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Theorem cusconngra 24499
Description: A complete (undirected simple) graph is connected. (Contributed by Alexander van der Vekens, 4-Dec-2017.)
Assertion
Ref Expression
cusconngra  |-  ( V ComplUSGrph  E  ->  V ConnGrph  E )

Proof of Theorem cusconngra
Dummy variables  f 
k  n  p are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cusisusgra 24281 . . 3  |-  ( V ComplUSGrph  E  ->  V USGrph  E )
2 usgrav 24161 . . 3  |-  ( V USGrph  E  ->  ( V  e. 
_V  /\  E  e.  _V ) )
31, 2syl 16 . 2  |-  ( V ComplUSGrph  E  ->  ( V  e. 
_V  /\  E  e.  _V ) )
4 simp-4l 765 . . . . . . . 8  |-  ( ( ( ( ( ( V  e.  _V  /\  E  e.  _V )  /\  V USGrph  E )  /\  k  e.  V )  /\  n  e.  ( V  \  { k } ) )  /\  {
n ,  k }  e.  ran  E )  ->  ( V  e. 
_V  /\  E  e.  _V ) )
5 simpr 461 . . . . . . . . . 10  |-  ( ( ( ( V  e. 
_V  /\  E  e.  _V )  /\  V USGrph  E
)  /\  k  e.  V )  ->  k  e.  V )
6 eldifi 3631 . . . . . . . . . 10  |-  ( n  e.  ( V  \  { k } )  ->  n  e.  V
)
75, 6anim12i 566 . . . . . . . . 9  |-  ( ( ( ( ( V  e.  _V  /\  E  e.  _V )  /\  V USGrph  E )  /\  k  e.  V )  /\  n  e.  ( V  \  {
k } ) )  ->  ( k  e.  V  /\  n  e.  V ) )
87adantr 465 . . . . . . . 8  |-  ( ( ( ( ( ( V  e.  _V  /\  E  e.  _V )  /\  V USGrph  E )  /\  k  e.  V )  /\  n  e.  ( V  \  { k } ) )  /\  {
n ,  k }  e.  ran  E )  ->  ( k  e.  V  /\  n  e.  V ) )
9 usgraf1o 24181 . . . . . . . . . . . . 13  |-  ( V USGrph  E  ->  E : dom  E -1-1-onto-> ran 
E )
109adantl 466 . . . . . . . . . . . 12  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  V USGrph  E )  ->  E : dom  E -1-1-onto-> ran  E
)
11 prcom 4111 . . . . . . . . . . . . . . 15  |-  { n ,  k }  =  { k ,  n }
1211eleq1i 2544 . . . . . . . . . . . . . 14  |-  ( { n ,  k }  e.  ran  E  <->  { k ,  n }  e.  ran  E )
13 f1ocnvfv2 6182 . . . . . . . . . . . . . 14  |-  ( ( E : dom  E -1-1-onto-> ran  E  /\  { k ,  n }  e.  ran  E )  ->  ( E `  ( `' E `  { k ,  n } ) )  =  { k ,  n } )
1412, 13sylan2b 475 . . . . . . . . . . . . 13  |-  ( ( E : dom  E -1-1-onto-> ran  E  /\  { n ,  k }  e.  ran  E )  ->  ( E `  ( `' E `  { k ,  n } ) )  =  { k ,  n } )
1514ex 434 . . . . . . . . . . . 12  |-  ( E : dom  E -1-1-onto-> ran  E  ->  ( { n ,  k }  e.  ran  E  ->  ( E `  ( `' E `  { k ,  n } ) )  =  { k ,  n } ) )
1610, 15syl 16 . . . . . . . . . . 11  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  V USGrph  E )  -> 
( { n ,  k }  e.  ran  E  ->  ( E `  ( `' E `  { k ,  n } ) )  =  { k ,  n } ) )
1716adantr 465 . . . . . . . . . 10  |-  ( ( ( ( V  e. 
