Theorem usgrres 26200

Description: A subgraph obtained by removing one vertex and all edges incident with this vertex from a simple graph (see uhgrspan1 26195) is a simple graph. (Contributed by Alexander van der Vekens, 2-Jan-2018.) (Revised by AV, 19-Dec-2021.)

Hypotheses

Ref	Expression
upgrres.v	|- V = (Vtx ` G )
upgrres.e	|- E = (iEdg ` G )
upgrres.f	|- F = { i e. dom E | N e/ ( E ` i ) }
upgrres.s	|- S = <. ( V \ { N } ) , ( E |` F ) >.

Assertion

Ref	Expression
usgrres	|- ( ( G e. USGraph /\ N e. V ) -> S e. USGraph )

Distinct variable groups:   i, E   i, N

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

Proof of Theorem usgrres

Dummy variables p x are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		upgrres.v	. . . . . 6 |- V = (Vtx ` G )
2		upgrres.e	. . . . . 6 |- E = (iEdg ` G )
3	1, 2	usgrf 26050	. . . . 5 |- ( G e. USGraph -> E : dom E -1-1-> { x e. ( ~P V \ { (/) } ) | ( # ` x ) = 2 } )
4		upgrres.f	. . . . . . 7 |- F = { i e. dom E | N e/ ( E ` i ) }
5	4	ssrab3 3688	. . . . . 6 |- F C_ dom E
6	5	a1i 11	. . . . 5 |- ( ( G e. USGraph /\ N e. V ) -> F C_ dom E )
7		f1ssres 6108	. . . . 5 |- ( ( E : dom E -1-1-> { x e. ( ~P V \ { (/) } ) | ( # ` x ) = 2 } /\ F C_ dom E ) -> ( E |` F ) : F -1-1-> { x e. ( ~P V \ { (/) } ) | ( # ` x ) = 2 } )
8	3, 6, 7	syl2an2r 876	. . . 4 |- ( ( G e. USGraph /\ N e. V ) -> ( E |` F ) : F -1-1-> { x e. ( ~P V \ { (/) } ) | ( # ` x ) = 2 } )
9		usgrumgr 26074	. . . . 5 |- ( G e. USGraph -> G e. UMGraph )
10	1, 2, 4	umgrreslem 26197	. . . . 5 |- ( ( G e. UMGraph /\ N e. V ) -> ran ( E |` F ) C_ { p e. ~P ( V \ { N } ) | ( # ` p ) = 2 } )
11	9, 10	sylan 488	. . . 4 |- ( ( G e. USGraph /\ N e. V ) -> ran ( E |` F ) C_ { p e. ~P ( V \ { N } ) | ( # ` p ) = 2 } )
12		f1ssr 6107	. . . 4 |- ( ( ( E |` F ) : F -1-1-> { x e. ( ~P V \ { (/) } ) | ( # ` x ) = 2 } /\ ran ( E |` F ) C_ { p e. ~P ( V \ { N } ) | ( # ` p ) = 2 } ) -> ( E |` F ) : F -1-1-> { p e. ~P ( V \ { N } ) | ( # ` p ) = 2 } )
13	8, 11, 12	syl2anc 693	. . 3 |- ( ( G e. USGraph /\ N e. V ) -> ( E |` F ) : F -1-1-> { p e. ~P ( V \ { N } ) | ( # ` p ) = 2 } )
14		ssdmres 5420	. . . . 5 |- ( F C_ dom E <-> dom ( E |` F ) = F )
15	5, 14	mpbi 220	. . . 4 |- dom ( E |` F ) = F
16		f1eq2 6097	. . . 4 |- ( dom ( E |` F ) = F -> ( ( E |` F ) : dom ( E |` F ) -1-1-> { p e. ~P ( V \ { N } ) | ( # ` p ) = 2 } <-> ( E |` F ) : F -1-1-> { p e. ~P ( V \ { N } ) | ( # ` p ) = 2 } ) )
17	15, 16	ax-mp 5	. . 3 |- ( ( E |` F ) : dom ( E |` F ) -1-1-> { p e. ~P ( V \ { N } ) | ( # ` p ) = 2 } <-> ( E |` F ) : F -1-1-> { p e. ~P ( V \ { N } ) | ( # ` p ) = 2 } )
18	13, 17	sylibr 224	. 2 |- ( ( G e. USGraph /\ N e. V ) -> ( E |` F ) : dom ( E |` F ) -1-1-> { p e. ~P ( V \ { N } ) | ( # ` p ) = 2 } )
19		upgrres.s	. . . 4 |- S = <. ( V \ { N } ) , ( E |` F ) >.
20		opex 4932	. . . 4 |- <. ( V \ { N } ) , ( E |` F ) >. e. _V
21	19, 20	eqeltri 2697	. . 3 |- S e. _V
22	1, 2, 4, 19	uhgrspan1lem2 26193	. . . . 5 |- (Vtx ` S ) = ( V \ { N } )
23	22	eqcomi 2631	. . . 4 |- ( V \ { N } ) = (Vtx ` S )
24	1, 2, 4, 19	uhgrspan1lem3 26194	. . . . 5 |- (iEdg ` S ) = ( E |` F )
25	24	eqcomi 2631	. . . 4 |- ( E |` F ) = (iEdg ` S )
26	23, 25	isusgrs 26051	. . 3 |- ( S e. _V -> ( S e. USGraph <-> ( E |` F ) : dom ( E |` F ) -1-1-> { p e. ~P ( V \ { N } ) | ( # ` p ) = 2 } ) )
27	21, 26	mp1i 13	. 2 |- ( ( G e. USGraph /\ N e. V ) -> ( S e. USGraph <-> ( E |` F ) : dom ( E |` F ) -1-1-> { p e. ~P ( V \ { N } ) | ( # ` p ) = 2 } ) )
28	18, 27	mpbird 247	1 |- ( ( G e. USGraph /\ N e. V ) -> S e. USGraph )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   e/ wnel 2897   {crab 2916   _Vcvv 3200   \ cdif 3571   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   {csn 4177   <.cop 4183   dom cdm 5114   ran crn 5115   |` cres 5116   -1-1->wf1 5885   ` cfv 5888   2c2 11070   #chash 13117  Vtxcvtx 25874  iEdgciedg 25875   UMGraph cumgr 25976   USGraph cusgr 26044

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  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-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-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  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-z 11378  df-uz 11688  df-fz 12327  df-hash 13118  df-vtx 25876  df-iedg 25877  df-uhgr 25953  df-upgr 25977  df-umgr 25978  df-usgr 26046

This theorem is referenced by: (None)