Theorem nbgrel 39410

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.)

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 variables g n v k are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-nbgr 39401	. . . . . 6 |- NeighbVtx = ( g e. _V , v e. (Vtx ` g ) |-> { n e. ( (Vtx ` g ) \ { v } ) | E. e e. (Edg ` g ) { v , n } C_ e } )
2	1	mpt2xeldm 6957	. . . . 5 |- ( K e. ( G NeighbVtx N ) -> ( G e. _V /\ N e. [_ G / g ]_ (Vtx ` g ) ) )
3		csbfv 5902	. . . . . . . . 9 |- [_ G / g ]_ (Vtx ` g ) = (Vtx ` G )
4		nbgrel.v	. . . . . . . . 9 |- V = (Vtx ` G )
5	3, 4	eqtr4i 2476	. . . . . . . 8 |- [_ G / g ]_ (Vtx ` g ) = V
6	5	eleq2i 2521	. . . . . . 7 |- ( N e. [_ G / g ]_ (Vtx ` g ) <-> N e. V )
7	6	biimpi 198	. . . . . 6 |- ( N e. [_ G / g ]_ (Vtx ` g ) -> N e. V )
8	7	adantl 468	. . . . 5 |- ( ( G e. _V /\ N e. [_ G / g ]_ (Vtx ` g ) ) -> N e. V )
9	2, 8	syl 17	. . . 4 |- ( K e. ( G NeighbVtx N ) -> N e. V )
10	9	a1i 11	. . 3 |- ( G e. W -> ( K e. ( G NeighbVtx N ) -> N e. V ) )
11	10	pm4.71rd 641	. 2 |- ( G e. W -> ( K e. ( G NeighbVtx N ) <-> ( N e. V /\ K e. ( G NeighbVtx N ) ) ) )
12		nbgrel.e	. . . . . . 7 |- E = (Edg ` G )
13	4, 12	nbgrval 39406	. . . . . 6 |- ( ( G e. W /\ N e. V ) -> ( G NeighbVtx N ) = { k e. ( V \ { N } ) | E. e e. E { N , k } C_ e } )
14	13	eleq2d 2514	. . . . 5 |- ( ( G e. W /\ N e. V ) -> ( K e. ( G NeighbVtx N ) <-> K e. { k e. ( V \ { N } ) | E. e e. E { N , k } C_ e } ) )
15		preq2 4052	. . . . . . . . 9 |- ( k = K -> { N , k } = { N , K } )
16	15	sseq1d 3459	. . . . . . . 8 |- ( k = K -> ( { N , k } C_ e <-> { N , K } C_ e ) )
17	16	rexbidv 2901	. . . . . . 7 |- ( k = K -> ( E. e e. E { N , k } C_ e <-> E. e e. E { N , K } C_ e ) )
18	17	elrab 3196	. . . . . 6 |- ( 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 ) )
19		eldifsn 4097	. . . . . . 7 |- ( K e. ( V \ { N } ) <-> ( K e. V /\ K =/= N ) )
20	19	anbi1i 701	. . . . . 6 |- ( ( 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 ) )
21		anass 655	. . . . . 6 |- ( ( ( 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 ) ) )
22	18, 20, 21	3bitri 275	. . . . 5 |- ( 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 ) ) )
23	14, 22	syl6bb 265	. . . 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 ) ) ) )
24	23	pm5.32da 647	. . 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 ) ) ) ) )
25		3anass 989	. . . 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 ) ) )
26		ancom 452	. . . . 5 |- ( ( K e. V /\ N e. V ) <-> ( N e. V /\ K e. V ) )
27	26	anbi1i 701	. . . 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 ) ) )
28		anass 655	. . . 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 ) ) ) )
29	25, 27, 28	3bitrri 276	. . 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 ) )
30	24, 29	syl6bb 265	. 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 ) ) )
31	11, 30	bitrd 257	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 188   /\ wa 371   /\ w3a 985   = wceq 1444   e. wcel 1887   =/= wne 2622   E.wrex 2738   {crab 2741   _Vcvv 3045   [_csb 3363   \ cdif 3401   C_ wss 3404   {csn 3968   {cpr 3970   ` cfv 5582  (class class class)co 6290  Vtxcvtx 39101  Edgcedga 39210   NeighbVtx cnbgr 39397

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-fal 1450  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-id 4749  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-iota 5546  df-fun 5584  df-fv 5590  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-1st 6793  df-2nd 6794  df-nbgr 39401

This theorem is referenced by:  nbgr2vtx1edg  39418  nbuhgr2vtx1edgblem  39419  nbuhgr2vtx1edgb  39420  nbgrisvtx  39427  nbgrsym  39437  isuvtxa  39467  iscplgredg  39485  cusgrexi  39507