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Theorem nbgraop1 24546
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, 17-Dec-2017.)
Assertion
Ref Expression
nbgraop1  |-  ( ( ( V  e.  Y  /\  E  e.  Z
)  /\  N  e.  V )  ->  ( Fun  E  ->  ( <. V ,  E >. Neighbors  N )  =  { n  e.  V  |  E. i  e.  dom  E ( E `
 i )  =  { N ,  n } } ) )
Distinct variable groups:    i, E, n    i, N, n    i, V, n    n, Y    n, Z
Allowed substitution hints:    Y( i)    Z( i)

Proof of Theorem nbgraop1
StepHypRef Expression
1 nbgraop 24544 . . 3  |-  ( ( ( V  e.  Y  /\  E  e.  Z
)  /\  N  e.  V )  ->  ( <. V ,  E >. Neighbors  N
)  =  { n  e.  V  |  { N ,  n }  e.  ran  E } )
2 elrnrexdmb 5938 . . . . 5  |-  ( Fun 
E  ->  ( { N ,  n }  e.  ran  E  <->  E. i  e.  dom  E { N ,  n }  =  ( E `  i ) ) )
3 eqcom 2391 . . . . . 6  |-  ( { N ,  n }  =  ( E `  i )  <->  ( E `  i )  =  { N ,  n }
)
43rexbii 2884 . . . . 5  |-  ( E. i  e.  dom  E { N ,  n }  =  ( E `  i )  <->  E. i  e.  dom  E ( E `
 i )  =  { N ,  n } )
52, 4syl6bb 261 . . . 4  |-  ( Fun 
E  ->  ( { N ,  n }  e.  ran  E  <->  E. i  e.  dom  E ( E `
 i )  =  { N ,  n } ) )
65rabbidv 3026 . . 3  |-  ( Fun 
E  ->  { n  e.  V  |  { N ,  n }  e.  ran  E }  =  { n  e.  V  |  E. i  e.  dom  E ( E `  i
)  =  { N ,  n } } )
71, 6sylan9eq 2443 . 2  |-  ( ( ( ( V  e.  Y  /\  E  e.  Z )  /\  N  e.  V )  /\  Fun  E )  ->  ( <. V ,  E >. Neighbors  N )  =  { n  e.  V  |  E. i  e.  dom  E ( E `
 i )  =  { N ,  n } } )
87ex 432 1  |-  ( ( ( V  e.  Y  /\  E  e.  Z
)  /\  N  e.  V )  ->  ( Fun  E  ->  ( <. V ,  E >. Neighbors  N )  =  { n  e.  V  |  E. i  e.  dom  E ( E `
 i )  =  { N ,  n } } ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1399    e. wcel 1826   E.wrex 2733   {crab 2736   {cpr 3946   <.cop 3950   dom cdm 4913   ran crn 4914   Fun wfun 5490   ` cfv 5496  (class class class)co 6196   Neighbors cnbgra 24538
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-rab 2741  df-v 3036  df-sbc 3253  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-br 4368  df-opab 4426  df-mpt 4427  df-id 4709  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-iota 5460  df-fun 5498  df-fn 5499  df-fv 5504  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-1st 6699  df-2nd 6700  df-nbgra 24541
This theorem is referenced by: (None)
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