Theorem dfnbgr2 26235

Description: Alternate definition of the neighbors of a vertex breaking up the subset relationship of an unordered pair. (Contributed by AV, 15-Nov-2020.) (Revised by AV, 21-Mar-2021.)

Hypotheses

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

Assertion

Ref	Expression
dfnbgr2	|- ( N e. V -> ( G NeighbVtx N ) = { n e. ( V \ { N } ) | E. e e. E ( N e. e /\ n e. e ) } )

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

Allowed substitution hint:   E( n)

Proof of Theorem dfnbgr2

Step	Hyp	Ref	Expression
1		nbgrval.v	. . 3 |- V = (Vtx ` G )
2		nbgrval.e	. . 3 |- E = (Edg ` G )
3	1, 2	nbgrval 26234	. 2 |- ( N e. V -> ( G NeighbVtx N ) = { n e. ( V \ { N } ) | E. e e. E { N , n } C_ e } )
4		vex 3203	. . . . . 6 |- n e. _V
5		prssg 4350	. . . . . 6 |- ( ( N e. V /\ n e. _V ) -> ( ( N e. e /\ n e. e ) <-> { N , n } C_ e ) )
6	4, 5	mpan2 707	. . . . 5 |- ( N e. V -> ( ( N e. e /\ n e. e ) <-> { N , n } C_ e ) )
7	6	bicomd 213	. . . 4 |- ( N e. V -> ( { N , n } C_ e <-> ( N e. e /\ n e. e ) ) )
8	7	rexbidv 3052	. . 3 |- ( N e. V -> ( E. e e. E { N , n } C_ e <-> E. e e. E ( N e. e /\ n e. e ) ) )
9	8	rabbidv 3189	. 2 |- ( N e. V -> { n e. ( V \ { N } ) | E. e e. E { N , n } C_ e } = { n e. ( V \ { N } ) | E. e e. E ( N e. e /\ n e. e ) } )
10	3, 9	eqtrd 2656	1 |- ( N e. V -> ( G NeighbVtx N ) = { n e. ( V \ { N } ) | E. e e. E ( N e. e /\ n e. 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  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: (None)