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Theorem constr3cyclp 26190
 Description: Construction of a 3-cycle from three given edges in a graph, containing an endpoint of one of these edges. (Contributed by Alexander van der Vekens, 17-Nov-2017.)
Hypotheses
Ref Expression
constr3cycl.f 𝐹 = {⟨0, (𝐸‘{𝐴, 𝐵})⟩, ⟨1, (𝐸‘{𝐵, 𝐶})⟩, ⟨2, (𝐸‘{𝐶, 𝐴})⟩}
constr3cycl.p 𝑃 = ({⟨0, 𝐴⟩, ⟨1, 𝐵⟩} ∪ {⟨2, 𝐶⟩, ⟨3, 𝐴⟩})
Assertion
Ref Expression
constr3cyclp ((𝑉 USGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉𝐶𝑉) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)) → (𝐹(𝑉 Cycles 𝐸)𝑃 ∧ (#‘𝐹) = 3 ∧ (𝑃‘0) = 𝐴))

Proof of Theorem constr3cyclp
StepHypRef Expression
1 constr3cycl.f . . 3 𝐹 = {⟨0, (𝐸‘{𝐴, 𝐵})⟩, ⟨1, (𝐸‘{𝐵, 𝐶})⟩, ⟨2, (𝐸‘{𝐶, 𝐴})⟩}
2 constr3cycl.p . . 3 𝑃 = ({⟨0, 𝐴⟩, ⟨1, 𝐵⟩} ∪ {⟨2, 𝐶⟩, ⟨3, 𝐴⟩})
31, 2constr3cycl 26189 . 2 ((𝑉 USGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉𝐶𝑉) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)) → (𝐹(𝑉 Cycles 𝐸)𝑃 ∧ (#‘𝐹) = 3))
4 simprl 790 . . . . 5 (((𝐴𝑉𝐵𝑉𝐶𝑉) ∧ (𝐹(𝑉 Cycles 𝐸)𝑃 ∧ (#‘𝐹) = 3)) → 𝐹(𝑉 Cycles 𝐸)𝑃)
5 simprr 792 . . . . 5 (((𝐴𝑉𝐵𝑉𝐶𝑉) ∧ (𝐹(𝑉 Cycles 𝐸)𝑃 ∧ (#‘𝐹) = 3)) → (#‘𝐹) = 3)
61, 2constr3lem4 26175 . . . . . . 7 ((𝐴𝑉𝐵𝑉𝐶𝑉) → (((𝑃‘0) = 𝐴 ∧ (𝑃‘1) = 𝐵) ∧ ((𝑃‘2) = 𝐶 ∧ (𝑃‘3) = 𝐴)))
76simplld 787 . . . . . 6 ((𝐴𝑉𝐵𝑉𝐶𝑉) → (𝑃‘0) = 𝐴)
87adantr 480 . . . . 5 (((𝐴𝑉𝐵𝑉𝐶𝑉) ∧ (𝐹(𝑉 Cycles 𝐸)𝑃 ∧ (#‘𝐹) = 3)) → (𝑃‘0) = 𝐴)
94, 5, 83jca 1235 . . . 4 (((𝐴𝑉𝐵𝑉𝐶𝑉) ∧ (𝐹(𝑉 Cycles 𝐸)𝑃 ∧ (#‘𝐹) = 3)) → (𝐹(𝑉 Cycles 𝐸)𝑃 ∧ (#‘𝐹) = 3 ∧ (𝑃‘0) = 𝐴))
109ex 449 . . 3 ((𝐴𝑉𝐵𝑉𝐶𝑉) → ((𝐹(𝑉 Cycles 𝐸)𝑃 ∧ (#‘𝐹) = 3) → (𝐹(𝑉 Cycles 𝐸)𝑃 ∧ (#‘𝐹) = 3 ∧ (𝑃‘0) = 𝐴)))
11103ad2ant2 1076 . 2 ((𝑉 USGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉𝐶𝑉) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)) → ((𝐹(𝑉 Cycles 𝐸)𝑃 ∧ (#‘𝐹) = 3) → (𝐹(𝑉 Cycles 𝐸)𝑃 ∧ (#‘𝐹) = 3 ∧ (𝑃‘0) = 𝐴)))
123, 11mpd 15 1 ((𝑉 USGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉𝐶𝑉) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)) → (𝐹(𝑉 Cycles 𝐸)𝑃 ∧ (#‘𝐹) = 3 ∧ (𝑃‘0) = 𝐴))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ∪ cun 3538  {cpr 4127  {ctp 4129  ⟨cop 4131   class class class wbr 4583  ◡ccnv 5037  ran crn 5039  ‘cfv 5804  (class class class)co 6549  0cc0 9815  1c1 9816  2c2 10947  3c3 10948  #chash 12979   USGrph cusg 25859   Cycles ccycl 26035 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-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-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-n0 11170  df-xnn0 11241  df-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335  df-hash 12980  df-word 13154  df-usgra 25862  df-wlk 26036  df-trail 26037  df-pth 26038  df-cycl 26041 This theorem is referenced by:  constr3cyclpe  26191
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