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Theorem nbgrval 39570
Description: The set of neighbors of a vertex  V in a graph  G. (Contributed by Alexander van der Vekens, 7-Oct-2017.) (Revised by AV, 24-Oct-2020.) (Revised by AV, 21-Mar-2021.)
Hypotheses
Ref Expression
nbgrval.v  |-  V  =  (Vtx `  G )
nbgrval.e  |-  E  =  (Edg `  G )
Assertion
Ref Expression
nbgrval  |-  ( N  e.  V  ->  ( G NeighbVtx  N )  =  {
n  e.  ( V 
\  { N }
)  |  E. e  e.  E  { N ,  n }  C_  e } )
Distinct variable groups:    e, E    e, G, n    e, N, n    e, V, n
Allowed substitution hint:    E( n)

Proof of Theorem nbgrval
Dummy variables  g 
k are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-nbgr 39565 . 2  |- NeighbVtx  =  ( g  e.  _V , 
k  e.  (Vtx `  g )  |->  { n  e.  ( (Vtx `  g
)  \  { k } )  |  E. e  e.  (Edg `  g
) { k ,  n }  C_  e } )
2 nbgrval.v . . . 4  |-  V  =  (Vtx `  G )
321vgrex 39257 . . 3  |-  ( N  e.  V  ->  G  e.  _V )
4 fveq2 5879 . . . . . 6  |-  ( g  =  G  ->  (Vtx `  g )  =  (Vtx
`  G ) )
54, 2syl6reqr 2524 . . . . 5  |-  ( g  =  G  ->  V  =  (Vtx `  g )
)
65eleq2d 2534 . . . 4  |-  ( g  =  G  ->  ( N  e.  V  <->  N  e.  (Vtx `  g ) ) )
76biimpac 494 . . 3  |-  ( ( N  e.  V  /\  g  =  G )  ->  N  e.  (Vtx `  g ) )
8 fvex 5889 . . . . 5  |-  (Vtx `  g )  e.  _V
98difexi 4546 . . . 4  |-  ( (Vtx
`  g )  \  { k } )  e.  _V
10 rabexg 4549 . . . 4  |-  ( ( (Vtx `  g )  \  { k } )  e.  _V  ->  { n  e.  ( (Vtx `  g
)  \  { k } )  |  E. e  e.  (Edg `  g
) { k ,  n }  C_  e }  e.  _V )
119, 10mp1i 13 . . 3  |-  ( ( N  e.  V  /\  ( g  =  G  /\  k  =  N ) )  ->  { n  e.  ( (Vtx `  g
)  \  { k } )  |  E. e  e.  (Edg `  g
) { k ,  n }  C_  e }  e.  _V )
124, 2syl6eqr 2523 . . . . . . 7  |-  ( g  =  G  ->  (Vtx `  g )  =  V )
1312adantr 472 . . . . . 6  |-  ( ( g  =  G  /\  k  =  N )  ->  (Vtx `  g )  =  V )
14 sneq 3969 . . . . . . 7  |-  ( k  =  N  ->  { k }  =  { N } )
1514adantl 473 . . . . . 6  |-  ( ( g  =  G  /\  k  =  N )  ->  { k }  =  { N } )
1613, 15difeq12d 3541 . . . . 5  |-  ( ( g  =  G  /\  k  =  N )  ->  ( (Vtx `  g
)  \  { k } )  =  ( V  \  { N } ) )
1716adantl 473 . . . 4  |-  ( ( N  e.  V  /\  ( g  =  G  /\  k  =  N ) )  ->  (
(Vtx `  g )  \  { k } )  =  ( V  \  { N } ) )
18 fveq2 5879 . . . . . . . 8  |-  ( g  =  G  ->  (Edg `  g )  =  (Edg
`  G ) )
19 nbgrval.e . . . . . . . 8  |-  E  =  (Edg `  G )
2018, 19syl6eqr 2523 . . . . . . 7  |-  ( g  =  G  ->  (Edg `  g )  =  E )
2120adantr 472 . . . . . 6  |-  ( ( g  =  G  /\  k  =  N )  ->  (Edg `  g )  =  E )
2221adantl 473 . . . . 5  |-  ( ( N  e.  V  /\  ( g  =  G  /\  k  =  N ) )  ->  (Edg `  g )  =  E )
23 preq1 4042 . . . . . . . 8  |-  ( k  =  N  ->  { k ,  n }  =  { N ,  n }
)
2423sseq1d 3445 . . . . . . 7  |-  ( k  =  N  ->  ( { k ,  n }  C_  e  <->  { N ,  n }  C_  e
) )
2524adantl 473 . . . . . 6  |-  ( ( g  =  G  /\  k  =  N )  ->  ( { k ,  n }  C_  e  <->  { N ,  n }  C_  e ) )
2625adantl 473 . . . . 5  |-  ( ( N  e.  V  /\  ( g  =  G  /\  k  =  N ) )  ->  ( { k ,  n }  C_  e  <->  { N ,  n }  C_  e
) )
2722, 26rexeqbidv 2988 . . . 4  |-  ( ( N  e.  V  /\  ( g  =  G  /\  k  =  N ) )  ->  ( E. e  e.  (Edg `  g ) { k ,  n }  C_  e 
<->  E. e  e.  E  { N ,  n }  C_  e ) )
2817, 27rabeqbidv 3026 . . 3  |-  ( ( N  e.  V  /\  ( g  =  G  /\  k  =  N ) )  ->  { n  e.  ( (Vtx `  g
)  \  { k } )  |  E. e  e.  (Edg `  g
) { k ,  n }  C_  e }  =  { n  e.  ( V  \  { N } )  |  E. e  e.  E  { N ,  n }  C_  e } )
293, 7, 11, 28ovmpt2dv2 6449 . 2  |-  ( N  e.  V  ->  ( NeighbVtx  =  ( g  e.  _V ,  k  e.  (Vtx `  g )  |->  { n  e.  ( (Vtx `  g
)  \  { k } )  |  E. e  e.  (Edg `  g
) { k ,  n }  C_  e } )  ->  ( G NeighbVtx  N )  =  {
n  e.  ( V 
\  { N }
)  |  E. e  e.  E  { N ,  n }  C_  e } ) )
301, 29mpi 20 1  |-  ( N  e.  V  ->  ( G NeighbVtx  N )  =  {
n  e.  ( V 
\  { N }
)  |  E. e  e.  E  { N ,  n }  C_  e } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 376    = wceq 1452    e. wcel 1904   E.wrex 2757   {crab 2760   _Vcvv 3031    \ cdif 3387    C_ wss 3390   {csn 3959   {cpr 3961   ` cfv 5589  (class class class)co 6308    |-> cmpt2 6310  Vtxcvtx 39251  Edgcedga 39371   NeighbVtx cnbgr 39561
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-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  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-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-iota 5553  df-fun 5591  df-fv 5597  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-nbgr 39565
This theorem is referenced by:  dfnbgr2  39571  dfnbgr3  39572  nbgrel  39574  nbuhgr  39575  nbupgr  39576  nbumgrvtx  39578  nbgr0vtxlem  39587  nbgrnself  39593
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