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Theorem uvtxnbgra 25300
Description: A universal vertex has all other vertices as neighbors. (Contributed by Alexander van der Vekens, 14-Oct-2017.)
Assertion
Ref Expression
uvtxnbgra  |-  ( ( V USGrph  E  /\  N  e.  ( V UnivVertex  E )
)  ->  ( <. V ,  E >. Neighbors  N )  =  ( V  \  { N } ) )

Proof of Theorem uvtxnbgra
Dummy variables  v  n  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 usgrav 25144 . . . 4  |-  ( V USGrph  E  ->  ( V  e. 
_V  /\  E  e.  _V ) )
2 uvtxel 25296 . . . 4  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  ( N  e.  ( V UnivVertex  E )  <->  ( N  e.  V  /\  A. v  e.  ( V  \  { N } ) { v ,  N }  e.  ran  E ) ) )
31, 2syl 17 . . 3  |-  ( V USGrph  E  ->  ( N  e.  ( V UnivVertex  E )  <->  ( N  e.  V  /\  A. v  e.  ( V 
\  { N }
) { v ,  N }  e.  ran  E ) ) )
4 nbusgra 25235 . . . . . 6  |-  ( V USGrph  E  ->  ( <. V ,  E >. Neighbors  N )  =  {
n  e.  V  |  { N ,  n }  e.  ran  E } )
54adantr 472 . . . . 5  |-  ( ( V USGrph  E  /\  ( N  e.  V  /\  A. v  e.  ( V 
\  { N }
) { v ,  N }  e.  ran  E ) )  ->  ( <. V ,  E >. Neighbors  N
)  =  { n  e.  V  |  { N ,  n }  e.  ran  E } )
6 preq2 4043 . . . . . . . . . 10  |-  ( n  =  x  ->  { N ,  n }  =  { N ,  x }
)
76eleq1d 2533 . . . . . . . . 9  |-  ( n  =  x  ->  ( { N ,  n }  e.  ran  E  <->  { N ,  x }  e.  ran  E ) )
87elrab 3184 . . . . . . . 8  |-  ( x  e.  { n  e.  V  |  { N ,  n }  e.  ran  E }  <->  ( x  e.  V  /\  { N ,  x }  e.  ran  E ) )
9 usgraedgrn 25187 . . . . . . . . . . . . . 14  |-  ( ( V USGrph  E  /\  { N ,  x }  e.  ran  E )  ->  N  =/=  x )
10 simpr 468 . . . . . . . . . . . . . . . 16  |-  ( ( N  =/=  x  /\  x  e.  V )  ->  x  e.  V )
11 elsni 3985 . . . . . . . . . . . . . . . . . . 19  |-  ( x  e.  { N }  ->  x  =  N )
1211eqcomd 2477 . . . . . . . . . . . . . . . . . 18  |-  ( x  e.  { N }  ->  N  =  x )
1312necon3ai 2668 . . . . . . . . . . . . . . . . 17  |-  ( N  =/=  x  ->  -.  x  e.  { N } )
1413adantr 472 . . . . . . . . . . . . . . . 16  |-  ( ( N  =/=  x  /\  x  e.  V )  ->  -.  x  e.  { N } )
1510, 14eldifd 3401 . . . . . . . . . . . . . . 15  |-  ( ( N  =/=  x  /\  x  e.  V )  ->  x  e.  ( V 
\  { N }
) )
1615ex 441 . . . . . . . . . . . . . 14  |-  ( N  =/=  x  ->  (
x  e.  V  ->  x  e.  ( V  \  { N } ) ) )
179, 16syl 17 . . . . . . . . . . . . 13  |-  ( ( V USGrph  E  /\  { N ,  x }  e.  ran  E )  ->  ( x  e.  V  ->  x  e.  ( V  \  { N } ) ) )
1817ex 441 . . . . . . . . . . . 12  |-  ( V USGrph  E  ->  ( { N ,  x }  e.  ran  E  ->  ( x  e.  V  ->  x  e.  ( V  \  { N } ) ) ) )
