Theorem constr3cyclpe 26191

Description: If there are three (different) vertices in a graph which are mutually connected by edges, there is a 3-cycle in the graph containing one of these vertices. (Contributed by Alexander van der Vekens, 17-Nov-2017.)

Assertion

Ref	Expression
constr3cyclpe	⊢ ((𝑉 USGrph 𝐸 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)) → ∃𝑓∃𝑝(𝑓(𝑉 Cycles 𝐸)𝑝 ∧ (#‘𝑓) = 3 ∧ (𝑝‘0) = 𝐴))

Distinct variable groups:   𝐴,𝑓,𝑝   𝐵,𝑓,𝑝   𝐶,𝑓,𝑝   𝑓,𝐸,𝑝   𝑓,𝑉,𝑝

Proof of Theorem constr3cyclpe

Step	Hyp	Ref	Expression
1		eqid 2610	. . 3 ⊢ {⟨0, (◡𝐸‘{𝐴, 𝐵})⟩, ⟨1, (◡𝐸‘{𝐵, 𝐶})⟩, ⟨2, (◡𝐸‘{𝐶, 𝐴})⟩} = {⟨0, (◡𝐸‘{𝐴, 𝐵})⟩, ⟨1, (◡𝐸‘{𝐵, 𝐶})⟩, ⟨2, (◡𝐸‘{𝐶, 𝐴})⟩}
2		eqid 2610	. . 3 ⊢ ({⟨0, 𝐴⟩, ⟨1, 𝐵⟩} ∪ {⟨2, 𝐶⟩, ⟨3, 𝐴⟩}) = ({⟨0, 𝐴⟩, ⟨1, 𝐵⟩} ∪ {⟨2, 𝐶⟩, ⟨3, 𝐴⟩})
3	1, 2	constr3cyclp 26190	. 2 ⊢ ((𝑉 USGrph 𝐸 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)) → ({⟨0, (◡𝐸‘{𝐴, 𝐵})⟩, ⟨1, (◡𝐸‘{𝐵, 𝐶})⟩, ⟨2, (◡𝐸‘{𝐶, 𝐴})⟩} (𝑉 Cycles 𝐸)({⟨0, 𝐴⟩, ⟨1, 𝐵⟩} ∪ {⟨2, 𝐶⟩, ⟨3, 𝐴⟩}) ∧ (#‘{⟨0, (◡𝐸‘{𝐴, 𝐵})⟩, ⟨1, (◡𝐸‘{𝐵, 𝐶})⟩, ⟨2, (◡𝐸‘{𝐶, 𝐴})⟩}) = 3 ∧ (({⟨0, 𝐴⟩, ⟨1, 𝐵⟩} ∪ {⟨2, 𝐶⟩, ⟨3, 𝐴⟩})‘0) = 𝐴))
4		tpex 6855	. . 3 ⊢ {⟨0, (◡𝐸‘{𝐴, 𝐵})⟩, ⟨1, (◡𝐸‘{𝐵, 𝐶})⟩, ⟨2, (◡𝐸‘{𝐶, 𝐴})⟩} ∈ V
5		prex 4836	. . . 4 ⊢ {⟨0, 𝐴⟩, ⟨1, 𝐵⟩} ∈ V
