Theorem nbupgruvtxres 26308

Description: The neighborhood of a universal vertex in a restricted pseudograph. (Contributed by Alexander van der Vekens, 2-Jan-2018.) (Revised by AV, 8-Nov-2020.)

Hypotheses

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

Assertion

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

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

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

Proof of Theorem nbupgruvtxres

Dummy variable n is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nbupgruvtxres.s	. . . . . . 7 |- S = <. ( V \ { N } ) , ( _I |` F ) >.
2		opex 4932	. . . . . . 7 |- <. ( V \ { N } ) , ( _I |` F ) >. e. _V
3	1, 2	eqeltri 2697	. . . . . 6 |- S e. _V
4		eqid 2622	. . . . . . 7 |- (Vtx ` S ) = (Vtx ` S )
5	4	nbgrssovtx 26260	. . . . . 6 |- ( S e. _V -> ( S NeighbVtx K ) C_ ( (Vtx ` S ) \ { K } ) )
6	3, 5	mp1i 13	. . . . 5 |- ( ( ( G e. UPGraph /\ N e. V ) /\ K e. ( V \ { N } ) ) -> ( S NeighbVtx K ) C_ ( (Vtx ` S ) \ { K } ) )
7		difpr 4334	. . . . . 6 |- ( V \ { N , K } ) = ( ( V \ { N } ) \ { K } )
8		nbupgruvtxres.v	. . . . . . . . . 10 |- V = (Vtx ` G )
9		nbupgruvtxres.e	. . . . . . . . . 10 |- E = (Edg ` G )
10		nbupgruvtxres.f	. . . . . . . . . 10 |- F = { e e. E | N e/ e }
11	8, 9, 10, 1	upgrres1lem2 26203	. . . . . . . . 9 |- (Vtx ` S ) = ( V \ { N } )
12	11	eqcomi 2631	. . . . . . . 8 |- ( V \ { N } ) = (Vtx ` S )
13	12	a1i 11	. . . . . . 7 |- ( ( ( G e. UPGraph /\ N e. V ) /\ K e. ( V \ { N } ) ) -> ( V \ { N } ) = (Vtx ` S ) )
14	13	difeq1d 3727	. . . . . 6 |- ( ( ( G e. UPGraph /\ N e. V ) /\ K e. ( V \ { N } ) ) -> ( ( V \ { N } ) \ { K } ) = ( (Vtx ` S ) \ { K } ) )
15	7, 14	syl5eq 2668	. . . . 5 |- ( ( ( G e. UPGraph /\ N e. V ) /\ K e. ( V \ { N } ) ) -> ( V \ { N , K } ) = ( (Vtx ` S ) \ { K } ) )
16	6, 15	sseqtr4d 3642	. . . 4 |- ( ( ( G e. UPGraph /\ N e. V ) /\ K e. ( V \ { N } ) ) -> ( S NeighbVtx K ) C_ ( V \ { N , K } ) )
17	16	adantr 481	. . 3 |- ( ( ( ( G e. UPGraph /\ N e. V ) /\ K e. ( V \ { N } ) ) /\ ( G NeighbVtx K ) = ( V \ { K } ) ) -> ( S NeighbVtx K ) C_ ( V \ { N , K } ) )
18		simpl 473	. . . . . . . 8 |- ( ( ( ( G e. UPGraph /\ N e. V ) /\ K e. ( V \ { N } ) ) /\ ( G NeighbVtx K ) = ( V \ { K } ) ) -> ( ( G e. UPGraph /\ N e. V ) /\ K e. ( V \ { N } ) ) )
19	18	anim1i 592	. . . . . . 7 |- ( ( ( ( ( G e. UPGraph /\ N e. V ) /\ K e. ( V \ { N } ) ) /\ ( G NeighbVtx K ) = ( V \ { K } ) ) /\ n e. ( V \ { N , K } ) ) -> ( ( ( G e. UPGraph /\ N e. V ) /\ K e. ( V \ { N } ) ) /\ n e. ( V \ { N , K } ) ) )
20		df-3an 1039	. . . . . . 7 |- ( ( ( G e. UPGraph /\ N e. V ) /\ K e. ( V \ { N } ) /\ n e. ( V \ { N , K } ) ) <-> ( ( ( G e. UPGraph /\ N e. V ) /\ K e. ( V \ { N } ) ) /\ n e. ( V \ { N , K } ) ) )
21	19, 20	sylibr 224	. . . . . 6 |- ( ( ( ( ( G e. UPGraph /\ N e. V ) /\ K e. ( V \ { N } ) ) /\ ( G NeighbVtx K ) = ( V \ { K } ) ) /\ n e. ( V \ { N , K } ) ) -> ( ( G e. UPGraph /\ N e. V ) /\ K e. ( V \ { N } ) /\ n e. ( V \ { N , K } ) ) )
