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Theorem isuvtx 23541
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 23479 . 2  |- UnivVertex  =  ( v  e.  _V , 
e  e.  _V  |->  { n  e.  v  | 
A. k  e.  ( v  \  { n } ) { k ,  n }  e.  ran  e } )
2 elex 3080 . . . 4  |-  ( V  e.  X  ->  V  e.  _V )
32adantr 465 . . 3  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  V  e.  _V )
4 elex 3080 . . . . 5  |-  ( E  e.  Y  ->  E  e.  _V )
54adantl 466 . . . 4  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  E  e.  _V )
65adantr 465 . . 3  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  v  =  V )  ->  E  e.  _V )
7 vex 3074 . . . 4  |-  v  e. 
_V
8 rabexg 4543 . . . 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 755 . . . 4  |-  ( ( ( V  e.  X  /\  E  e.  Y
)  /\  ( v  =  V  /\  e  =  E ) )  -> 
v  =  V )
11 difeq1 3568 . . . . . . 7  |-  ( v  =  V  ->  (
v  \  { n } )  =  ( V  \  { n } ) )
1211adantr 465 . . . . . 6  |-  ( ( v  =  V  /\  e  =  E )  ->  ( v  \  {
n } )  =  ( V  \  {
n } ) )
13 rneq 5166 . . . . . . . 8  |-  ( e  =  E  ->  ran  e  =  ran  E )
1413eleq2d 2521 . . . . . . 7  |-  ( e  =  E  ->  ( { k ,  n }  e.  ran  e  <->  { k ,  n }  e.  ran  E ) )
1514adantl 466 . . . . . 6  |-  ( ( v  =  V  /\  e  =  E )  ->  ( { k ,  n }  e.  ran  e 
<->  { k ,  n }  e.  ran  E ) )
1612, 15raleqbidv 3030 . . . . 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 466 . . . 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 3066 . . 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 6327 . 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 369    = wceq 1370    e. wcel 1758   A.wral 2795   {crab 2799   _Vcvv 3071    \ cdif 3426   {csn 3978   {cpr 3980   ran crn 4942  (class class class)co 6193    |-> cmpt2 6195   UnivVertex cuvtx 23476
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4514  ax-nul 4522  ax-pr 4632
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3073  df-sbc 3288  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-br 4394  df-opab 4452  df-id 4737  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-iota 5482  df-fun 5521  df-fv 5527  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-uvtx 23479
This theorem is referenced by:  uvtxel  23542  uvtxisvtx  23543  uvtx0  23544  uvtx01vtx  23545  cusgrauvtxb  23549
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