Theorem upgrres1 26205

Description: A pseudograph obtained by removing one vertex and all edges incident with this vertex is a pseudograph. Remark: This graph is not a subgraph of the original graph in the sense of df-subgr 26160 since the domains of the edge functions may not be compatible. (Contributed by AV, 8-Nov-2020.)

Hypotheses

Ref	Expression
upgrres1.v	|- V = (Vtx ` G )
upgrres1.e	|- E = (Edg ` G )
upgrres1.f	|- F = { e e. E | N e/ e }
upgrres1.s	|- S = <. ( V \ { N } ) , ( _I |` F ) >.

Assertion

Ref	Expression
upgrres1	|- ( ( G e. UPGraph /\ N e. V ) -> S e. UPGraph )

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

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

Proof of Theorem upgrres1

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

Step	Hyp	Ref	Expression
1		f1oi 6174	. . . . 5 |- ( _I |` F ) : F -1-1-onto-> F
2		f1of 6137	. . . . 5 |- ( ( _I |` F ) : F -1-1-onto-> F -> ( _I |` F ) : F --> F )
3	1, 2	mp1i 13	. . . 4 |- ( ( G e. UPGraph /\ N e. V ) -> ( _I |` F ) : F --> F )
4	3	ffdmd 6063	. . 3 |- ( ( G e. UPGraph /\ N e. V ) -> ( _I |` F ) : dom ( _I |` F ) --> F )
5		upgrres1.f	. . . . 5 |- F = { e e. E | N e/ e }
6		simpr 477	. . . . . . . . . . 11 |- ( ( ( G e. UPGraph /\ N e. V ) /\ e e. E ) -> e e. E )
7	6	adantr 481	. . . . . . . . . 10 |- ( ( ( ( G e. UPGraph /\ N e. V ) /\ e e. E ) /\ N e/ e ) -> e e. E )
8		upgrres1.e	. . . . . . . . . . . . 13 |- E = (Edg ` G )
9	8	eleq2i 2693	. . . . . . . . . . . 12 |- ( e e. E <-> e e. (Edg ` G ) )
10		edgupgr 26029	. . . . . . . . . . . . 13 |- ( ( G e. UPGraph /\ e e. (Edg ` G ) ) -> ( e e. ~P (Vtx ` G ) /\ e =/= (/) /\ ( # ` e ) <_ 2 ) )
11		elpwi 4168	. . . . . . . . . . . . . . 15 |- ( e e. ~P (Vtx ` G ) -> e C_ (Vtx ` G ) )
12		upgrres1.v	. . . . . . . . . . . . . . 15 |- V = (Vtx ` G )
13	11, 12	syl6sseqr 3652	. . . . . . . . . . . . . 14 |- ( e e. ~P (Vtx ` G ) -> e C_ V )
14	13	3ad2ant1 1082	. . . . . . . . . . . . 13 |- ( ( e e. ~P (Vtx ` G ) /\ e =/= (/) /\ ( # ` e ) <_ 2 ) -> e C_ V )
15	10, 14	syl 17	. . . . . . . . . . . 12 |- ( ( G e. UPGraph /\ e e. (Edg ` G ) ) -> e C_ V )
16	9, 15	sylan2b 492	. . . . . . . . . . 11 |- ( ( G e. UPGraph /\ e e. E ) -> e C_ V )
17	16	ad4ant13 1292	. . . . . . . . . 10 |- ( ( ( ( G e. UPGraph /\ N e. V ) /\ e e. E ) /\ N e/ e ) -> e C_ V )
18		simpr 477	. . . . . . . . . 10 |- ( ( ( ( G e. UPGraph /\ N e. V ) /\ e e. E ) /\ N e/ e ) -> N e/ e )
19		elpwdifsn 4319	. . . . . . . . . 10 |- ( ( e e. E /\ e C_ V /\ N e/ e ) -> e e. ~P ( V \ { N } ) )
20	7, 17, 18, 19	syl3anc 1326	. . . . . . . . 9 |- ( ( ( ( G e. UPGraph /\ N e. V ) /\ e e. E ) /\ N e/ e ) -> e e. ~P ( V \ { N } ) )
