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Theorem dfnbgr3 39418
Description: Alternate definition of the neighbors of a vertex using the edge function instead of the edges themselves [see also nbgraop1 25165]. (Contributed by Alexander van der Vekens, 17-Dec-2017.) (Revised by AV, 25-Oct-2020.)
Hypotheses
Ref Expression
dfnbgr3.v  |-  V  =  (Vtx `  G )
dfnbgr3.i  |-  I  =  (iEdg `  G )
Assertion
Ref Expression
dfnbgr3  |-  ( ( G  e.  W  /\  N  e.  V  /\  Fun  I )  ->  ( G NeighbVtx  N )  =  {
n  e.  ( V 
\  { N }
)  |  E. i  e.  dom  I { N ,  n }  C_  (
I `  i ) } )
Distinct variable groups:    n, G    i, I, n    i, N, n    n, V    n, W
Allowed substitution hints:    G( i)    V( i)    W( i)

Proof of Theorem dfnbgr3
Dummy variable  e is distinct from all other variables.
StepHypRef Expression
1 dfnbgr3.v . . . 4  |-  V  =  (Vtx `  G )
2 eqid 2453 . . . 4  |-  (Edg `  G )  =  (Edg
`  G )
31, 2nbgrval 39416 . . 3  |-  ( ( G  e.  W  /\  N  e.  V )  ->  ( G NeighbVtx  N )  =  { n  e.  ( V  \  { N } )  |  E. e  e.  (Edg `  G
) { N ,  n }  C_  e } )
433adant3 1029 . 2  |-  ( ( G  e.  W  /\  N  e.  V  /\  Fun  I )  ->  ( G NeighbVtx  N )  =  {
n  e.  ( V 
\  { N }
)  |  E. e  e.  (Edg `  G ) { N ,  n }  C_  e } )
5 edgaval 39222 . . . . . . 7  |-  ( G  e.  W  ->  (Edg `  G )  =  ran  (iEdg `  G ) )
6 dfnbgr3.i . . . . . . . . 9  |-  I  =  (iEdg `  G )
76eqcomi 2462 . . . . . . . 8  |-  (iEdg `  G )  =  I
87rneqi 5064 . . . . . . 7  |-  ran  (iEdg `  G )  =  ran  I
95, 8syl6eq 2503 . . . . . 6  |-  ( G  e.  W  ->  (Edg `  G )  =  ran  I )
109rexeqdv 2996 . . . . 5  |-  ( G  e.  W  ->  ( E. e  e.  (Edg `  G ) { N ,  n }  C_  e  <->  E. e  e.  ran  I { N ,  n }  C_  e ) )
11103ad2ant1 1030 . . . 4  |-  ( ( G  e.  W  /\  N  e.  V  /\  Fun  I )  ->  ( E. e  e.  (Edg `  G ) { N ,  n }  C_  e  <->  E. e  e.  ran  I { N ,  n }  C_  e ) )
12 funfn 5614 . . . . . . 7  |-  ( Fun  I  <->  I  Fn  dom  I )
1312biimpi 198 . . . . . 6  |-  ( Fun  I  ->  I  Fn  dom  I )
14133ad2ant3 1032 . . . . 5  |-  ( ( G  e.  W  /\  N  e.  V  /\  Fun  I )  ->  I  Fn  dom  I )
15 sseq2 3456 . . . . . 6  |-  ( e  =  ( I `  i )  ->  ( { N ,  n }  C_  e  <->  { N ,  n }  C_  ( I `  i ) ) )
1615rexrn 6029 . . . . 5  |-  ( I  Fn  dom  I  -> 
( E. e  e. 
ran  I { N ,  n }  C_  e  <->  E. i  e.  dom  I { N ,  n }  C_  ( I `  i
) ) )
1714, 16syl 17 . . . 4  |-  ( ( G  e.  W  /\  N  e.  V  /\  Fun  I )  ->  ( E. e  e.  ran  I { N ,  n }  C_  e  <->  E. i  e.  dom  I { N ,  n }  C_  (
I `  i )
) )
1811, 17bitrd 257 . . 3  |-  ( ( G  e.  W  /\  N  e.  V  /\  Fun  I )  ->  ( E. e  e.  (Edg `  G ) { N ,  n }  C_  e  <->  E. i  e.  dom  I { N ,  n }  C_  ( I `  i
) ) )
1918rabbidv 3038 . 2  |-  ( ( G  e.  W  /\  N  e.  V  /\  Fun  I )  ->  { n  e.  ( V  \  { N } )  |  E. e  e.  (Edg `  G
) { N ,  n }  C_  e }  =  { n  e.  ( V  \  { N } )  |  E. i  e.  dom  I { N ,  n }  C_  ( I `  i
) } )
204, 19eqtrd 2487 1  |-  ( ( G  e.  W  /\  N  e.  V  /\  Fun  I )  ->  ( G NeighbVtx  N )  =  {
n  e.  ( V 
\  { N }
)  |  E. i  e.  dom  I { N ,  n }  C_  (
I `  i ) } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ w3a 986    = wceq 1446    e. wcel 1889   E.wrex 2740   {crab 2743    \ cdif 3403    C_ wss 3406   {csn 3970   {cpr 3972   dom cdm 4837   ran crn 4838   Fun wfun 5579    Fn wfn 5580   ` cfv 5585  (class class class)co 6295  Vtxcvtx 39111  iEdgciedg 39112  Edgcedga 39220   NeighbVtx cnbgr 39407
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-sep 4528  ax-nul 4537  ax-pr 4642  ax-un 6588
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-ral 2744  df-rex 2745  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3734  df-if 3884  df-sn 3971  df-pr 3973  df-op 3977  df-uni 4202  df-br 4406  df-opab 4465  df-mpt 4466  df-id 4752  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-iota 5549  df-fun 5587  df-fn 5588  df-fv 5593  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-edga 39221  df-nbgr 39411
This theorem is referenced by: (None)
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