Theorem nbuhgr 26239

Description: The set of neighbors of a vertex in a hypergraph. This version of nbgrval 26234 (with N being an arbitrary set instead of being a vertex) only holds for classes whose edges are subsets of the set of vertices (hypergraphs!). (Contributed by AV, 26-Oct-2020.) (Proof shortened by AV, 15-Nov-2020.)

Hypotheses

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

Assertion

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

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

Allowed substitution hint:   E( n)

Proof of Theorem nbuhgr

Step	Hyp	Ref	Expression
1		nbgrel.v	. . . 4 |- V = (Vtx ` G )
2		nbgrel.e	. . . 4 |- E = (Edg ` G )
3	1, 2	nbgrval 26234	. . 3 |- ( N e. V -> ( G NeighbVtx N ) = { n e. ( V \ { N } ) | E. e e. E { N , n } C_ e } )
4	3	a1d 25	. 2 |- ( N e. V -> ( ( G e. UHGraph /\ N e. X ) -> ( G NeighbVtx N ) = { n e. ( V \ { N } ) | E. e e. E { N , n } C_ e } ) )
5		df-nel 2898	. . . . . 6 |- ( N e/ V <-> -. N e. V )
6	1	nbgrnvtx0 26237	. . . . . 6 |- ( N e/ V -> ( G NeighbVtx N ) = (/) )
7	5, 6	sylbir 225	. . . . 5 |- ( -. N e. V -> ( G NeighbVtx N ) = (/) )
8	7	adantr 481	. . . 4 |- ( ( -. N e. V /\ ( G e. UHGraph /\ N e. X ) ) -> ( G NeighbVtx N ) = (/) )
9		simpl 473	. . . . . . . . . . . 12 |- ( ( G e. UHGraph /\ N e. X ) -> G e. UHGraph )
10	9	adantr 481	. . . . . . . . . . 11 |- ( ( ( G e. UHGraph /\ N e. X ) /\ n e. ( V \ { N } ) ) -> G e. UHGraph )
11	2	eleq2i 2693	. . . . . . . . . . . 12 |- ( e e. E <-> e e. (Edg ` G ) )
12	11	biimpi 206	. . . . . . . . . . 11 |- ( e e. E -> e e. (Edg ` G ) )
13		edguhgr 26024	. . . . . . . . . . 11 |- ( ( G e. UHGraph /\ e e. (Edg ` G ) ) -> e e. ~P (Vtx ` G ) )
14	10, 12, 13	syl2an 494	. . . . . . . . . 10 |- ( ( ( ( G e. UHGraph /\ N e. X ) /\ n e. ( V \ { N } ) ) /\ e e. E ) -> e e. ~P (Vtx ` G ) )
15		selpw 4165	. . . . . . . . . . . 12 |- ( e e. ~P (Vtx ` G ) <-> e C_ (Vtx ` G ) )
16	1	eqcomi 2631	. . . . . . . . . . . . 13 |- (Vtx ` G ) = V
17	16	sseq2i 3630	. . . . . . . . . . . 12 |- ( e C_ (Vtx ` G ) <-> e C_ V )
18	15, 17	bitri 264	. . . . . . . . . . 11 |- ( e e. ~P (Vtx ` G ) <-> e C_ V )
19		sstr 3611	. . . . . . . . . . . . . . 15 |- ( ( { N , n } C_ e /\ e C_ V ) -> { N , n } C_ V )
20		vex 3203	. . . . . . . . . . . . . . . . 17 |- n e. _V
21		prssg 4350	. . . . . . . . . . . . . . . . . 18 |- ( ( N e. X /\ n e. _V ) -> ( ( N e. V /\ n e. V ) <-> { N , n } C_ V ) )
22	21	bicomd 213	. . . . . . . . . . . . . . . . 17 |- ( ( N e. X /\ n e. _V ) -> ( { N , n } C_ V <-> ( N e. V /\ n e. V ) ) )
23	20, 22	mpan2 707	. . . . . . . . . . . . . . . 16 |- ( N e. X -> ( { N , n } C_ V <-> ( N e. V /\ n e. V ) ) )
24		simpl 473	. . . . . . . . . . . . . . . 16 |- ( ( N e. V /\ n e. V ) -> N e. V )
