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Theorem nbgrel 39410
Description: Characterization of a neighbor of a vertex  V in a graph  G. (Contributed by Alexander van der Vekens and Mario Carneiro, 9-Oct-2017.) (Revised by AV, 26-Oct-2020.)
Hypotheses
Ref Expression
nbgrel.v  |-  V  =  (Vtx `  G )
nbgrel.e  |-  E  =  (Edg `  G )
Assertion
Ref Expression
nbgrel  |-  ( G  e.  W  ->  ( K  e.  ( G NeighbVtx  N )  <->  ( ( K  e.  V  /\  N  e.  V )  /\  K  =/=  N  /\  E. e  e.  E  { N ,  K }  C_  e
) ) )
Distinct variable groups:    e, E    e, G    e, K    e, N    e, V    e, W

Proof of Theorem nbgrel
Dummy variables  g  n  v  k are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-nbgr 39401 . . . . . 6  |- NeighbVtx  =  ( g  e.  _V , 
v  e.  (Vtx `  g )  |->  { n  e.  ( (Vtx `  g
)  \  { v } )  |  E. e  e.  (Edg `  g
) { v ,  n }  C_  e } )
21mpt2xeldm 6957 . . . . 5  |-  ( K  e.  ( G NeighbVtx  N )  ->  ( G  e. 
_V  /\  N  e.  [_ G  /  g ]_ (Vtx `  g ) ) )
3 csbfv 5902 . . . . . . . . 9  |-  [_ G  /  g ]_ (Vtx `  g )  =  (Vtx
`  G )
4 nbgrel.v . . . . . . . . 9  |-  V  =  (Vtx `  G )
53, 4eqtr4i 2476 . . . . . . . 8  |-  [_ G  /  g ]_ (Vtx `  g )  =  V
65eleq2i 2521 . . . . . . 7  |-  ( N  e.  [_ G  / 
g ]_ (Vtx `  g
)  <->  N  e.  V
)
76biimpi 198 . . . . . 6  |-  ( N  e.  [_ G  / 
g ]_ (Vtx `  g
)  ->  N  e.  V )
87adantl 468 . . . . 5  |-  ( ( G  e.  _V  /\  N  e.  [_ G  / 
g ]_ (Vtx `  g
) )  ->  N  e.  V )
92, 8syl 17 . . . 4  |-  ( K  e.  ( G NeighbVtx  N )  ->  N  e.  V
)
109a1i 11 . . 3  |-  ( G  e.  W  ->  ( K  e.  ( G NeighbVtx  N )  ->  N  e.  V ) )
1110pm4.71rd 641 . 2  |-  ( G  e.  W  ->  ( K  e.  ( G NeighbVtx  N )  <->  ( N  e.  V  /\  K  e.  ( G NeighbVtx  N )
) ) )
12 nbgrel.e . . . . . . 7  |-  E  =  (Edg `  G )
134, 12nbgrval 39406 . . . . . 6  |-  ( ( G  e.  W  /\  N  e.  V )  ->  ( G NeighbVtx  N )  =  { k  e.  ( V  \  { N } )  |  E. e  e.  E  { N ,  k }  C_  e } )
1413eleq2d 2514 . . . . 5  |-  ( ( G  e.  W  /\  N  e.  V )  ->  ( K  e.  ( G NeighbVtx  N )  <->  K  e.  { k  e.  ( V 
\  { N }
)  |  E. e  e.  E  { N ,  k }  C_  e } ) )
15 preq2 4052 . . . . . . . . 9  |-  ( k  =  K  ->  { N ,  k }  =  { N ,  K }
)
1615sseq1d 3459 . . . . . . . 8  |-  ( k  =  K  ->  ( { N ,  k } 
C_  e  <->  { N ,  K }  C_  e
) )
1716rexbidv 2901 . . . . . . 7  |-  ( k  =  K  ->  ( E. e  e.  E  { N ,  k } 
C_  e  <->  E. e  e.  E  { N ,  K }  C_  e
) )
1817elrab 3196 . . . . . 6  |-  ( K  e.  { k  e.  ( V  \  { N } )  |  E. e  e.  E  { N ,  k }  C_  e }  <->  ( K  e.  ( V  \  { N } )  /\  E. e  e.  E  { N ,  K }  C_  e ) )
19 eldifsn 4097 . . . . . . 7  |-  ( K  e.  ( V  \  { N } )  <->  ( K  e.  V  /\  K  =/= 
N ) )
2019anbi1i 701 . . . . . 6  |-  ( ( K  e.  ( V 
\  { N }
)  /\  E. e  e.  E  { N ,  K }  C_  e
)  <->  ( ( K  e.  V  /\  K  =/=  N )  /\  E. e  e.  E  { N ,  K }  C_  e ) )