21		simpl 473	. . . . . . . . . . . 12 |- ( ( G e. UPGraph /\ N e. V ) -> G e. UPGraph )
22	9	biimpi 206	. . . . . . . . . . . 12 |- ( e e. E -> e e. (Edg ` G ) )
23	10	simp2d 1074	. . . . . . . . . . . 12 |- ( ( G e. UPGraph /\ e e. (Edg ` G ) ) -> e =/= (/) )
24	21, 22, 23	syl2an 494	. . . . . . . . . . 11 |- ( ( ( G e. UPGraph /\ N e. V ) /\ e e. E ) -> e =/= (/) )
25	24	adantr 481	. . . . . . . . . 10 |- ( ( ( ( G e. UPGraph /\ N e. V ) /\ e e. E ) /\ N e/ e ) -> e =/= (/) )
26		nelsn 4212	. . . . . . . . . 10 |- ( e =/= (/) -> -. e e. { (/) } )
27	25, 26	syl 17	. . . . . . . . 9 |- ( ( ( ( G e. UPGraph /\ N e. V ) /\ e e. E ) /\ N e/ e ) -> -. e e. { (/) } )
28	20, 27	eldifd 3585	. . . . . . . 8 |- ( ( ( ( G e. UPGraph /\ N e. V ) /\ e e. E ) /\ N e/ e ) -> e e. ( ~P ( V \ { N } ) \ { (/) } ) )
29	28	ex 450	. . . . . . 7 |- ( ( ( G e. UPGraph /\ N e. V ) /\ e e. E ) -> ( N e/ e -> e e. ( ~P ( V \ { N } ) \ { (/) } ) ) )
30	29	ralrimiva 2966	. . . . . 6 |- ( ( G e. UPGraph /\ N e. V ) -> A. e e. E ( N e/ e -> e e. ( ~P ( V \ { N } ) \ { (/) } ) ) )
31		rabss 3679	. . . . . 6 |- ( { e e. E | N e/ e } C_ ( ~P ( V \ { N } ) \ { (/) } ) <-> A. e e. E ( N e/ e -> e e. ( ~P ( V \ { N } ) \ { (/) } ) ) )
32	30, 31	sylibr 224	. . . . 5 |- ( ( G e. UPGraph /\ N e. V ) -> { e e. E | N e/ e } C_ ( ~P ( V \ { N } ) \ { (/) } ) )
33	5, 32	syl5eqss 3649	. . . 4 |- ( ( G e. UPGraph /\ N e. V ) -> F C_ ( ~P ( V \ { N } ) \ { (/) } ) )
34		elrabi 3359	. . . . . . 7 |- ( p e. { e e. E | N e/ e } -> p e. E )
35		edgval 25941	. . . . . . . . . . . 12 |- (Edg ` G ) = ran (iEdg ` G )
36	8, 35	eqtri 2644	. . . . . . . . . . 11 |- E = ran (iEdg ` G )
37	36	eleq2i 2693	. . . . . . . . . 10 |- ( p e. E <-> p e. ran (iEdg ` G ) )
38		eqid 2622	. . . . . . . . . . . . 13 |- (iEdg ` G ) = (iEdg ` G )
39	12, 38	upgrf 25981	. . . . . . . . . . . 12 |- ( G e. UPGraph -> (iEdg ` G ) : dom (iEdg ` G ) --> { x e. ( ~P V \ { (/) } ) | ( # ` x ) <_ 2 } )
40		frn 6053	. . . . . . . . . . . 12 |- ( (iEdg ` G ) : dom (iEdg ` G ) --> { x e. ( ~P V \ { (/) } ) | ( # ` x ) <_ 2 } -> ran (iEdg ` G ) C_ { x e. ( ~P V \ { (/) } ) | ( # ` x ) <_ 2 } )
41	39, 40	syl 17	. . . . . . . . . . 11 |- ( G e. UPGraph -> ran (iEdg ` G ) C_ { x e. ( ~P V \ { (/) } ) | ( # ` x ) <_ 2 } )
42	41	sseld 3602	. . . . . . . . . 10 |- ( G e. UPGraph -> ( p e. ran (iEdg ` G ) -> p e. { x e. ( ~P V \ { (/) } ) | ( # ` x ) <_ 2 } ) )
43	37, 42	syl5bi 232	. . . . . . . . 9 |- ( G e. UPGraph -> ( p e. E -> p e. { x e. ( ~P V \ { (/) } ) | ( # ` x ) <_ 2 } ) )
44		fveq2 6191	. . . . . . . . . . . 12 |- ( x = p -> ( # ` x ) = ( # ` p ) )
