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Theorem nbgraop1 24403
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 24401 . . 3  |-  ( ( ( V  e.  Y  /\  E  e.  Z
)  /\  N  e.  V )  ->  ( <. V ,  E >. Neighbors  N
)  =  { n  e.  V  |  { N ,  n }  e.  ran  E } )
2 elrnrexdmb 6021 . . . . 5  |-  ( Fun 
E  ->  ( { N ,  n }  e.  ran  E  <->  E. i  e.  dom  E { N ,  n }  =  ( E `  i ) ) )
3 eqcom 2452 . . . . . 6  |-  ( { N ,  n }  =  ( E `  i )  <->  ( E `  i )  =  { N ,  n }
)
43rexbii 2945 . . . . 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 3087 . . 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 2504 . 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 434 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 369    = wceq 1383    e. wcel 1804   E.wrex 2794   {crab 2797   {cpr 4016   <.cop 4020   dom cdm 4989   ran crn 4990   Fun wfun 5572   ` cfv 5578  (class class class)co 6281   Neighbors cnbgra 24395
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-sbc 3314  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-br 4438  df-opab 4496  df-mpt 4497  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-iota 5541  df-fun 5580  df-fn 5581  df-fv 5586  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-1st 6785  df-2nd 6786  df-nbgra 24398
This theorem is referenced by: (None)
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