6		prex 4836	. . . 4 ⊢ {⟨2, 𝐶⟩, ⟨3, 𝐴⟩} ∈ V
7	5, 6	unex 6854	. . 3 ⊢ ({⟨0, 𝐴⟩, ⟨1, 𝐵⟩} ∪ {⟨2, 𝐶⟩, ⟨3, 𝐴⟩}) ∈ V
8		breq12 4588	. . . 4 ⊢ ((𝑓 = {⟨0, (◡𝐸‘{𝐴, 𝐵})⟩, ⟨1, (◡𝐸‘{𝐵, 𝐶})⟩, ⟨2, (◡𝐸‘{𝐶, 𝐴})⟩} ∧ 𝑝 = ({⟨0, 𝐴⟩, ⟨1, 𝐵⟩} ∪ {⟨2, 𝐶⟩, ⟨3, 𝐴⟩})) → (𝑓(𝑉 Cycles 𝐸)𝑝 ↔ {⟨0, (◡𝐸‘{𝐴, 𝐵})⟩, ⟨1, (◡𝐸‘{𝐵, 𝐶})⟩, ⟨2, (◡𝐸‘{𝐶, 𝐴})⟩} (𝑉 Cycles 𝐸)({⟨0, 𝐴⟩, ⟨1, 𝐵⟩} ∪ {⟨2, 𝐶⟩, ⟨3, 𝐴⟩})))
9		fveq2 6103	. . . . . 6 ⊢ (𝑓 = {⟨0, (◡𝐸‘{𝐴, 𝐵})⟩, ⟨1, (◡𝐸‘{𝐵, 𝐶})⟩, ⟨2, (◡𝐸‘{𝐶, 𝐴})⟩} → (#‘𝑓) = (#‘{⟨0, (◡𝐸‘{𝐴, 𝐵})⟩, ⟨1, (◡𝐸‘{𝐵, 𝐶})⟩, ⟨2, (◡𝐸‘{𝐶, 𝐴})⟩}))
10	9	eqeq1d 2612	. . . . 5 ⊢ (𝑓 = {⟨0, (◡𝐸‘{𝐴, 𝐵})⟩, ⟨1, (◡𝐸‘{𝐵, 𝐶})⟩, ⟨2, (◡𝐸‘{𝐶, 𝐴})⟩} → ((#‘𝑓) = 3 ↔ (#‘{⟨0, (◡𝐸‘{𝐴, 𝐵})⟩, ⟨1, (◡𝐸‘{𝐵, 𝐶})⟩, ⟨2, (◡𝐸‘{𝐶, 𝐴})⟩}) = 3))
11	10	adantr 480	. . . 4 ⊢ ((𝑓 = {⟨0, (◡𝐸‘{𝐴, 𝐵})⟩, ⟨1, (◡𝐸‘{𝐵, 𝐶})⟩, ⟨2, (◡𝐸‘{𝐶, 𝐴})⟩} ∧ 𝑝 = ({⟨0, 𝐴⟩, ⟨1, 𝐵⟩} ∪ {⟨2, 𝐶⟩, ⟨3, 𝐴⟩})) → ((#‘𝑓) = 3 ↔ (#‘{⟨0, (◡𝐸‘{𝐴, 𝐵})⟩, ⟨1, (◡𝐸‘{𝐵, 𝐶})⟩, ⟨2, (◡𝐸‘{𝐶, 𝐴})⟩}) = 3))
12		fveq1 6102	. . . . . 6 ⊢ (𝑝 = ({⟨0, 𝐴⟩, ⟨1, 𝐵⟩} ∪ {⟨2, 𝐶⟩, ⟨3, 𝐴⟩}) → (𝑝‘0) = (({⟨0, 𝐴⟩, ⟨1, 𝐵⟩} ∪ {⟨2, 𝐶⟩, ⟨3, 𝐴⟩})‘0))
13	12	eqeq1d 2612	. . . . 5 ⊢ (𝑝 = ({⟨0, 𝐴⟩, ⟨1, 𝐵⟩} ∪ {⟨2, 𝐶⟩, ⟨3, 𝐴⟩}) → ((𝑝‘0) = 𝐴 ↔ (({⟨0, 𝐴⟩, ⟨1, 𝐵⟩} ∪ {⟨2, 𝐶⟩, ⟨3, 𝐴⟩})‘0) = 𝐴))