22		dif32 3891	. . . . . . . . . . . . 13 |- ( ( V \ { N } ) \ { K } ) = ( ( V \ { K } ) \ { N } )
23	7, 22	eqtri 2644	. . . . . . . . . . . 12 |- ( V \ { N , K } ) = ( ( V \ { K } ) \ { N } )
24	23	eleq2i 2693	. . . . . . . . . . 11 |- ( n e. ( V \ { N , K } ) <-> n e. ( ( V \ { K } ) \ { N } ) )
25		eldifsn 4317	. . . . . . . . . . 11 |- ( n e. ( ( V \ { K } ) \ { N } ) <-> ( n e. ( V \ { K } ) /\ n =/= N ) )
26	24, 25	bitri 264	. . . . . . . . . 10 |- ( n e. ( V \ { N , K } ) <-> ( n e. ( V \ { K } ) /\ n =/= N ) )
27	26	simplbi 476	. . . . . . . . 9 |- ( n e. ( V \ { N , K } ) -> n e. ( V \ { K } ) )
28		eleq2 2690	. . . . . . . . 9 |- ( ( G NeighbVtx K ) = ( V \ { K } ) -> ( n e. ( G NeighbVtx K ) <-> n e. ( V \ { K } ) ) )
29	27, 28	syl5ibr 236	. . . . . . . 8 |- ( ( G NeighbVtx K ) = ( V \ { K } ) -> ( n e. ( V \ { N , K } ) -> n e. ( G NeighbVtx K ) ) )
30	29	adantl 482	. . . . . . 7 |- ( ( ( ( G e. UPGraph /\ N e. V ) /\ K e. ( V \ { N } ) ) /\ ( G NeighbVtx K ) = ( V \ { K } ) ) -> ( n e. ( V \ { N , K } ) -> n e. ( G NeighbVtx K ) ) )
31	30	imp 445	. . . . . 6 |- ( ( ( ( ( G e. UPGraph /\ N e. V ) /\ K e. ( V \ { N } ) ) /\ ( G NeighbVtx K ) = ( V \ { K } ) ) /\ n e. ( V \ { N , K } ) ) -> n e. ( G NeighbVtx K ) )
32	8, 9, 10, 1	nbupgrres 26266	. . . . . 6 |- ( ( ( G e. UPGraph /\ N e. V ) /\ K e. ( V \ { N } ) /\ n e. ( V \ { N , K } ) ) -> ( n e. ( G NeighbVtx K ) -> n e. ( S NeighbVtx K ) ) )
33	21, 31, 32	sylc 65	. . . . 5 |- ( ( ( ( ( G e. UPGraph /\ N e. V ) /\ K e. ( V \ { N } ) ) /\ ( G NeighbVtx K ) = ( V \ { K } ) ) /\ n e. ( V \ { N , K } ) ) -> n e. ( S NeighbVtx K ) )
34	33	ralrimiva 2966	. . . 4 |- ( ( ( ( G e. UPGraph /\ N e. V ) /\ K e. ( V \ { N } ) ) /\ ( G NeighbVtx K ) = ( V \ { K } ) ) -> A. n e. ( V \ { N , K } ) n e. ( S NeighbVtx K ) )
35		dfss3 3592	. . . 4 |- ( ( V \ { N , K } ) C_ ( S NeighbVtx K ) <-> A. n e. ( V \ { N , K } ) n e. ( S NeighbVtx K ) )
36	34, 35	sylibr 224	. . 3 |- ( ( ( ( G e. UPGraph /\ N e. V ) /\ K e. ( V \ { N } ) ) /\ ( G NeighbVtx K ) = ( V \ { K } ) ) -> ( V \ { N , K } ) C_ ( S NeighbVtx K ) )
37	17, 36	eqssd 3620	. 2 |- ( ( ( ( G e. UPGraph /\ N e. V ) /\ K e. ( V \ { N } ) ) /\ ( G NeighbVtx K ) = ( V \ { K } ) ) -> ( S NeighbVtx K ) = ( V \ { N , K } ) )
38	37	ex 450	1 |- ( ( ( G e. UPGraph /\ N e. V ) /\ K e. ( V \ { N } ) ) -> ( ( G NeighbVtx K ) = ( V \ { K } ) -> ( S NeighbVtx K ) = ( V \ { N , K } ) ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   e/ wnel 2897   A.wral 2912   {crab 2916   _Vcvv 3200   \ cdif 3571   C_ wss 3574   {csn 4177   {cpr 4179   <.cop 4183   _I cid 5023   |` cres 5116   ` cfv 5888  (class class class)co 6650  Vtxcvtx 25874  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-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-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:  uvtxupgrres  26309