Theorem nbupgrres 26266

Description: The neighborhood of a vertex in a restricted pseudograph (not necessarily valid for a hypergraph, because N, K and M could be connected by one edge, so M is a neighbor of K in the original graph, but not in the restricted graph, because the edge between M and K, also incident with N, was removed). (Contributed by Alexander van der Vekens, 2-Jan-2018.) (Revised by AV, 8-Nov-2020.)

Hypotheses

Ref	Expression
nbupgrres.v	|- V = (Vtx ` G )
nbupgrres.e	|- E = (Edg ` G )
nbupgrres.f	|- F = { e e. E | N e/ e }
nbupgrres.s	|- S = <. ( V \ { N } ) , ( _I |` F ) >.

Assertion

Ref	Expression
nbupgrres	|- ( ( ( G e. UPGraph /\ N e. V ) /\ K e. ( V \ { N } ) /\ M e. ( V \ { N , K } ) ) -> ( M e. ( G NeighbVtx K ) -> M e. ( S NeighbVtx K ) ) )

Distinct variable groups:   e, E   e, G   e, K   e, N   e, M   e, V

Allowed substitution hints:   S( e)   F( e)

Proof of Theorem nbupgrres

Step	Hyp	Ref	Expression
1		simp1l 1085	. . . . . 6 |- ( ( ( G e. UPGraph /\ N e. V ) /\ K e. ( V \ { N } ) /\ M e. ( V \ { N , K } ) ) -> G e. UPGraph )
2		eldifi 3732	. . . . . . 7 |- ( K e. ( V \ { N } ) -> K e. V )
3	2	3ad2ant2 1083	. . . . . 6 |- ( ( ( G e. UPGraph /\ N e. V ) /\ K e. ( V \ { N } ) /\ M e. ( V \ { N , K } ) ) -> K e. V )
4		eldifsn 4317	. . . . . . . . 9 |- ( M e. ( ( V \ { N } ) \ { K } ) <-> ( M e. ( V \ { N } ) /\ M =/= K ) )
5		eldifi 3732	. . . . . . . . . 10 |- ( M e. ( V \ { N } ) -> M e. V )
6	5	anim1i 592	. . . . . . . . 9 |- ( ( M e. ( V \ { N } ) /\ M =/= K ) -> ( M e. V /\ M =/= K ) )
7	4, 6	sylbi 207	. . . . . . . 8 |- ( M e. ( ( V \ { N } ) \ { K } ) -> ( M e. V /\ M =/= K ) )
8		difpr 4334	. . . . . . . 8 |- ( V \ { N , K } ) = ( ( V \ { N } ) \ { K } )
9	7, 8	eleq2s 2719	. . . . . . 7 |- ( M e. ( V \ { N , K } ) -> ( M e. V /\ M =/= K ) )
10	9	3ad2ant3 1084	. . . . . 6 |- ( ( ( G e. UPGraph /\ N e. V ) /\ K e. ( V \ { N } ) /\ M e. ( V \ { N , K } ) ) -> ( M e. V /\ M =/= K ) )
11		nbupgrres.v	. . . . . . 7 |- V = (Vtx ` G )
12		nbupgrres.e	. . . . . . 7 |- E = (Edg ` G )
13	11, 12	nbupgrel 26241	. . . . . 6 |- ( ( ( G e. UPGraph /\ K e. V ) /\ ( M e. V /\ M =/= K ) ) -> ( M e. ( G NeighbVtx K ) <-> { M , K } e. E ) )
14	1, 3, 10, 13	syl21anc 1325	. . . . 5 |- ( ( ( G e. UPGraph /\ N e. V ) /\ K e. ( V \ { N } ) /\ M e. ( V \ { N , K } ) ) -> ( M e. ( G NeighbVtx K ) <-> { M , K } e. E ) )
15	14	biimpa 501	. . . 4 |- ( ( ( ( G e. UPGraph /\ N e. V ) /\ K e. ( V \ { N } ) /\ M e. ( V \ { N , K } ) ) /\ M e. ( G NeighbVtx K ) ) -> { M , K } e. E )
16	8	eleq2i 2693	. . . . . . . . . 10 |- ( M e. ( V \ { N , K } ) <-> M e. ( ( V \ { N } ) \ { K } ) )
