‘𝑓) = 3)) → 𝑓(CycleS‘𝐺)𝑝)
5 simpr 476 . . . . . 6 ((𝑓(CycleS‘𝐺)𝑝 ∧ (#‘𝑓) = 3) → (#‘𝑓) = 3)
65adantl 481 . . . . 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 41347 . . . . . 6 ((𝐺 ∈ UPGraph ∧ 𝑓(CycleS‘𝐺)𝑝 ∧ (#‘𝑓) = 3) → ∃𝑎𝑉𝑏𝑉𝑐𝑉 (({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸 ∧ {𝑐, 𝑎} ∈ 𝐸) ∧ (𝑎𝑏𝑏𝑐𝑐𝑎)))
10 simpl 472 . . . . . . . . 9 ((({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸 ∧ {𝑐, 𝑎} ∈ 𝐸) ∧ (𝑎𝑏𝑏𝑐𝑐𝑎)) → ({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸 ∧ {𝑐, 𝑎} ∈ 𝐸))
1110reximi 2994 . . . . . . . 8 (∃𝑐𝑉 (({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸 ∧ {𝑐, 𝑎} ∈ 𝐸) ∧ (𝑎𝑏𝑏𝑐𝑐𝑎)) → ∃𝑐𝑉 ({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸 ∧ {𝑐, 𝑎} ∈ 𝐸))
1211reximi 2994 . . . . . . 7 (∃𝑏𝑉𝑐𝑉 (({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸 ∧ {𝑐, 𝑎} ∈ 𝐸) ∧ (𝑎𝑏𝑏𝑐𝑐𝑎)) → ∃𝑏𝑉𝑐𝑉 ({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸 ∧ {𝑐, 𝑎} ∈ 𝐸))
1312reximi 2994 . . . . . 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 1318 . . . 4 ((𝐺 ∈ UMGraph ∧ (𝑓(CycleS‘𝐺)𝑝 ∧ (#‘𝑓) = 3)) → ∃𝑎𝑉𝑏𝑉𝑐𝑉 ({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸 ∧ {𝑐, 𝑎} ∈ 𝐸))
1615ex 449 . . 3 (𝐺 ∈ UMGraph → ((𝑓(CycleS‘𝐺)𝑝 ∧ (#‘𝑓) = 3) → ∃𝑎𝑉𝑏𝑉𝑐𝑉 ({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸 ∧ {𝑐, 𝑎} ∈ 𝐸)))
1716exlimdvv 1849 . 2 (𝐺 ∈ UMGraph → (∃𝑓𝑝(𝑓(CycleS‘𝐺)𝑝 ∧ (#‘𝑓) = 3) → ∃𝑎𝑉𝑏𝑉𝑐𝑉 ({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸 ∧ {𝑐, 𝑎} ∈ 𝐸)))
18 simplll 794 . . . . . 6 ((((𝐺 ∈ UMGraph ∧ (𝑎𝑉𝑏𝑉)) ∧ 𝑐𝑉) ∧ ({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸 ∧ {𝑐, 𝑎} ∈ 𝐸)) → 𝐺 ∈ UMGraph )
19 df-3an 1033 . . . . . . . 8 ((𝑎𝑉𝑏𝑉𝑐𝑉) ↔ ((𝑎𝑉𝑏𝑉) ∧ 𝑐𝑉))
2019biimpri 217 . . . . . . 7 (((𝑎𝑉𝑏𝑉) ∧ 𝑐𝑉) → (𝑎𝑉𝑏𝑉𝑐𝑉))
2120ad4ant23 1289 . . . . . 6 ((((𝐺 ∈ UMGraph ∧ (𝑎𝑉𝑏𝑉)) ∧ 𝑐𝑉) ∧ ({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸 ∧ {𝑐, 𝑎} ∈ 𝐸)) → (𝑎𝑉𝑏𝑉𝑐𝑉))
22 simpr 476 . . . . . 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 41350 . . . . . . 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 1051 . . . . . . . 8 ((𝑓(CycleS‘𝐺)𝑝 ∧ (#‘𝑓) = 3 ∧ (𝑝‘0) = 𝑎) → (𝑓(CycleS‘𝐺)𝑝 ∧ (#‘𝑓) = 3))
25242eximi 1753 . . . . . . 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 1318 . . . . 5 ((((𝐺 ∈ UMGraph ∧ (𝑎𝑉𝑏𝑉)) ∧ 𝑐𝑉) ∧ ({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸 ∧ {𝑐, 𝑎} ∈ 𝐸)) → ∃𝑓𝑝(𝑓(CycleS‘𝐺)𝑝 ∧ (#‘𝑓) = 3))
2827ex 449 . . . 4 (((𝐺 ∈ UMGraph ∧ (𝑎𝑉𝑏𝑉)) ∧ 𝑐𝑉) → (({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸 ∧ {𝑐, 𝑎} ∈ 𝐸) → ∃𝑓𝑝(𝑓(CycleS‘𝐺)𝑝 ∧ (#‘𝑓) = 3)))
2928rexlimdva 3013 . . 3 ((𝐺 ∈ UMGraph ∧ (𝑎𝑉𝑏𝑉)) → (∃𝑐𝑉 ({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸 ∧ {𝑐, 𝑎} ∈ 𝐸) → ∃𝑓𝑝(𝑓(CycleS‘𝐺)𝑝 ∧ (#‘𝑓) = 3)))
3029rexlimdvva 3020 . 2 (𝐺 ∈ UMGraph → (∃𝑎𝑉𝑏𝑉𝑐𝑉 ({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑐} ∈ 𝐸 ∧ {𝑐, 𝑎} ∈ 𝐸) → ∃𝑓𝑝(𝑓(CycleS‘𝐺)𝑝 ∧ (#‘𝑓) = 3)))
3117, 30impbid 201 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 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475  ∃wex 1695   ∈ wcel 1977   ≠ wne 2780  ∃wrex 2897  {cpr 4127   class class class wbr 4583  ‘cfv 5804  0cc0 9815  3c3 10948  #chash 12979  Vtxcvtx 25673   UPGraph cupgr 25747   UMGraph cumgr 25748  Edgcedga 25792  CycleSccycls 40991 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-ifp 1007  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335  df-hash 12980  df-word 13154  df-concat 13156  df-s1 13157  df-s2 13444  df-s3 13445  df-s4 13446  df-uhgr 25724  df-upgr 25749  df-umgr 25750  df-edga 25793  df-1wlks 40800  df-wlks 40801  df-trls 40901  df-pths 40923  df-cycls 40993 This theorem is referenced by: (None)
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