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Theorem nbgraop 25230
Description: The set of neighbors of an element of the first component of an ordered pair, especially of a vertex in a graph. (Contributed by Alexander van der Vekens, 7-Oct-2017.)
Assertion
Ref Expression
nbgraop  |-  ( ( ( V  e.  Y  /\  E  e.  Z
)  /\  N  e.  V )  ->  ( <. V ,  E >. Neighbors  N
)  =  { n  e.  V  |  { N ,  n }  e.  ran  E } )
Distinct variable groups:    n, V    n, E    n, N    n, Y    n, Z

Proof of Theorem nbgraop
Dummy variables  g 
k are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-nbgra 25227 . 2  |- Neighbors  =  ( g  e.  _V , 
k  e.  ( 1st `  g )  |->  { n  e.  ( 1st `  g
)  |  { k ,  n }  e.  ran  ( 2nd `  g
) } )
2 opex 4664 . . . 4  |-  <. V ,  E >.  e.  _V
32a1i 11 . . 3  |-  ( ( ( V  e.  Y  /\  E  e.  Z
)  /\  N  e.  V )  ->  <. V ,  E >.  e.  _V )
4 op1stg 6824 . . . . . . . 8  |-  ( ( V  e.  Y  /\  E  e.  Z )  ->  ( 1st `  <. V ,  E >. )  =  V )
54eqcomd 2477 . . . . . . 7  |-  ( ( V  e.  Y  /\  E  e.  Z )  ->  V  =  ( 1st `  <. V ,  E >. ) )
65eleq2d 2534 . . . . . 6  |-  ( ( V  e.  Y  /\  E  e.  Z )  ->  ( N  e.  V  <->  N  e.  ( 1st `  <. V ,  E >. )
) )
76biimpa 492 . . . . 5  |-  ( ( ( V  e.  Y  /\  E  e.  Z
)  /\  N  e.  V )  ->  N  e.  ( 1st `  <. V ,  E >. )
)
87adantr 472 . . . 4  |-  ( ( ( ( V  e.  Y  /\  E  e.  Z )  /\  N  e.  V )  /\  g  =  <. V ,  E >. )  ->  N  e.  ( 1st `  <. V ,  E >. ) )
9 fveq2 5879 . . . . 5  |-  ( g  =  <. V ,  E >.  ->  ( 1st `  g
)  =  ( 1st `  <. V ,  E >. ) )
109adantl 473 . . . 4  |-  ( ( ( ( V  e.  Y  /\  E  e.  Z )  /\  N  e.  V )  /\  g  =  <. V ,  E >. )  ->  ( 1st `  g )  =  ( 1st `  <. V ,  E >. ) )
118, 10eleqtrrd 2552 . . 3  |-  ( ( ( ( V  e.  Y  /\  E  e.  Z )  /\  N  e.  V )  /\  g  =  <. V ,  E >. )  ->  N  e.  ( 1st `  g ) )
12 fvex 5889 . . . 4  |-  ( 1st `  g )  e.  _V
13 rabexg 4549 . . . 4  |-  ( ( 1st `  g )  e.  _V  ->  { n  e.  ( 1st `  g
)  |  { k ,  n }  e.  ran  ( 2nd `  g
) }  e.  _V )
1412, 13mp1i 13 . . 3  |-  ( ( ( ( V  e.  Y  /\  E  e.  Z )  /\  N  e.  V )  /\  (
g  =  <. V ,  E >.  /\  k  =  N ) )  ->  { n  e.  ( 1st `  g )  |  { k ,  n }  e.  ran  ( 2nd `  g ) }  e.  _V )
159, 4sylan9eq 2525 . . . . . . . . 9  |-  ( ( g  =  <. V ,  E >.  /\  ( V  e.  Y  /\  E  e.  Z ) )  -> 
( 1st `  g
)  =  V )
1615ex 441 . . . . . . . 8  |-  ( g  =  <. V ,  E >.  ->  ( ( V  e.  Y  /\  E  e.  Z )  ->  ( 1st `  g )  =  V ) )
1716adantr 472 . . . . . . 7  |-  ( ( g  =  <. V ,  E >.  /\  k  =  N )  ->  (
( V  e.  Y  /\  E  e.  Z
)  ->  ( 1st `  g )  =  V ) )
1817com12 31 . . . . . 6  |-  ( ( V  e.  Y  /\  E  e.  Z )  ->  ( ( g  = 
<. V ,  E >.  /\  k  =  N )  ->  ( 1st `  g
)  =  V ) )
1918adantr 472 . . . . 5  |-  ( ( ( V  e.  Y  /\  E  e.  Z
)  /\  N  e.  V )  ->  (
( g  =  <. V ,  E >.  /\  k  =  N )  ->  ( 1st `  g )  =  V ) )
2019imp 436 . . . 4  |-  ( ( ( ( V  e.  Y  /\  E  e.  Z )  /\  N  e.  V )  /\  (
g  =  <. V ,  E >.  /\  k  =  N ) )  -> 
( 1st `  g
)  =  V )
