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Theorem uvtxaval 39509
Description: The set of all universal vertices. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 29-Oct-2020.)
Hypothesis
Ref Expression
isuvtxa.v  |-  V  =  (Vtx `  G )
Assertion
Ref Expression
uvtxaval  |-  ( G  e.  W  ->  (UnivVtx `  G )  =  {
v  e.  V  |  A. n  e.  ( V  \  { v } ) n  e.  ( G NeighbVtx  v ) } )
Distinct variable groups:    n, G, v    n, V, v
Allowed substitution hints:    W( v, n)

Proof of Theorem uvtxaval
Dummy variable  g is distinct from all other variables.
StepHypRef Expression
1 df-uvtxa 39453 . . 3  |- UnivVtx  =  ( g  e.  _V  |->  { v  e.  (Vtx `  g )  |  A. n  e.  ( (Vtx `  g )  \  {
v } ) n  e.  ( g NeighbVtx  v
) } )
21a1i 11 . 2  |-  ( G  e.  W  -> UnivVtx  =  ( g  e.  _V  |->  { v  e.  (Vtx `  g )  |  A. n  e.  ( (Vtx `  g )  \  {
v } ) n  e.  ( g NeighbVtx  v
) } ) )
3 fveq2 5888 . . . . 5  |-  ( g  =  G  ->  (Vtx `  g )  =  (Vtx
`  G ) )
4 isuvtxa.v . . . . 5  |-  V  =  (Vtx `  G )
53, 4syl6eqr 2514 . . . 4  |-  ( g  =  G  ->  (Vtx `  g )  =  V )
65difeq1d 3562 . . . . 5  |-  ( g  =  G  ->  (
(Vtx `  g )  \  { v } )  =  ( V  \  { v } ) )
7 oveq1 6322 . . . . . 6  |-  ( g  =  G  ->  (
g NeighbVtx  v )  =  ( G NeighbVtx  v ) )
87eleq2d 2525 . . . . 5  |-  ( g  =  G  ->  (
n  e.  ( g NeighbVtx  v )  <->  n  e.  ( G NeighbVtx  v ) ) )
96, 8raleqbidv 3013 . . . 4  |-  ( g  =  G  ->  ( A. n  e.  (
(Vtx `  g )  \  { v } ) n  e.  ( g NeighbVtx  v )  <->  A. n  e.  ( V  \  {
v } ) n  e.  ( G NeighbVtx  v ) ) )
105, 9rabeqbidv 3052 . . 3  |-  ( g  =  G  ->  { v  e.  (Vtx `  g
)  |  A. n  e.  ( (Vtx `  g
)  \  { v } ) n  e.  ( g NeighbVtx  v ) }  =  { v  e.  V  |  A. n  e.  ( V  \  { v } ) n  e.  ( G NeighbVtx  v ) } )
1110adantl 472 . 2  |-  ( ( G  e.  W  /\  g  =  G )  ->  { v  e.  (Vtx
`  g )  | 
A. n  e.  ( (Vtx `  g )  \  { v } ) n  e.  ( g NeighbVtx  v ) }  =  { v  e.  V  |  A. n  e.  ( V  \  { v } ) n  e.  ( G NeighbVtx  v ) } )
12 elex 3066 . 2  |-  ( G  e.  W  ->  G  e.  _V )
13 fvex 5898 . . . . 5  |-  (Vtx `  G )  e.  _V
144, 13eqeltri 2536 . . . 4  |-  V  e. 
_V
1514rabex 4568 . . 3  |-  { v  e.  V  |  A. n  e.  ( V  \  { v } ) n  e.  ( G NeighbVtx  v ) }  e.  _V
1615a1i 11 . 2  |-  ( G  e.  W  ->  { v  e.  V  |  A. n  e.  ( V  \  { v } ) n  e.  ( G NeighbVtx  v ) }  e.  _V )
172, 11, 12, 16fvmptd 5977 1  |-  ( G  e.  W  ->  (UnivVtx `  G )  =  {
v  e.  V  |  A. n  e.  ( V  \  { v } ) n  e.  ( G NeighbVtx  v ) } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1455    e. wcel 1898   A.wral 2749   {crab 2753   _Vcvv 3057    \ cdif 3413   {csn 3980    |-> cmpt 4475   ` cfv 5601  (class class class)co 6315  Vtxcvtx 39147   NeighbVtx cnbgr 39447  UnivVtxcuvtxa 39448
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-sep 4539  ax-nul 4548  ax-pr 4653
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-ral 2754  df-rex 2755  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-if 3894  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4213  df-br 4417  df-opab 4476  df-mpt 4477  df-id 4768  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-iota 5565  df-fun 5603  df-fv 5609  df-ov 6318  df-uvtxa 39453
This theorem is referenced by:  uvtxael  39510  uvtxaisvtx  39511  uvtxa0  39516  isuvtxa  39517  uvtxa01vtx0  39519  uvtxusgr  39525
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