Theorem nbgrel 26238

Description: Characterization of a neighbor of a vertex V in a graph G. (Contributed by Alexander van der Vekens and Mario Carneiro, 9-Oct-2017.) (Revised by AV, 26-Oct-2020.) (Proof shortened by AV, 6-Jun-2021.)

Hypotheses

Ref	Expression
nbgrel.v	|- V = (Vtx ` G )
nbgrel.e	|- E = (Edg ` G )

Assertion

Ref	Expression
nbgrel	|- ( G e. W -> ( K e. ( G NeighbVtx N ) <-> ( ( K e. V /\ N e. V ) /\ K =/= N /\ E. e e. E { N , K } C_ e ) ) )

Distinct variable groups:   e, E   e, G   e, K   e, N   e, V   e, W

Proof of Theorem nbgrel

Dummy variable k is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nbgrcl 26233	. . . . 5 |- ( K e. ( G NeighbVtx N ) -> N e. (Vtx ` G ) )
2		nbgrel.v	. . . . 5 |- V = (Vtx ` G )
3	1, 2	syl6eleqr 2712	. . . 4 |- ( K e. ( G NeighbVtx N ) -> N e. V )
4	3	a1i 11	. . 3 |- ( G e. W -> ( K e. ( G NeighbVtx N ) -> N e. V ) )
5	4	pm4.71rd 667	. 2 |- ( G e. W -> ( K e. ( G NeighbVtx N ) <-> ( N e. V /\ K e. ( G NeighbVtx N ) ) ) )
6		nbgrel.e	. . . . . . . 8 |- E = (Edg ` G )
7	2, 6	nbgrval 26234	. . . . . . 7 |- ( N e. V -> ( G NeighbVtx N ) = { k e. ( V \ { N } ) | E. e e. E { N , k } C_ e } )
8	7	eleq2d 2687	. . . . . 6 |- ( N e. V -> ( K e. ( G NeighbVtx N ) <-> K e. { k e. ( V \ { N } ) | E. e e. E { N , k } C_ e } ) )
9		preq2 4269	. . . . . . . . . 10 |- ( k = K -> { N , k } = { N , K } )
10	9	sseq1d 3632	. . . . . . . . 9 |- ( k = K -> ( { N , k } C_ e <-> { N , K } C_ e ) )
11	10	rexbidv 3052	. . . . . . . 8 |- ( k = K -> ( E. e e. E { N , k } C_ e <-> E. e e. E { N , K } C_ e ) )
12	11	elrab 3363	. . . . . . 7 |- ( K e. { k e. ( V \ { N } ) | E. e e. E { N , k } C_ e } <-> ( K e. ( V \ { N } ) /\ E. e e. E { N , K } C_ e ) )
13		eldifsn 4317	. . . . . . . 8 |- ( K e. ( V \ { N } ) <-> ( K e. V /\ K =/= N ) )
14	13	anbi1i 731	. . . . . . 7 |- ( ( K e. ( V \ { N } ) /\ E. e e. E { N , K } C_ e ) <-> ( ( K e. V /\ K =/= N ) /\ E. e e. E { N , K } C_ e ) )
15		anass 681	. . . . . . 7 |- ( ( ( K e. V /\ K =/= N ) /\ E. e e. E { N , K } C_ e ) <-> ( K e. V /\ ( K =/= N /\ E. e e. E { N , K } C_ e ) ) )
16	12, 14, 15	3bitri 286	. . . . . 6 |- ( K e. { k e. ( V \ { N } ) | E. e e. E { N , k } C_ e } <-> ( K e. V /\ ( K =/= N /\ E. e e. E { N , K } C_ e ) ) )
17	8, 16	syl6bb 276	. . . . 5 |- ( N e. V -> ( K e. ( G NeighbVtx N ) <-> ( K e. V /\ ( K =/= N /\ E. e e. E { N , K } C_ e ) ) ) )
18	17	adantl 482	. . . 4 |- ( ( G e. W /\ N e. V ) -> ( K e. ( G NeighbVtx N ) <-> ( K e. V /\ ( K =/= N /\ E. e e. E { N , K } C_ e ) ) ) )
19	18	pm5.32da 673	. . 3 |- ( G e. W -> ( ( N e. V /\ K e. ( G NeighbVtx N ) ) <-> ( N e. V /\ ( K e. V /\ ( K =/= N /\ E. e e. E { N , K } C_ e ) ) ) ) )
20		3anass 1042	. . . 4 |- ( ( ( K e. V /\ N e. V ) /\ K =/= N /\ E. e e. E { N , K } C_ e ) <-> ( ( K e. V /\ N e. V ) /\ ( K =/= N /\ E. e e. E { N , K } C_ e ) ) )
21		ancom 466	. . . . 5 |- ( ( K e. V /\ N e. V ) <-> ( N e. V /\ K e. V ) )
22	21	anbi1i 731	. . . 4 |- ( ( ( K e. V /\ N e. V ) /\ ( K =/= N /\ E. e e. E { N , K } C_ e ) ) <-> ( ( N e. V /\ K e. V ) /\ ( K =/= N /\ E. e e. E { N , K } C_ e ) ) )
23		anass 681	. . . 4 |- ( ( ( N e. V /\ K e. V ) /\ ( K =/= N /\ E. e e. E { N , K } C_ e ) ) <-> ( N e. V /\ ( K e. V /\ ( K =/= N /\ E. e e. E { N , K } C_ e ) ) ) )
24	20, 22, 23	3bitrri 287	. . 3 |- ( ( N e. V /\ ( K e. V /\ ( K =/= N /\ E. e e. E { N , K } C_ e ) ) ) <-> ( ( K e. V /\ N e. V ) /\ K =/= N /\ E. e e. E { N , K } C_ e ) )
25	19, 24	syl6bb 276	. 2 |- ( G e. W -> ( ( N e. V /\ K e. ( G NeighbVtx N ) ) <-> ( ( K e. V /\ N e. V ) /\ K =/= N /\ E. e e. E { N , K } C_ e ) ) )
26	5, 25	bitrd 268	1 |- ( G e. W -> ( K e. ( G NeighbVtx N ) <-> ( ( K e. V /\ N e. V ) /\ K =/= N /\ E. e e. E { N , K } C_ e ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913   {crab 2916   \ cdif 3571   C_ wss 3574   {csn 4177   {cpr 4179   ` cfv 5888  (class class class)co 6650  Vtxcvtx 25874  Edgcedg 25939   NeighbVtx cnbgr 26224

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-nbgr 26228

This theorem is referenced by:  nbgr2vtx1edg  26246  nbuhgr2vtx1edgblem  26247  nbuhgr2vtx1edgb  26248  nbgrisvtx  26255  nbgrsym  26265  isuvtxa  26295  iscplgredg  26313  cusgrexi  26339  structtocusgr  26342