1918com13 82 . . . . . . . . . . 11  |-  ( x  e.  V  ->  ( { N ,  x }  e.  ran  E  ->  ( V USGrph  E  ->  x  e.  ( V  \  { N } ) ) ) )
2019imp 436 . . . . . . . . . 10  |-  ( ( x  e.  V  /\  { N ,  x }  e.  ran  E )  -> 
( V USGrph  E  ->  x  e.  ( V  \  { N } ) ) )
2120com12 31 . . . . . . . . 9  |-  ( V USGrph  E  ->  ( ( x  e.  V  /\  { N ,  x }  e.  ran  E )  ->  x  e.  ( V  \  { N } ) ) )
2221adantr 472 . . . . . . . 8  |-  ( ( V USGrph  E  /\  ( N  e.  V  /\  A. v  e.  ( V 
\  { N }
) { v ,  N }  e.  ran  E ) )  ->  (
( x  e.  V  /\  { N ,  x }  e.  ran  E )  ->  x  e.  ( V  \  { N } ) ) )
238, 22syl5bi 225 . . . . . . 7  |-  ( ( V USGrph  E  /\  ( N  e.  V  /\  A. v  e.  ( V 
\  { N }
) { v ,  N }  e.  ran  E ) )  ->  (
x  e.  { n  e.  V  |  { N ,  n }  e.  ran  E }  ->  x  e.  ( V  \  { N } ) ) )
24 eldifi 3544 . . . . . . . . . 10  |-  ( x  e.  ( V  \  { N } )  ->  x  e.  V )
2524adantl 473 . . . . . . . . 9  |-  ( ( ( V USGrph  E  /\  ( N  e.  V  /\  A. v  e.  ( V  \  { N } ) { v ,  N }  e.  ran  E ) )  /\  x  e.  ( V  \  { N } ) )  ->  x  e.  V )
26 preq1 4042 . . . . . . . . . . . . . . . . 17  |-  ( v  =  x  ->  { v ,  N }  =  { x ,  N } )
2726eleq1d 2533 . . . . . . . . . . . . . . . 16  |-  ( v  =  x  ->  ( { v ,  N }  e.  ran  E  <->  { x ,  N }  e.  ran  E ) )
2827rspcva 3134 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  ( V 
\  { N }
)  /\  A. v  e.  ( V  \  { N } ) { v ,  N }  e.  ran  E )  ->  { x ,  N }  e.  ran  E )
29 prcom 4041 . . . . . . . . . . . . . . . . . 18  |-  { x ,  N }  =  { N ,  x }
3029eleq1i 2540 . . . . . . . . . . . . . . . . 17  |-  ( { x ,  N }  e.  ran  E  <->  { N ,  x }  e.  ran  E )
3130biimpi 199 . . . . . . . . . . . . . . . 16  |-  ( { x ,  N }  e.  ran  E  ->  { N ,  x }  e.  ran  E )
32312a1d 26 . . . . . . . . . . . . . . 15  |-  ( { x ,  N }  e.  ran  E  ->  ( V USGrph  E  ->  ( N  e.  V  ->  { N ,  x }  e.  ran  E ) ) )
3328, 32syl 17 . . . . . . . . . . . . . 14  |-  ( ( x  e.  ( V 
\  { N }
)  /\  A. v  e.  ( V  \  { N } ) { v ,  N }  e.  ran  E )  ->  ( V USGrph  E  ->  ( N  e.  V  ->  { N ,  x }  e.  ran  E ) ) )
3433ex 441 . . . . . . . . . . . . 13  |-  ( x  e.  ( V  \  { N } )  -> 
( A. v  e.  ( V  \  { N } ) { v ,  N }  e.  ran  E  ->  ( V USGrph  E  ->  ( N  e.  V  ->  { N ,  x }  e.  ran  E ) ) ) )
3534com14 90 . . . . . . . . . . . 12  |-  ( N  e.  V  ->  ( A. v  e.  ( V  \  { N }
) { v ,  N }  e.  ran  E  ->  ( V USGrph  E  ->  ( x  e.  ( V  \  { N } )  ->  { N ,  x }  e.  ran  E ) ) ) )
3635imp 436 . . . . . . . . . . 11  |-  ( ( N  e.  V  /\  A. v  e.  ( V 
\  { N }
) { v ,  N }  e.  ran  E )  ->  ( V USGrph  E  ->  ( x  e.  ( V  \  { N } )  ->  { N ,  x }  e.  ran  E ) ) )
