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Theorem uvtxel 21490
Description: An element of the set of all universal vertices. (Contributed by Alexander van der Vekens, 12-Oct-2017.)
Assertion
Ref Expression
uvtxel  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  ( N  e.  ( V UnivVertex  E )  <->  ( N  e.  V  /\  A. k  e.  ( V  \  { N } ) { k ,  N }  e.  ran  E ) ) )
Distinct variable groups:    k, V    k, E    k, N
Allowed substitution hints:    X( k)    Y( k)

Proof of Theorem uvtxel
Dummy variable  n is distinct from all other variables.
StepHypRef Expression
1 isuvtx 21489 . . 3  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  ( V UnivVertex  E )  =  { n  e.  V  |  A. k  e.  ( V  \  { n } ) { k ,  n }  e.  ran  E } )
21eleq2d 2502 . 2  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  ( N  e.  ( V UnivVertex  E )  <->  N  e.  { n  e.  V  |  A. k  e.  ( V  \  { n }
) { k ,  n }  e.  ran  E } ) )
3 sneq 3817 . . . . 5  |-  ( n  =  N  ->  { n }  =  { N } )
43difeq2d 3457 . . . 4  |-  ( n  =  N  ->  ( V  \  { n }
)  =  ( V 
\  { N }
) )
5 preq2 3876 . . . . 5  |-  ( n  =  N  ->  { k ,  n }  =  { k ,  N } )
65eleq1d 2501 . . . 4  |-  ( n  =  N  ->  ( { k ,  n }  e.  ran  E  <->  { k ,  N }  e.  ran  E ) )
74, 6raleqbidv 2908 . . 3  |-  ( n  =  N  ->  ( A. k  e.  ( V  \  { n }
) { k ,  n }  e.  ran  E  <->  A. k  e.  ( V  \  { N }
) { k ,  N }  e.  ran  E ) )
87elrab 3084 . 2  |-  ( N  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 ) )
92, 8syl6bb 253 1  |-  ( ( V  e.  X  /\  E  e.  Y )  ->  ( N  e.  ( V UnivVertex  E )  <->  ( N  e.  V  /\  A. k  e.  ( V  \  { N } ) { k ,  N }  e.  ran  E ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2697   {crab 2701    \ cdif 3309   {csn 3806   {cpr 3807   ran crn 4871  (class class class)co 6073   UnivVertex cuvtx 21424
This theorem is referenced by:  uvtxnbgra  21494
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-iota 5410  df-fun 5448  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-uvtx 21427
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