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Theorem uvtxael 39624
Description: A universal vertex, i.e. an element of the set of all universal vertices. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 29-Oct-2020.)
Hypothesis
Ref Expression
uvtxael.v  |-  V  =  (Vtx `  G )
Assertion
Ref Expression
uvtxael  |-  ( G  e.  W  ->  ( N  e.  (UnivVtx `  G
)  <->  ( N  e.  V  /\  A. n  e.  ( V  \  { N } ) n  e.  ( G NeighbVtx  N )
) ) )
Distinct variable groups:    n, G    n, N    n, V
Allowed substitution hint:    W( n)

Proof of Theorem uvtxael
Dummy variable  v is distinct from all other variables.
StepHypRef Expression
1 uvtxael.v . . . 4  |-  V  =  (Vtx `  G )
21uvtxaval 39623 . . 3  |-  ( G  e.  W  ->  (UnivVtx `  G )  =  {
v  e.  V  |  A. n  e.  ( V  \  { v } ) n  e.  ( G NeighbVtx  v ) } )
32eleq2d 2534 . 2  |-  ( G  e.  W  ->  ( N  e.  (UnivVtx `  G
)  <->  N  e.  { v  e.  V  |  A. n  e.  ( V  \  { v } ) n  e.  ( G NeighbVtx  v ) } ) )
4 sneq 3969 . . . . 5  |-  ( v  =  N  ->  { v }  =  { N } )
54difeq2d 3540 . . . 4  |-  ( v  =  N  ->  ( V  \  { v } )  =  ( V 
\  { N }
) )
6 oveq2 6316 . . . . 5  |-  ( v  =  N  ->  ( G NeighbVtx  v )  =  ( G NeighbVtx  N ) )
76eleq2d 2534 . . . 4  |-  ( v  =  N  ->  (
n  e.  ( G NeighbVtx  v )  <->  n  e.  ( G NeighbVtx  N ) ) )
85, 7raleqbidv 2987 . . 3  |-  ( v  =  N  ->  ( A. n  e.  ( V  \  { v } ) n  e.  ( G NeighbVtx  v )  <->  A. n  e.  ( V  \  { N } ) n  e.  ( G NeighbVtx  N )
) )
98elrab 3184 . 2  |-  ( N  e.  { v  e.  V  |  A. n  e.  ( V  \  {
v } ) n  e.  ( G NeighbVtx  v ) }  <->  ( N  e.  V  /\  A. n  e.  ( V  \  { N } ) n  e.  ( G NeighbVtx  N )
) )
103, 9syl6bb 269 1  |-  ( G  e.  W  ->  ( N  e.  (UnivVtx `  G
)  <->  ( N  e.  V  /\  A. n  e.  ( V  \  { N } ) n  e.  ( G NeighbVtx  N )
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 376    = wceq 1452    e. wcel 1904   A.wral 2756   {crab 2760    \ cdif 3387   {csn 3959   ` cfv 5589  (class class class)co 6308  Vtxcvtx 39251   NeighbVtx cnbgr 39561  UnivVtxcuvtxa 39562
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-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  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-csb 3350  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-mpt 4456  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-uvtxa 39567
This theorem is referenced by:  vtxnbuvtx  39627  uvtx2vtx1edg  39635  uvtx2vtx1edgb  39636  uvtxnbgrb  39638  iscplgrnb  39648  cplgr1v  39662  cusgrexi  39672
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