3736impcom 437 . . . . . . . . . 10  |-  ( ( V USGrph  E  /\  ( N  e.  V  /\  A. v  e.  ( V 
\  { N }
) { v ,  N }  e.  ran  E ) )  ->  (
x  e.  ( V 
\  { N }
)  ->  { N ,  x }  e.  ran  E ) )
3837imp 436 . . . . . . . . 9  |-  ( ( ( V USGrph  E  /\  ( N  e.  V  /\  A. v  e.  ( V  \  { N } ) { v ,  N }  e.  ran  E ) )  /\  x  e.  ( V  \  { N } ) )  ->  { N ,  x }  e.  ran  E )
3925, 38, 8sylanbrc 677 . . . . . . . 8  |-  ( ( ( V USGrph  E  /\  ( N  e.  V  /\  A. v  e.  ( V  \  { N } ) { v ,  N }  e.  ran  E ) )  /\  x  e.  ( V  \  { N } ) )  ->  x  e.  { n  e.  V  |  { N ,  n }  e.  ran  E } )
4039ex 441 . . . . . . 7  |-  ( ( V USGrph  E  /\  ( N  e.  V  /\  A. v  e.  ( V 
\  { N }
) { v ,  N }  e.  ran  E ) )  ->  (
x  e.  ( V 
\  { N }
)  ->  x  e.  { n  e.  V  |  { N ,  n }  e.  ran  E } ) )
4123, 40impbid 195 . . . . . 6  |-  ( ( V USGrph  E  /\  ( N  e.  V  /\  A. v  e.  ( V 
\  { N }
) { v ,  N }  e.  ran  E ) )  ->  (
x  e.  { n  e.  V  |  { N ,  n }  e.  ran  E }  <->  x  e.  ( V  \  { N } ) ) )
4241eqrdv 2469 . . . . 5  |-  ( ( V USGrph  E  /\  ( N  e.  V  /\  A. v  e.  ( V 
\  { N }
) { v ,  N }  e.  ran  E ) )  ->  { n  e.  V  |  { N ,  n }  e.  ran  E }  =  ( V  \  { N } ) )
435, 42eqtrd 2505 . . . 4  |-  ( ( V USGrph  E  /\  ( N  e.  V  /\  A. v  e.  ( V 
\  { N }
) { v ,  N }  e.  ran  E ) )  ->  ( <. V ,  E >. Neighbors  N
)  =  ( V 
\  { N }
) )
4443ex 441 . . 3  |-  ( V USGrph  E  ->  ( ( N  e.  V  /\  A. v  e.  ( V  \  { N } ) { v ,  N }  e.  ran  E )  ->  ( <. V ,  E >. Neighbors  N )  =  ( V  \  { N } ) ) )
453, 44sylbid 223 . 2  |-  ( V USGrph  E  ->  ( N  e.  ( V UnivVertex  E )  ->  ( <. V ,  E >. Neighbors  N )  =  ( V  \  { N } ) ) )
4645imp 436 1  |-  ( ( V USGrph  E  /\  N  e.  ( V UnivVertex  E )
)  ->  ( <. V ,  E >. Neighbors  N )  =  ( V  \  { N } ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 189    /\ wa 376    = wceq 1452    e. wcel 1904    =/= wne 2641   A.wral 2756   {crab 2760   _Vcvv 3031    \ cdif 3387   {csn 3959   {cpr 3961   <.cop 3965   class class class wbr 4395   ran crn 4840  (class class class)co 6308   USGrph cusg 25136   Neighbors cnbgra 25224   UnivVertex cuvtx 25226
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-card 8391  df-cda 8616  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-nn 10632  df-2 10690  df-n0 10894  df-z 10962  df-uz 11183  df-fz 11811  df-hash 12554  df-usgra 25139  df-nbgra 25227  df-uvtx 25229
This theorem is referenced by:  uvtxnm1nbgra  25301  uvtxnbgravtx  25302  uvtxnb  25304  usgrauvtxvd  40180
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