Theorem dfnbgr3 26236

Description: Alternate definition of the neighbors of a vertex using the edge function instead of the edges themselves (see also nbgrval 26234). (Contributed by Alexander van der Vekens, 17-Dec-2017.) (Revised by AV, 25-Oct-2020.) (Revised by AV, 21-Mar-2021.)

Hypotheses

Ref	Expression
dfnbgr3.v	|- V = (Vtx ` G )
dfnbgr3.i	|- I = (iEdg ` G )

Assertion

Ref	Expression
dfnbgr3	|- ( ( 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

Allowed substitution hints:   G( i)   V( i)

Proof of Theorem dfnbgr3

Dummy variable e is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfnbgr3.v	. . . 4 |- V = (Vtx ` G )
2		eqid 2622	. . . 4 |- (Edg ` G ) = (Edg ` G )
3	1, 2	nbgrval 26234	. . 3 |- ( N e. V -> ( G NeighbVtx N ) = { n e. ( V \ { N } ) | E. e e. (Edg ` G ) { N , n } C_ e } )
4	3	adantr 481	. 2 |- ( ( N e. V /\ Fun I ) -> ( G NeighbVtx N ) = { n e. ( V \ { N } ) | E. e e. (Edg ` G ) { N , n } C_ e } )
5		edgval 25941	. . . . . 6 |- (Edg ` G ) = ran (iEdg ` G )
6		dfnbgr3.i	. . . . . . . 8 |- I = (iEdg ` G )
7	6	eqcomi 2631	. . . . . . 7 |- (iEdg ` G ) = I
8	7	rneqi 5352	. . . . . 6 |- ran (iEdg ` G ) = ran I
9	5, 8	eqtri 2644	. . . . 5 |- (Edg ` G ) = ran I
10	9	rexeqi 3143	. . . 4 |- ( E. e e. (Edg ` G ) { N , n } C_ e <-> E. e e. ran I { N , n } C_ e )
11		funfn 5918	. . . . . . 7 |- ( Fun I <-> I Fn dom I )
12	11	biimpi 206	. . . . . 6 |- ( Fun I -> I Fn dom I )
13	12	adantl 482	. . . . 5 |- ( ( N e. V /\ Fun I ) -> I Fn dom I )
14		sseq2 3627	. . . . . 6 |- ( e = ( I ` i ) -> ( { N , n } C_ e <-> { N , n } C_ ( I ` i ) ) )
15	14	rexrn 6361	. . . . 5 |- ( I Fn dom I -> ( E. e e. ran I { N , n } C_ e <-> E. i e. dom I { N , n } C_ ( I ` i ) ) )
16	13, 15	syl 17	. . . 4 |- ( ( N e. V /\ Fun I ) -> ( E. e e. ran I { N , n } C_ e <-> E. i e. dom I { N , n } C_ ( I ` i ) ) )
17	10, 16	syl5bb 272	. . 3 |- ( ( N e. V /\ Fun I ) -> ( E. e e. (Edg ` G ) { N , n } C_ e <-> E. i e. dom I { N , n } C_ ( I ` i ) ) )
18	17	rabbidv 3189	. 2 |- ( ( 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 ) } )
19	4, 18	eqtrd 2656	1 |- ( ( N e. V /\ Fun I ) -> ( G NeighbVtx N ) = { n e. ( V \ { N } ) | E. i e. dom I { N , n } C_ ( I ` i ) } )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   E.wrex 2913   {crab 2916   \ cdif 3571   C_ wss 3574   {csn 4177   {cpr 4179   dom cdm 5114   ran crn 5115   Fun wfun 5882   Fn wfn 5883   ` cfv 5888  (class class class)co 6650  Vtxcvtx 25874  iEdgciedg 25875  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-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-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-iota 5851  df-fun 5890  df-fn 5891  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-edg 25940  df-nbgr 26228

This theorem is referenced by: (None)