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Theorem 3v3e3cycl 21640
Description: If and only if there is a 3-cycle in a graph, there are three (different) vertices in the graph which are mutually connected by edges. (Contributed by Alexander van der Vekens, 14-Nov-2017.)
Assertion
Ref Expression
3v3e3cycl  |-  ( V USGrph  E  ->  ( E. f E. p ( f ( V Cycles  E ) p  /\  ( # `  f
)  =  3 )  <->  E. a  e.  V  E. b  e.  V  E. c  e.  V  ( { a ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E  /\  { c ,  a }  e.  ran  E
) ) )
Distinct variable groups:    E, a,
b, c, f, p    V, a, b, c, f, p

Proof of Theorem 3v3e3cycl
StepHypRef Expression
1 usgrafun 21366 . . 3  |-  ( V USGrph  E  ->  Fun  E )
2 19.41vv 1925 . . . . 5  |-  ( E. f E. p ( ( f ( V Cycles  E ) p  /\  ( # `  f )  =  3 )  /\  Fun  E )  <->  ( E. f E. p ( f ( V Cycles  E ) p  /\  ( # `  f )  =  3 )  /\  Fun  E
) )
3 simpr 448 . . . . . . 7  |-  ( ( ( f ( V Cycles  E ) p  /\  ( # `  f )  =  3 )  /\  Fun  E )  ->  Fun  E )
4 simpll 731 . . . . . . 7  |-  ( ( ( f ( V Cycles  E ) p  /\  ( # `  f )  =  3 )  /\  Fun  E )  ->  f
( V Cycles  E )
p )
5 simplr 732 . . . . . . 7  |-  ( ( ( f ( V Cycles  E ) p  /\  ( # `  f )  =  3 )  /\  Fun  E )  ->  ( # `
 f )  =  3 )
6 3v3e3cycl1 21619 . . . . . . 7  |-  ( ( Fun  E  /\  f
( V Cycles  E )
p  /\  ( # `  f
)  =  3 )  ->  E. a  e.  V  E. b  e.  V  E. c  e.  V  ( { a ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E  /\  { c ,  a }  e.  ran  E
) )
73, 4, 5, 6syl3anc 1184 . . . . . 6  |-  ( ( ( f ( V Cycles  E ) p  /\  ( # `  f )  =  3 )  /\  Fun  E )  ->  E. a  e.  V  E. b  e.  V  E. c  e.  V  ( {
a ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E  /\  { c ,  a }  e.  ran  E ) )
87exlimivv 1645 . . . . 5  |-  ( E. f E. p ( ( f ( V Cycles  E ) p  /\  ( # `  f )  =  3 )  /\  Fun  E )  ->  E. a  e.  V  E. b  e.  V  E. c  e.  V  ( {
a ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E  /\  { c ,  a }  e.  ran  E ) )
92, 8sylbir 205 . . . 4  |-  ( ( E. f E. p
( f ( V Cycles  E ) p  /\  ( # `  f )  =  3 )  /\  Fun  E )  ->  E. a  e.  V  E. b  e.  V  E. c  e.  V  ( {
a ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E  /\  { c ,  a }  e.  ran  E ) )
109expcom 425 . . 3  |-  ( Fun 
E  ->  ( E. f E. p ( f ( V Cycles  E ) p  /\  ( # `  f )  =  3 )  ->  E. a  e.  V  E. b  e.  V  E. c  e.  V  ( {
a ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E  /\  { c ,  a }  e.  ran  E ) ) )
111, 10syl 16 . 2  |-  ( V USGrph  E  ->  ( E. f E. p ( f ( V Cycles  E ) p  /\  ( # `  f
)  =  3 )  ->  E. a  e.  V  E. b  e.  V  E. c  e.  V  ( { a ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E  /\  { c ,  a }  e.  ran  E
) ) )
12 3v3e3cycl2 21639 . 2  |-  ( V USGrph  E  ->  ( E. a  e.  V  E. b  e.  V  E. c  e.  V  ( {
a ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E  /\  { c ,  a }  e.  ran  E )  ->  E. f E. p
( f ( V Cycles  E ) p  /\  ( # `  f )  =  3 ) ) )
1311, 12impbid 184 1  |-  ( V USGrph  E  ->  ( E. f E. p ( f ( V Cycles  E ) p  /\  ( # `  f
)  =  3 )  <->  E. a  e.  V  E. b  e.  V  E. c  e.  V  ( { a ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E  /\  { c ,  a }  e.  ran  E
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936   E.wex 1550    = wceq 1652    e. wcel 1725   E.wrex 2698   {cpr 3807   class class class wbr 4204   ran crn 4870   Fun wfun 5439   ` cfv 5445  (class class class)co 6072   3c3 10039   #chash 11606   USGrph cusg 21353   Cycles ccycl 21503
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4692  ax-cnex 9035  ax-resscn 9036  ax-1cn 9037  ax-icn 9038  ax-addcl 9039  ax-addrcl 9040  ax-mulcl 9041  ax-mulrcl 9042  ax-mulcom 9043  ax-addass 9044  ax-mulass 9045  ax-distr 9046  ax-i2m1 9047  ax-1ne0 9048  ax-1rid 9049  ax-rnegex 9050  ax-rrecex 9051  ax-cnre 9052  ax-pre-lttri 9053  ax-pre-lttrn 9054  ax-pre-ltadd 9055  ax-pre-mulgt0 9056
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4837  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-ov 6075  df-oprab 6076  df-mpt2 6077  df-1st 6340  df-2nd 6341  df-riota 6540  df-recs 6624  df-rdg 6659  df-1o 6715  df-oadd 6719  df-er 6896  df-map 7011  df-pm 7012  df-en 7101  df-dom 7102  df-sdom 7103  df-fin 7104  df-card 7815  df-cda 8037  df-pnf 9111  df-mnf 9112  df-xr 9113  df-ltxr 9114  df-le 9115  df-sub 9282  df-neg 9283  df-nn 9990  df-2 10047  df-3 10048  df-n0 10211  df-z 10272  df-uz 10478  df-fz 11033  df-fzo 11124  df-hash 11607  df-word 11711  df-usgra 21355  df-wlk 21504  df-trail 21505  df-pth 21506  df-cycl 21509
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