17		eldifsn 4317	. . . . . . . . . . 11 |- ( M e. ( V \ { N } ) <-> ( M e. V /\ M =/= N ) )
18	17	anbi1i 731	. . . . . . . . . 10 |- ( ( M e. ( V \ { N } ) /\ M =/= K ) <-> ( ( M e. V /\ M =/= N ) /\ M =/= K ) )
19	16, 4, 18	3bitri 286	. . . . . . . . 9 |- ( M e. ( V \ { N , K } ) <-> ( ( M e. V /\ M =/= N ) /\ M =/= K ) )
20		simpr 477	. . . . . . . . . . 11 |- ( ( M e. V /\ M =/= N ) -> M =/= N )
21	20	necomd 2849	. . . . . . . . . 10 |- ( ( M e. V /\ M =/= N ) -> N =/= M )
22	21	adantr 481	. . . . . . . . 9 |- ( ( ( M e. V /\ M =/= N ) /\ M =/= K ) -> N =/= M )
23	19, 22	sylbi 207	. . . . . . . 8 |- ( M e. ( V \ { N , K } ) -> N =/= M )
24	23	3ad2ant3 1084	. . . . . . 7 |- ( ( ( G e. UPGraph /\ N e. V ) /\ K e. ( V \ { N } ) /\ M e. ( V \ { N , K } ) ) -> N =/= M )
25		eldifsn 4317	. . . . . . . . 9 |- ( K e. ( V \ { N } ) <-> ( K e. V /\ K =/= N ) )
26		simpr 477	. . . . . . . . . 10 |- ( ( K e. V /\ K =/= N ) -> K =/= N )
27	26	necomd 2849	. . . . . . . . 9 |- ( ( K e. V /\ K =/= N ) -> N =/= K )
28	25, 27	sylbi 207	. . . . . . . 8 |- ( K e. ( V \ { N } ) -> N =/= K )
29	28	3ad2ant2 1083	. . . . . . 7 |- ( ( ( G e. UPGraph /\ N e. V ) /\ K e. ( V \ { N } ) /\ M e. ( V \ { N , K } ) ) -> N =/= K )
30	24, 29	nelprd 4203	. . . . . 6 |- ( ( ( G e. UPGraph /\ N e. V ) /\ K e. ( V \ { N } ) /\ M e. ( V \ { N , K } ) ) -> -. N e. { M , K } )
31		df-nel 2898	. . . . . 6 |- ( N e/ { M , K } <-> -. N e. { M , K } )
32	30, 31	sylibr 224	. . . . 5 |- ( ( ( G e. UPGraph /\ N e. V ) /\ K e. ( V \ { N } ) /\ M e. ( V \ { N , K } ) ) -> N e/ { M , K } )
33	32	adantr 481	. . . 4 |- ( ( ( ( G e. UPGraph /\ N e. V ) /\ K e. ( V \ { N } ) /\ M e. ( V \ { N , K } ) ) /\ M e. ( G NeighbVtx K ) ) -> N e/ { M , K } )
34		neleq2 2903	. . . . 5 |- ( e = { M , K } -> ( N e/ e <-> N e/ { M , K } ) )
35		nbupgrres.f	. . . . 5 |- F = { e e. E | N e/ e }
36	34, 35	elrab2 3366	. . . 4 |- ( { M , K } e. F <-> ( { M , K } e. E /\ N e/ { M , K } ) )
37	15, 33, 36	sylanbrc 698	. . 3 |- ( ( ( ( G e. UPGraph /\ N e. V ) /\ K e. ( V \ { N } ) /\ M e. ( V \ { N , K } ) ) /\ M e. ( G NeighbVtx K ) ) -> { M , K } e. F )
38		nbupgrres.s	. . . . . . . 8 |- S = <. ( V \ { N } ) , ( _I |` F ) >.
39	11, 12, 35, 38	upgrres1 26205	. . . . . . 7 |- ( ( G e. UPGraph /\ N e. V ) -> S e. UPGraph )
40	39	3ad2ant1 1082	. . . . . 6 |- ( ( ( G e. UPGraph /\ N e. V ) /\ K e. ( V \ { N } ) /\ M e. ( V \ { N , K } ) ) -> S e. UPGraph )
41		simp2 1062	. . . . . 6 |- ( ( ( G e. UPGraph /\ N e. V ) /\ K e. ( V \ { N } ) /\ M e. ( V \ { N , K } ) ) -> K e. ( V \ { N } ) )
42	16, 4	sylbb 209	. . . . . . 7 |- ( M e. ( V \ { N , K } ) -> ( M e. ( V \ { N } ) /\ M =/= K ) )
