MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nbgrasym Structured version   Unicode version

Theorem nbgrasym 24143
Description: A vertex in a graph is a neighbor of a second vertex if and only if the second vertex is a neighbor of the first vertex. (Contributed by Alexander van der Vekens, 12-Oct-2017.)
Assertion
Ref Expression
nbgrasym  |-  ( V USGrph  E  ->  ( N  e.  ( <. V ,  E >. Neighbors  K )  <->  K  e.  ( <. V ,  E >. Neighbors  N ) ) )

Proof of Theorem nbgrasym
StepHypRef Expression
1 usgrav 24042 . 2  |-  ( V USGrph  E  ->  ( V  e. 
_V  /\  E  e.  _V ) )
2 3ancoma 980 . . . . 5  |-  ( ( K  e.  V  /\  N  e.  V  /\  { K ,  N }  e.  ran  E )  <->  ( N  e.  V  /\  K  e.  V  /\  { K ,  N }  e.  ran  E ) )
3 prcom 4105 . . . . . . 7  |-  { K ,  N }  =  { N ,  K }
43eleq1i 2544 . . . . . 6  |-  ( { K ,  N }  e.  ran  E  <->  { N ,  K }  e.  ran  E )
543anbi3i 1189 . . . . 5  |-  ( ( N  e.  V  /\  K  e.  V  /\  { K ,  N }  e.  ran  E )  <->  ( N  e.  V  /\  K  e.  V  /\  { N ,  K }  e.  ran  E ) )
62, 5bitri 249 . . . 4  |-  ( ( K  e.  V  /\  N  e.  V  /\  { K ,  N }  e.  ran  E )  <->  ( N  e.  V  /\  K  e.  V  /\  { N ,  K }  e.  ran  E ) )
76a1i 11 . . 3  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  ( ( K  e.  V  /\  N  e.  V  /\  { K ,  N }  e.  ran  E )  <->  ( N  e.  V  /\  K  e.  V  /\  { N ,  K }  e.  ran  E ) ) )
8 nbgrael 24130 . . 3  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  ( N  e.  (
<. V ,  E >. Neighbors  K
)  <->  ( K  e.  V  /\  N  e.  V  /\  { K ,  N }  e.  ran  E ) ) )
9 nbgrael 24130 . . 3  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  ( K  e.  (
<. V ,  E >. Neighbors  N
)  <->  ( N  e.  V  /\  K  e.  V  /\  { N ,  K }  e.  ran  E ) ) )
107, 8, 93bitr4d 285 . 2  |-  ( ( V  e.  _V  /\  E  e.  _V )  ->  ( N  e.  (
<. V ,  E >. Neighbors  K
)  <->  K  e.  ( <. V ,  E >. Neighbors  N
) ) )
111, 10syl 16 1  |-  ( V USGrph  E  ->  ( N  e.  ( <. V ,  E >. Neighbors  K )  <->  K  e.  ( <. V ,  E >. Neighbors  N ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    e. wcel 1767   _Vcvv 3113   {cpr 4029   <.cop 4033   class class class wbr 4447   ran crn 5000  (class class class)co 6284   USGrph cusg 24034   Neighbors cnbgra 24121
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-8 1769  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-pow 4625  ax-pr 4686  ax-un 6576
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-csb 3436  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-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fv 5596  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-1st 6784  df-2nd 6785  df-usgra 24037  df-nbgra 24124
This theorem is referenced by:  uvtxnbgravtx  24199
  Copyright terms: Public domain W3C validator