21 preq1 4042 . . . . . . 7  |-  ( k  =  N  ->  { k ,  n }  =  { N ,  n }
)
2221adantl 473 . . . . . 6  |-  ( ( g  =  <. V ,  E >.  /\  k  =  N )  ->  { k ,  n }  =  { N ,  n }
)
2322adantl 473 . . . . 5  |-  ( ( ( ( V  e.  Y  /\  E  e.  Z )  /\  N  e.  V )  /\  (
g  =  <. V ,  E >.  /\  k  =  N ) )  ->  { k ,  n }  =  { N ,  n } )
24 fveq2 5879 . . . . . . . . . . . 12  |-  ( g  =  <. V ,  E >.  ->  ( 2nd `  g
)  =  ( 2nd `  <. V ,  E >. ) )
25 op2ndg 6825 . . . . . . . . . . . 12  |-  ( ( V  e.  Y  /\  E  e.  Z )  ->  ( 2nd `  <. V ,  E >. )  =  E )
2624, 25sylan9eq 2525 . . . . . . . . . . 11  |-  ( ( g  =  <. V ,  E >.  /\  ( V  e.  Y  /\  E  e.  Z ) )  -> 
( 2nd `  g
)  =  E )
2726ex 441 . . . . . . . . . 10  |-  ( g  =  <. V ,  E >.  ->  ( ( V  e.  Y  /\  E  e.  Z )  ->  ( 2nd `  g )  =  E ) )
2827adantr 472 . . . . . . . . 9  |-  ( ( g  =  <. V ,  E >.  /\  k  =  N )  ->  (
( V  e.  Y  /\  E  e.  Z
)  ->  ( 2nd `  g )  =  E ) )
2928com12 31 . . . . . . . 8  |-  ( ( V  e.  Y  /\  E  e.  Z )  ->  ( ( g  = 
<. V ,  E >.  /\  k  =  N )  ->  ( 2nd `  g
)  =  E ) )
3029adantr 472 . . . . . . 7  |-  ( ( ( V  e.  Y  /\  E  e.  Z
)  /\  N  e.  V )  ->  (
( g  =  <. V ,  E >.  /\  k  =  N )  ->  ( 2nd `  g )  =  E ) )
3130imp 436 . . . . . 6  |-  ( ( ( ( V  e.  Y  /\  E  e.  Z )  /\  N  e.  V )  /\  (
g  =  <. V ,  E >.  /\  k  =  N ) )  -> 
( 2nd `  g
)  =  E )
3231rneqd 5068 . . . . 5  |-  ( ( ( ( V  e.  Y  /\  E  e.  Z )  /\  N  e.  V )  /\  (
g  =  <. V ,  E >.  /\  k  =  N ) )  ->  ran  ( 2nd `  g
)  =  ran  E
)
3323, 32eleq12d 2543 . . . 4  |-  ( ( ( ( V  e.  Y  /\  E  e.  Z )  /\  N  e.  V )  /\  (
g  =  <. V ,  E >.  /\  k  =  N ) )  -> 
( { k ,  n }  e.  ran  ( 2nd `  g )  <->  { N ,  n }  e.  ran  E ) )
3420, 33rabeqbidv 3026 . . 3  |-  ( ( ( ( V  e.  Y  /\  E  e.  Z )  /\  N  e.  V )  /\  (
g  =  <. V ,  E >.  /\  k  =  N ) )  ->  { n  e.  ( 1st `  g )  |  { k ,  n }  e.  ran  ( 2nd `  g ) }  =  { n  e.  V  |  { N ,  n }  e.  ran  E }
)
353, 11, 14, 34ovmpt2dv2 6449 . 2  |-  ( ( ( V  e.  Y  /\  E  e.  Z
)  /\  N  e.  V )  ->  ( Neighbors  =  ( g  e.  _V ,  k  e.  ( 1st `  g )  |->  { n  e.  ( 1st `  g )  |  {
k ,  n }  e.  ran  ( 2nd `  g
) } )  -> 
( <. V ,  E >. Neighbors  N )  =  {
n  e.  V  |  { N ,  n }  e.  ran  E } ) )
361, 35mpi 20 1  |-  ( ( ( V  e.  Y  /\  E  e.  Z
)  /\  N  e.  V )  ->  ( <. V ,  E >. Neighbors  N
)  =  { n  e.  V  |  { N ,  n }  e.  ran  E } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 376    = wceq 1452    e. wcel 1904   {crab 2760   _Vcvv 3031   {cpr 3961   <.cop 3965   ran crn 4840   ` cfv 5589  (class class class)co 6308    |-> cmpt2 6310   1stc1st 6810   2ndc2nd 6811   Neighbors cnbgra 25224
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-iota 5553  df-fun 5591  df-fv 5597  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-1st 6812  df-2nd 6813  df-nbgra 25227
This theorem is referenced by:  nbgraop1  25232  nbgrael  25233  nbusgra  25235  rusgraprop3  25750
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