21 anass 655 . . . . . 6  |-  ( ( ( K  e.  V  /\  K  =/=  N
)  /\  E. e  e.  E  { N ,  K }  C_  e
)  <->  ( K  e.  V  /\  ( K  =/=  N  /\  E. e  e.  E  { N ,  K }  C_  e ) ) )
2218, 20, 213bitri 275 . . . . 5  |-  ( K  e.  { k  e.  ( V  \  { N } )  |  E. e  e.  E  { N ,  k }  C_  e }  <->  ( K  e.  V  /\  ( K  =/=  N  /\  E. e  e.  E  { N ,  K }  C_  e ) ) )
2314, 22syl6bb 265 . . . 4  |-  ( ( G  e.  W  /\  N  e.  V )  ->  ( K  e.  ( G NeighbVtx  N )  <->  ( K  e.  V  /\  ( K  =/=  N  /\  E. e  e.  E  { N ,  K }  C_  e ) ) ) )
2423pm5.32da 647 . . 3  |-  ( G  e.  W  ->  (
( N  e.  V  /\  K  e.  ( G NeighbVtx  N ) )  <->  ( N  e.  V  /\  ( K  e.  V  /\  ( K  =/=  N  /\  E. e  e.  E  { N ,  K }  C_  e ) ) ) ) )
25 3anass 989 . . . 4  |-  ( ( ( K  e.  V  /\  N  e.  V
)  /\  K  =/=  N  /\  E. e  e.  E  { N ,  K }  C_  e )  <-> 
( ( K  e.  V  /\  N  e.  V )  /\  ( K  =/=  N  /\  E. e  e.  E  { N ,  K }  C_  e ) ) )
26 ancom 452 . . . . 5  |-  ( ( K  e.  V  /\  N  e.  V )  <->  ( N  e.  V  /\  K  e.  V )
)
2726anbi1i 701 . . . 4  |-  ( ( ( K  e.  V  /\  N  e.  V
)  /\  ( K  =/=  N  /\  E. e  e.  E  { N ,  K }  C_  e
) )  <->  ( ( N  e.  V  /\  K  e.  V )  /\  ( K  =/=  N  /\  E. e  e.  E  { N ,  K }  C_  e ) ) )
28 anass 655 . . . 4  |-  ( ( ( N  e.  V  /\  K  e.  V
)  /\  ( K  =/=  N  /\  E. e  e.  E  { N ,  K }  C_  e
) )  <->  ( N  e.  V  /\  ( K  e.  V  /\  ( K  =/=  N  /\  E. e  e.  E  { N ,  K }  C_  e ) ) ) )
2925, 27, 283bitrri 276 . . 3  |-  ( ( N  e.  V  /\  ( K  e.  V  /\  ( K  =/=  N  /\  E. e  e.  E  { N ,  K }  C_  e ) ) )  <-> 
( ( K  e.  V  /\  N  e.  V )  /\  K  =/=  N  /\  E. e  e.  E  { N ,  K }  C_  e
) )
3024, 29syl6bb 265 . 2  |-  ( G  e.  W  ->  (
( N  e.  V  /\  K  e.  ( G NeighbVtx  N ) )  <->  ( ( K  e.  V  /\  N  e.  V )  /\  K  =/=  N  /\  E. e  e.  E  { N ,  K }  C_  e ) ) )
3111, 30bitrd 257 1  |-  ( G  e.  W  ->  ( K  e.  ( G NeighbVtx  N )  <->  ( ( K  e.  V  /\  N  e.  V )  /\  K  =/=  N  /\  E. e  e.  E  { N ,  K }  C_  e
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    /\ w3a 985    = wceq 1444    e. wcel 1887    =/= wne 2622   E.wrex 2738   {crab 2741   _Vcvv 3045   [_csb 3363    \ cdif 3401    C_ wss 3404   {csn 3968   {cpr 3970   ` cfv 5582  (class class class)co 6290  Vtxcvtx 39101  Edgcedga 39210   NeighbVtx cnbgr 39397
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-fal 1450  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-id 4749  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-iota 5546  df-fun 5584  df-fv 5590  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-1st 6793  df-2nd 6794  df-nbgr 39401
This theorem is referenced by:  nbgr2vtx1edg  39418  nbuhgr2vtx1edgblem  39419  nbuhgr2vtx1edgb  39420  nbgrisvtx  39427  nbgrsym  39437  isuvtxa  39467  iscplgredg  39485  cusgrexi  39507
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