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Theorem 3v3e3cycl 28411
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 28240 . . 3  |-  ( V USGrph  E  ->  Fun  E )
2 19.41v 1854 . . . . . . 7  |-  ( E. p ( ( f ( V Cycles  E ) p  /\  ( # `  f )  =  3 )  /\  Fun  E
)  <->  ( E. p
( f ( V Cycles  E ) p  /\  ( # `  f )  =  3 )  /\  Fun  E ) )
32exbii 1572 . . . . . 6  |-  ( 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 ) )
4 19.41v 1854 . . . . . 6  |-  ( 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
) )
53, 4bitri 240 . . . . 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
) )
6 simpr 447 . . . . . . . 8  |-  ( ( ( f ( V Cycles  E ) p  /\  ( # `  f )  =  3 )  /\  Fun  E )  ->  Fun  E )
7 simpll 730 . . . . . . . 8  |-  ( ( ( f ( V Cycles  E ) p  /\  ( # `  f )  =  3 )  /\  Fun  E )  ->  f
( V Cycles  E )
p )
8 simplr 731 . . . . . . . 8  |-  ( ( ( f ( V Cycles  E ) p  /\  ( # `  f )  =  3 )  /\  Fun  E )  ->  ( # `
 f )  =  3 )
9 3v3e3cycl1 28390 . . . . . . . 8  |-  ( ( 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
) )
106, 7, 8, 9syl3anc 1182 . . . . . . 7  |-  ( ( ( 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 ) )
11102eximi 1567 . . . . . 6  |-  ( E. f E. p ( ( f ( V Cycles  E ) p  /\  ( # `  f )  =  3 )  /\  Fun  E )  ->  E. f E. p 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 id 19 . . . . . . 7  |-  ( 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. 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 ) )
1312exlimivv 1625 . . . . . 6  |-  ( E. f E. p 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. 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 ) )
1411, 13syl 15 . . . . 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 ) )
155, 14sylbir 204 . . . 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 ) )
1615expcom 424 . . 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 ) ) )
171, 16syl 15 . 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
) ) )
18 3v3e3cycl2 28410 . 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 ) ) )
1917, 18impbid 183 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 176    /\ wa 358    /\ w3a 934   E.wex 1531    = wceq 1632    e. wcel 1696   E.wrex 2557   {cpr 3654   class class class wbr 4039   ran crn 4706   Fun wfun 5265   ` cfv 5271  (class class class)co 5874   3c3 9812   #chash 11353   USGrph cusg 28227   Cycles ccycl 28318
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-map 6790  df-pm 6791  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-card 7588  df-cda 7810  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-3 9821  df-n0 9982  df-z 10041  df-uz 10247  df-fz 10799  df-fzo 10887  df-hash 11354  df-word 11425  df-usgra 28229  df-wlk 28319  df-trail 28320  df-pth 28321  df-cycl 28324
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