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Theorem isuvtx 24693
Description: The set of all universal vertices. (Contributed by Alexander van der Vekens, 12-Oct-2017.)
Assertion
Ref Expression
isuvtx  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  ( V UnivVertex  E )  =  { n  e.  V  |  A. k  e.  ( V  \  { n } ) { k ,  n }  e.  ran  E } )
Distinct variable groups:    k, V, n    k, E, n    n, X    n, Y
Allowed substitution hints:    X( k)    Y( k)

Proof of Theorem isuvtx
Dummy variables  e 
v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-uvtx 24627 . 2  |- UnivVertex  =  ( v  e.  _V , 
e  e.  _V  |->  { n  e.  v  | 
A. k  e.  ( v  \  { n } ) { k ,  n }  e.  ran  e } )
2 elex 3115 . . . 4  |-  ( V  e.  X  ->  V  e.  _V )
32adantr 463 . . 3  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  V  e.  _V )
4 elex 3115 . . . . 5  |-  ( E  e.  Y  ->  E  e.  _V )
54adantl 464 . . . 4  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  E  e.  _V )
65adantr 463 . . 3  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  v  =  V )  ->  E  e.  _V )
7 vex 3109 . . . 4  |-  v  e. 
_V
8 rabexg 4587 . . . 4  |-  ( v  e.  _V  ->  { n  e.  v  |  A. k  e.  ( v  \  { n } ) { k ,  n }  e.  ran  e }  e.  _V )
97, 8mp1i 12 . . 3  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( v  =  V  /\  e  =  E ) )  ->  { n  e.  v  |  A. k  e.  ( v  \  { n } ) { k ,  n }  e.  ran  e }  e.  _V )
10 simprl 754 . . . 4  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( v  =  V  /\  e  =  E ) )  -> 
v  =  V )
11 difeq1 3601 . . . . . . 7  |-  ( v  =  V  ->  (
v  \  { n } )  =  ( V  \  { n } ) )
1211adantr 463 . . . . . 6  |-  ( ( v  =  V  /\  e  =  E )  ->  ( v  \  {
n } )  =  ( V  \  {
n } ) )
13 rneq 5217 . . . . . . . 8  |-  ( e  =  E  ->  ran  e  =  ran  E )
1413eleq2d 2524 . . . . . . 7  |-  ( e  =  E  ->  ( { k ,  n }  e.  ran  e  <->  { k ,  n }  e.  ran  E ) )
1514adantl 464 . . . . . 6  |-  ( ( v  =  V  /\  e  =  E )  ->  ( { k ,  n }  e.  ran  e 
<->  { k ,  n }  e.  ran  E ) )
1612, 15raleqbidv 3065 . . . . 5  |-  ( ( v  =  V  /\  e  =  E )  ->  ( A. k  e.  ( v  \  {
n } ) { k ,  n }  e.  ran  e  <->  A. k  e.  ( V  \  {
n } ) { k ,  n }  e.  ran  E ) )
1716adantl 464 . . . 4  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( v  =  V  /\  e  =  E ) )  -> 
( A. k  e.  ( v  \  {
n } ) { k ,  n }  e.  ran  e  <->  A. k  e.  ( V  \  {
n } ) { k ,  n }  e.  ran  E ) )
1810, 17rabeqbidv 3101 . . 3  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( v  =  V  /\  e  =  E ) )  ->  { n  e.  v  |  A. k  e.  ( v  \  { n } ) { k ,  n }  e.  ran  e }  =  {
n  e.  V  |  A. k  e.  ( V  \  { n }
) { k ,  n }  e.  ran  E } )
193, 6, 9, 18ovmpt2dv2 6409 . 2  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  ( UnivVertex  =  ( v  e.  _V ,  e  e. 
_V  |->  { n  e.  v  |  A. k  e.  ( v  \  {
n } ) { k ,  n }  e.  ran  e } )  ->  ( V UnivVertex  E )  =  { n  e.  V  |  A. k  e.  ( V  \  {
n } ) { k ,  n }  e.  ran  E } ) )
201, 19mpi 17 1  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  ( V UnivVertex  E )  =  { n  e.  V  |  A. k  e.  ( V  \  { n } ) { k ,  n }  e.  ran  E } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1398    e. wcel 1823   A.wral 2804   {crab 2808   _Vcvv 3106    \ cdif 3458   {csn 4016   {cpr 4018   ran crn 4989  (class class class)co 6270    |-> cmpt2 6272   UnivVertex cuvtx 24624
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-iota 5534  df-fun 5572  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-uvtx 24627
This theorem is referenced by:  uvtxel  24694  uvtxisvtx  24695  uvtx0  24696  uvtx01vtx  24697  cusgrauvtxb  24701
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