25	23, 24	syl6bi 243	. . . . . . . . . . . . . . 15 |- ( N e. X -> ( { N , n } C_ V -> N e. V ) )
26	19, 25	syl5com 31	. . . . . . . . . . . . . 14 |- ( ( { N , n } C_ e /\ e C_ V ) -> ( N e. X -> N e. V ) )
27	26	ex 450	. . . . . . . . . . . . 13 |- ( { N , n } C_ e -> ( e C_ V -> ( N e. X -> N e. V ) ) )
28	27	com13 88	. . . . . . . . . . . 12 |- ( N e. X -> ( e C_ V -> ( { N , n } C_ e -> N e. V ) ) )
29	28	ad3antlr 767	. . . . . . . . . . 11 |- ( ( ( ( G e. UHGraph /\ N e. X ) /\ n e. ( V \ { N } ) ) /\ e e. E ) -> ( e C_ V -> ( { N , n } C_ e -> N e. V ) ) )
30	18, 29	syl5bi 232	. . . . . . . . . 10 |- ( ( ( ( G e. UHGraph /\ N e. X ) /\ n e. ( V \ { N } ) ) /\ e e. E ) -> ( e e. ~P (Vtx ` G ) -> ( { N , n } C_ e -> N e. V ) ) )
31	14, 30	mpd 15	. . . . . . . . 9 |- ( ( ( ( G e. UHGraph /\ N e. X ) /\ n e. ( V \ { N } ) ) /\ e e. E ) -> ( { N , n } C_ e -> N e. V ) )
32	31	rexlimdva 3031	. . . . . . . 8 |- ( ( ( G e. UHGraph /\ N e. X ) /\ n e. ( V \ { N } ) ) -> ( E. e e. E { N , n } C_ e -> N e. V ) )
33	32	con3rr3 151	. . . . . . 7 |- ( -. N e. V -> ( ( ( G e. UHGraph /\ N e. X ) /\ n e. ( V \ { N } ) ) -> -. E. e e. E { N , n } C_ e ) )
34	33	expdimp 453	. . . . . 6 |- ( ( -. N e. V /\ ( G e. UHGraph /\ N e. X ) ) -> ( n e. ( V \ { N } ) -> -. E. e e. E { N , n } C_ e ) )
35	34	ralrimiv 2965	. . . . 5 |- ( ( -. N e. V /\ ( G e. UHGraph /\ N e. X ) ) -> A. n e. ( V \ { N } ) -. E. e e. E { N , n } C_ e )
36		rabeq0 3957	. . . . 5 |- ( { n e. ( V \ { N } ) | E. e e. E { N , n } C_ e } = (/) <-> A. n e. ( V \ { N } ) -. E. e e. E { N , n } C_ e )
37	35, 36	sylibr 224	. . . 4 |- ( ( -. N e. V /\ ( G e. UHGraph /\ N e. X ) ) -> { n e. ( V \ { N } ) | E. e e. E { N , n } C_ e } = (/) )
38	8, 37	eqtr4d 2659	. . 3 |- ( ( -. N e. V /\ ( G e. UHGraph /\ N e. X ) ) -> ( G NeighbVtx N ) = { n e. ( V \ { N } ) | E. e e. E { N , n } C_ e } )
39	38	ex 450	. 2 |- ( -. N e. V -> ( ( G e. UHGraph /\ N e. X ) -> ( G NeighbVtx N ) = { n e. ( V \ { N } ) | E. e e. E { N , n } C_ e } ) )
40	4, 39	pm2.61i 176	1 |- ( ( G e. UHGraph /\ N e. X ) -> ( G NeighbVtx N ) = { n e. ( V \ { N } ) | E. e e. E { N , n } C_ e } )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   e/ wnel 2897   A.wral 2912   E.wrex 2913   {crab 2916   _Vcvv 3200   \ cdif 3571   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   {csn 4177   {cpr 4179   ` cfv 5888  (class class class)co 6650  Vtxcvtx 25874  Edgcedg 25939   UHGraph cuhgr 25951   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-nel 2898  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-pw 4160  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-fn 5891  df-f 5892  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-edg 25940  df-uhgr 25953  df-nbgr 26228

This theorem is referenced by:  uhgrnbgr0nb  26250