14	13	adantl 481	. . . 4 ⊢ ((𝑓 = {⟨0, (◡𝐸‘{𝐴, 𝐵})⟩, ⟨1, (◡𝐸‘{𝐵, 𝐶})⟩, ⟨2, (◡𝐸‘{𝐶, 𝐴})⟩} ∧ 𝑝 = ({⟨0, 𝐴⟩, ⟨1, 𝐵⟩} ∪ {⟨2, 𝐶⟩, ⟨3, 𝐴⟩})) → ((𝑝‘0) = 𝐴 ↔ (({⟨0, 𝐴⟩, ⟨1, 𝐵⟩} ∪ {⟨2, 𝐶⟩, ⟨3, 𝐴⟩})‘0) = 𝐴))
15	8, 11, 14	3anbi123d 1391	. . 3 ⊢ ((𝑓 = {⟨0, (◡𝐸‘{𝐴, 𝐵})⟩, ⟨1, (◡𝐸‘{𝐵, 𝐶})⟩, ⟨2, (◡𝐸‘{𝐶, 𝐴})⟩} ∧ 𝑝 = ({⟨0, 𝐴⟩, ⟨1, 𝐵⟩} ∪ {⟨2, 𝐶⟩, ⟨3, 𝐴⟩})) → ((𝑓(𝑉 Cycles 𝐸)𝑝 ∧ (#‘𝑓) = 3 ∧ (𝑝‘0) = 𝐴) ↔ ({⟨0, (◡𝐸‘{𝐴, 𝐵})⟩, ⟨1, (◡𝐸‘{𝐵, 𝐶})⟩, ⟨2, (◡𝐸‘{𝐶, 𝐴})⟩} (𝑉 Cycles 𝐸)({⟨0, 𝐴⟩, ⟨1, 𝐵⟩} ∪ {⟨2, 𝐶⟩, ⟨3, 𝐴⟩}) ∧ (#‘{⟨0, (◡𝐸‘{𝐴, 𝐵})⟩, ⟨1, (◡𝐸‘{𝐵, 𝐶})⟩, ⟨2, (◡𝐸‘{𝐶, 𝐴})⟩}) = 3 ∧ (({⟨0, 𝐴⟩, ⟨1, 𝐵⟩} ∪ {⟨2, 𝐶⟩, ⟨3, 𝐴⟩})‘0) = 𝐴)))
16	4, 7, 15	spc2ev 3274	. 2 ⊢ (({⟨0, (◡𝐸‘{𝐴, 𝐵})⟩, ⟨1, (◡𝐸‘{𝐵, 𝐶})⟩, ⟨2, (◡𝐸‘{𝐶, 𝐴})⟩} (𝑉 Cycles 𝐸)({⟨0, 𝐴⟩, ⟨1, 𝐵⟩} ∪ {⟨2, 𝐶⟩, ⟨3, 𝐴⟩}) ∧ (#‘{⟨0, (◡𝐸‘{𝐴, 𝐵})⟩, ⟨1, (◡𝐸‘{𝐵, 𝐶})⟩, ⟨2, (◡𝐸‘{𝐶, 𝐴})⟩}) = 3 ∧ (({⟨0, 𝐴⟩, ⟨1, 𝐵⟩} ∪ {⟨2, 𝐶⟩, ⟨3, 𝐴⟩})‘0) = 𝐴) → ∃𝑓∃𝑝(𝑓(𝑉 Cycles 𝐸)𝑝 ∧ (#‘𝑓) = 3 ∧ (𝑝‘0) = 𝐴))
17	3, 16	syl 17	1 ⊢ ((𝑉 USGrph 𝐸 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ∧ ({𝐴, 𝐵} ∈ ran 𝐸 ∧ {𝐵, 𝐶} ∈ ran 𝐸 ∧ {𝐶, 𝐴} ∈ ran 𝐸)) → ∃𝑓∃𝑝(𝑓(𝑉 Cycles 𝐸)𝑝 ∧ (#‘𝑓) = 3 ∧ (𝑝‘0) = 𝐴))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475  ∃wex 1695   ∈ 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:  3cyclfrgra  26542