43	42	3ad2ant3 1084	. . . . . 6 |- ( ( ( G e. UPGraph /\ N e. V ) /\ K e. ( V \ { N } ) /\ M e. ( V \ { N , K } ) ) -> ( M e. ( V \ { N } ) /\ M =/= K ) )
44	40, 41, 43	jca31 557	. . . . 5 |- ( ( ( G e. UPGraph /\ N e. V ) /\ K e. ( V \ { N } ) /\ M e. ( V \ { N , K } ) ) -> ( ( S e. UPGraph /\ K e. ( V \ { N } ) ) /\ ( M e. ( V \ { N } ) /\ M =/= K ) ) )
45	44	adantr 481	. . . 4 |- ( ( ( ( G e. UPGraph /\ N e. V ) /\ K e. ( V \ { N } ) /\ M e. ( V \ { N , K } ) ) /\ M e. ( G NeighbVtx K ) ) -> ( ( S e. UPGraph /\ K e. ( V \ { N } ) ) /\ ( M e. ( V \ { N } ) /\ M =/= K ) ) )
46	11, 12, 35, 38	upgrres1lem2 26203	. . . . . 6 |- (Vtx ` S ) = ( V \ { N } )
47	46	eqcomi 2631	. . . . 5 |- ( V \ { N } ) = (Vtx ` S )
48		edgval 25941	. . . . . 6 |- (Edg ` S ) = ran (iEdg ` S )
49	11, 12, 35, 38	upgrres1lem3 26204	. . . . . . 7 |- (iEdg ` S ) = ( _I |` F )
50	49	rneqi 5352	. . . . . 6 |- ran (iEdg ` S ) = ran ( _I |` F )
51		rnresi 5479	. . . . . 6 |- ran ( _I |` F ) = F
52	48, 50, 51	3eqtrri 2649	. . . . 5 |- F = (Edg ` S )
53	47, 52	nbupgrel 26241	. . . 4 |- ( ( ( S e. UPGraph /\ K e. ( V \ { N } ) ) /\ ( M e. ( V \ { N } ) /\ M =/= K ) ) -> ( M e. ( S NeighbVtx K ) <-> { M , K } e. F ) )
54	45, 53	syl 17	. . 3 |- ( ( ( ( G e. UPGraph /\ N e. V ) /\ K e. ( V \ { N } ) /\ M e. ( V \ { N , K } ) ) /\ M e. ( G NeighbVtx K ) ) -> ( M e. ( S NeighbVtx K ) <-> { M , K } e. F ) )
55	37, 54	mpbird 247	. 2 |- ( ( ( ( G e. UPGraph /\ N e. V ) /\ K e. ( V \ { N } ) /\ M e. ( V \ { N , K } ) ) /\ M e. ( G NeighbVtx K ) ) -> M e. ( S NeighbVtx K ) )
56	55	ex 450	1 |- ( ( ( G e. UPGraph /\ N e. V ) /\ K e. ( V \ { N } ) /\ M e. ( V \ { N , K } ) ) -> ( M e. ( G NeighbVtx K ) -> M e. ( S NeighbVtx K ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   e/ wnel 2897   {crab 2916   \ cdif 3571   {csn 4177   {cpr 4179   <.cop 4183   _I cid 5023   ran crn 5115   |` cres 5116   ` cfv 5888  (class class class)co 6650  Vtxcvtx 25874  iEdgciedg 25875  Edgcedg 25939   UPGraph cupgr 25975   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-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  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-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  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-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  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-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-fz 12327  df-hash 13118  df-vtx 25876  df-iedg 25877  df-edg 25940  df-upgr 25977  df-nbgr 26228

This theorem is referenced by:  nbupgruvtxres  26308