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Theorem isuvtx 24164
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 24098 . 2  |- UnivVertex  =  ( v  e.  _V , 
e  e.  _V  |->  { n  e.  v  | 
A. k  e.  ( v  \  { n } ) { k ,  n }  e.  ran  e } )
2 elex 3122 . . . 4  |-  ( V  e.  X  ->  V  e.  _V )
32adantr 465 . . 3  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  V  e.  _V )
4 elex 3122 . . . . 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 3116 . . . 4  |-  v  e. 
_V
8 rabexg 4597 . . . 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 3615 . . . . . . 7  |-  ( v  =  V  ->  (
v  \  { n } )  =  ( V  \  { n } ) )
1211adantr 465 . . . . . 6  |-  ( ( v  =  V  /\  e  =  E )  ->  ( v  \  {
n } )  =  ( V  \  {
n } ) )
13 rneq 5226 . . . . . . . 8  |-  ( e  =  E  ->  ran  e  =  ran  E )
1413eleq2d 2537 . . . . . . 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 3072 . . . . 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 3108 . . 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 6418 . 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 1379    e. wcel 1767   A.wral 2814   {crab 2818   _Vcvv 3113    \ cdif 3473   {csn 4027   {cpr 4029   ran crn 5000  (class class class)co 6282    |-> cmpt2 6284   UnivVertex cuvtx 24095
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-iota 5549  df-fun 5588  df-fv 5594  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-uvtx 24098
This theorem is referenced by:  uvtxel  24165  uvtxisvtx  24166  uvtx0  24167  uvtx01vtx  24168  cusgrauvtxb  24172
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