_V  /\  E  e.  _V )  /\  V USGrph  E
)  /\  k  e.  V )  ->  ( { n ,  k }  e.  ran  E  ->  ( E `  ( `' E `  { k ,  n } ) )  =  { k ,  n } ) )
1817adantr 465 . . . . . . . . 9  |-  ( ( ( ( ( V  e.  _V  /\  E  e.  _V )  /\  V USGrph  E )  /\  k  e.  V )  /\  n  e.  ( V  \  {
k } ) )  ->  ( { n ,  k }  e.  ran  E  ->  ( E `  ( `' E `  { k ,  n } ) )  =  { k ,  n } ) )
1918imp 429 . . . . . . . 8  |-  ( ( ( ( ( ( V  e.  _V  /\  E  e.  _V )  /\  V USGrph  E )  /\  k  e.  V )  /\  n  e.  ( V  \  { k } ) )  /\  {
n ,  k }  e.  ran  E )  ->  ( E `  ( `' E `  { k ,  n } ) )  =  { k ,  n } )
20 1pthon2v 24418 . . . . . . . 8  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  ( k  e.  V  /\  n  e.  V
)  /\  ( E `  ( `' E `  { k ,  n } ) )  =  { k ,  n } )  ->  E. f E. p  f (
k ( V PathOn  E
) n ) p )
214, 8, 19, 20syl3anc 1228 . . . . . . 7  |-  ( ( ( ( ( ( V  e.  _V  /\  E  e.  _V )  /\  V USGrph  E )  /\  k  e.  V )  /\  n  e.  ( V  \  { k } ) )  /\  {
n ,  k }  e.  ran  E )  ->  E. f E. p  f ( k ( V PathOn  E ) n ) p )
2221ex 434 . . . . . 6  |-  ( ( ( ( ( V  e.  _V  /\  E  e.  _V )  /\  V USGrph  E )  /\  k  e.  V )  /\  n  e.  ( V  \  {
k } ) )  ->  ( { n ,  k }  e.  ran  E  ->  E. f E. p  f (
k ( V PathOn  E
) n ) p ) )
2322ralimdva 2875 . . . . 5  |-  ( ( ( ( V  e. 
_V  /\  E  e.  _V )  /\  V USGrph  E
)  /\  k  e.  V )  ->  ( A. n  e.  ( V  \  { k } ) { n ,  k }  e.  ran  E  ->  A. n  e.  ( V  \  { k } ) E. f E. p  f (
k ( V PathOn  E
) n ) p ) )
2423ralimdva 2875 . . . 4  |-  ( ( ( V  e.  _V  /\  E  e.  _V )  /\  V USGrph  E )  -> 
( A. k  e.  V  A. n  e.  ( V  \  {
k } ) { n ,  k }  e.  ran  E  ->  A. k  e.  V  A. n  e.  ( V  \  { k } ) E. f E. p  f ( k ( V PathOn  E ) n ) p ) )
2524expimpd 603 . . 3  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  ( ( V USGrph  E  /\  A. k  e.  V  A. n  e.  ( V  \  { k } ) { n ,  k }  e.  ran  E )  ->  A. k  e.  V  A. n  e.  ( V  \  {
k } ) E. f E. p  f ( k ( V PathOn  E ) n ) p ) )
26 iscusgra 24279 . . 3  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  ( V ComplUSGrph  E  <->  ( V USGrph  E  /\  A. k  e.  V  A. n  e.  ( V  \  {
k } ) { n ,  k }  e.  ran  E ) ) )
27 isconngra1 24496 . . 3  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  ( V ConnGrph  E  <->  A. k  e.  V  A. n  e.  ( V  \  {
k } ) E. f E. p  f ( k ( V PathOn  E ) n ) p ) )
2825, 26, 273imtr4d 268 . 2  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  ( V ComplUSGrph  E  ->  V ConnGrph  E ) )
293, 28mpcom 36 1  |-  ( V ComplUSGrph  E  ->  V ConnGrph  E )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379   E.wex 1596    e. wcel 1767   A.wral 2817   _Vcvv 3118    \ cdif 3478   {csn 4033   {cpr 4035   class class class wbr 4453   `'ccnv 5004   dom cdm 5005   ran crn 5006   -1-1-onto->wf1o 5593   ` cfv 5594  (class class class)co 6295   USGrph cusg 24153   ComplUSGrph ccusgra 24241   PathOn cpthon 24327   ConnGrph cconngra 24492
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-1o 7142  df-oadd 7146  df-er 7323  df-map 7434  df-pm 7435  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-card 8332  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-nn 10549  df-2 10606  df-n0 10808  df-z 10877  df-uz 11095  df-fz 11685  df-fzo 11805  df-hash 12386  df-word 12523  df-usgra 24156  df-cusgra 24244  df-wlk 24331  df-trail 24332  df-pth 24333  df-wlkon 24337  df-pthon 24339  df-conngra 24493
This theorem is referenced by: (None)
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