Theorem nbgrval 26234

Description: The set of neighbors of a vertex V in a graph G. (Contributed by Alexander van der Vekens, 7-Oct-2017.) (Revised by AV, 24-Oct-2020.) (Revised by AV, 21-Mar-2021.)

Hypotheses

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

Assertion

Ref	Expression
nbgrval	|- ( N e. V -> ( G NeighbVtx N ) = { n e. ( V \ { N } ) | E. e e. E { N , n } C_ e } )

Distinct variable groups:   e, E   e, G, n   e, N, n   e, V, n

Allowed substitution hint:   E( n)

Proof of Theorem nbgrval

Dummy variables g k are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-nbgr 26228	. 2 |- NeighbVtx = ( g e. _V , k e. (Vtx ` g ) |-> { n e. ( (Vtx ` g ) \ { k } ) | E. e e. (Edg ` g ) { k , n } C_ e } )
2		nbgrval.v	. . . 4 |- V = (Vtx ` G )
3	2	1vgrex 25882	. . 3 |- ( N e. V -> G e. _V )
4		fveq2 6191	. . . . . 6 |- ( g = G -> (Vtx ` g ) = (Vtx ` G ) )
5	4, 2	syl6reqr 2675	. . . . 5 |- ( g = G -> V = (Vtx ` g ) )
6	5	eleq2d 2687	. . . 4 |- ( g = G -> ( N e. V <-> N e. (Vtx ` g ) ) )
7	6	biimpac 503	. . 3 |- ( ( N e. V /\ g = G ) -> N e. (Vtx ` g ) )
8		fvex 6201	. . . . 5 |- (Vtx ` g ) e. _V
9	8	difexi 4809	. . . 4 |- ( (Vtx ` g ) \ { k } ) e. _V
10		rabexg 4812	. . . 4 |- ( ( (Vtx ` g ) \ { k } ) e. _V -> { n e. ( (Vtx ` g ) \ { k } ) | E. e e. (Edg ` g ) { k , n } C_ e } e. _V )
11	9, 10	mp1i 13	. . 3 |- ( ( N e. V /\ ( g = G /\ k = N ) ) -> { n e. ( (Vtx ` g ) \ { k } ) | E. e e. (Edg ` g ) { k , n } C_ e } e. _V )
12	4, 2	syl6eqr 2674	. . . . . . 7 |- ( g = G -> (Vtx ` g ) = V )
13	12	adantr 481	. . . . . 6 |- ( ( g = G /\ k = N ) -> (Vtx ` g ) = V )
14		sneq 4187	. . . . . . 7 |- ( k = N -> { k } = { N } )
15	14	adantl 482	. . . . . 6 |- ( ( g = G /\ k = N ) -> { k } = { N } )
16	13, 15	difeq12d 3729	. . . . 5 |- ( ( g = G /\ k = N ) -> ( (Vtx ` g ) \ { k } ) = ( V \ { N } ) )
17	16	adantl 482	. . . 4 |- ( ( N e. V /\ ( g = G /\ k = N ) ) -> ( (Vtx ` g ) \ { k } ) = ( V \ { N } ) )
18		fveq2 6191	. . . . . . . 8 |- ( g = G -> (Edg ` g ) = (Edg ` G ) )
19		nbgrval.e	. . . . . . . 8 |- E = (Edg ` G )
20	18, 19	syl6eqr 2674	. . . . . . 7 |- ( g = G -> (Edg ` g ) = E )
21	20	adantr 481	. . . . . 6 |- ( ( g = G /\ k = N ) -> (Edg ` g ) = E )
22	21	adantl 482	. . . . 5 |- ( ( N e. V /\ ( g = G /\ k = N ) ) -> (Edg ` g ) = E )
23		preq1 4268	. . . . . . . 8 |- ( k = N -> { k , n } = { N , n } )
24	23	sseq1d 3632	. . . . . . 7 |- ( k = N -> ( { k , n } C_ e <-> { N , n } C_ e ) )
25	24	adantl 482	. . . . . 6 |- ( ( g = G /\ k = N ) -> ( { k , n } C_ e <-> { N , n } C_ e ) )
26	25	adantl 482	. . . . 5 |- ( ( N e. V /\ ( g = G /\ k = N ) ) -> ( { k , n } C_ e <-> { N , n } C_ e ) )
27	22, 26	rexeqbidv 3153	. . . 4 |- ( ( N e. V /\ ( g = G /\ k = N ) ) -> ( E. e e. (Edg ` g ) { k , n } C_ e <-> E. e e. E { N , n } C_ e ) )
28	17, 27	rabeqbidv 3195	. . 3 |- ( ( N e. V /\ ( g = G /\ k = N ) ) -> { n e. ( (Vtx ` g ) \ { k } ) | E. e e. (Edg ` g ) { k , n } C_ e } = { n e. ( V \ { N } ) | E. e e. E { N , n } C_ e } )
29	3, 7, 11, 28	ovmpt2dv2 6794	. 2 |- ( N e. V -> ( NeighbVtx = ( g e. _V , k e. (Vtx ` g ) |-> { n e. ( (Vtx ` g ) \ { k } ) | E. e e. (Edg ` g ) { k , n } C_ e } ) -> ( G NeighbVtx N ) = { n e. ( V \ { N } ) | E. e e. E { N , n } C_ e } ) )
30	1, 29	mpi 20	1 |- ( N e. V -> ( G NeighbVtx N ) = { n e. ( V \ { N } ) | E. e e. E { N , n } C_ e } )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   E.wrex 2913   {crab 2916   _Vcvv 3200   \ cdif 3571   C_ wss 3574   {csn 4177   {cpr 4179   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652  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

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  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-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-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-nbgr 26228

This theorem is referenced by:  dfnbgr2  26235  dfnbgr3  26236  nbgrel  26238  nbuhgr  26239  nbupgr  26240  nbumgrvtx  26242  nbgr0vtxlem  26251  nbgrnself  26257