45	44	breq1d 4663	. . . . . . . . . . 11 |- ( x = p -> ( ( # ` x ) <_ 2 <-> ( # ` p ) <_ 2 ) )
46	45	elrab 3363	. . . . . . . . . 10 |- ( p e. { x e. ( ~P V \ { (/) } ) | ( # ` x ) <_ 2 } <-> ( p e. ( ~P V \ { (/) } ) /\ ( # ` p ) <_ 2 ) )
47	46	simprbi 480	. . . . . . . . 9 |- ( p e. { x e. ( ~P V \ { (/) } ) | ( # ` x ) <_ 2 } -> ( # ` p ) <_ 2 )
48	43, 47	syl6 35	. . . . . . . 8 |- ( G e. UPGraph -> ( p e. E -> ( # ` p ) <_ 2 ) )
49	48	adantr 481	. . . . . . 7 |- ( ( G e. UPGraph /\ N e. V ) -> ( p e. E -> ( # ` p ) <_ 2 ) )
50	34, 49	syl5com 31	. . . . . 6 |- ( p e. { e e. E | N e/ e } -> ( ( G e. UPGraph /\ N e. V ) -> ( # ` p ) <_ 2 ) )
51	50, 5	eleq2s 2719	. . . . 5 |- ( p e. F -> ( ( G e. UPGraph /\ N e. V ) -> ( # ` p ) <_ 2 ) )
52	51	impcom 446	. . . 4 |- ( ( ( G e. UPGraph /\ N e. V ) /\ p e. F ) -> ( # ` p ) <_ 2 )
53	33, 52	ssrabdv 3681	. . 3 |- ( ( G e. UPGraph /\ N e. V ) -> F C_ { p e. ( ~P ( V \ { N } ) \ { (/) } ) | ( # ` p ) <_ 2 } )
54	4, 53	fssd 6057	. 2 |- ( ( G e. UPGraph /\ N e. V ) -> ( _I |` F ) : dom ( _I |` F ) --> { p e. ( ~P ( V \ { N } ) \ { (/) } ) | ( # ` p ) <_ 2 } )
55		upgrres1.s	. . . 4 |- S = <. ( V \ { N } ) , ( _I |` F ) >.
56		opex 4932	. . . 4 |- <. ( V \ { N } ) , ( _I |` F ) >. e. _V
57	55, 56	eqeltri 2697	. . 3 |- S e. _V
58	12, 8, 5, 55	upgrres1lem2 26203	. . . . 5 |- (Vtx ` S ) = ( V \ { N } )
59	58	eqcomi 2631	. . . 4 |- ( V \ { N } ) = (Vtx ` S )
60	12, 8, 5, 55	upgrres1lem3 26204	. . . . 5 |- (iEdg ` S ) = ( _I |` F )
61	60	eqcomi 2631	. . . 4 |- ( _I |` F ) = (iEdg ` S )
62	59, 61	isupgr 25979	. . 3 |- ( S e. _V -> ( S e. UPGraph <-> ( _I |` F ) : dom ( _I |` F ) --> { p e. ( ~P ( V \ { N } ) \ { (/) } ) | ( # ` p ) <_ 2 } ) )
63	57, 62	mp1i 13	. 2 |- ( ( G e. UPGraph /\ N e. V ) -> ( S e. UPGraph <-> ( _I |` F ) : dom ( _I |` F ) --> { p e. ( ~P ( V \ { N } ) \ { (/) } ) | ( # ` p ) <_ 2 } ) )
64	54, 63	mpbird 247	1 |- ( ( G e. UPGraph /\ N e. V ) -> S e. UPGraph )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ 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   (/)c0 3915   ~Pcpw 4158   {csn 4177   <.cop 4183   class class class wbr 4653   _I cid 5023   dom cdm 5114   ran crn 5115   |` cres 5116   -->wf 5884   -1-1-onto->wf1o 5887   ` cfv 5888   <_ cle 10075   2c2 11070   #chash 13117  Vtxcvtx 25874  iEdgciedg 25875  Edgcedg 25939   UPGraph cupgr 25975

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-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-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-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-1st 7168  df-2nd 7169  df-vtx 25876  df-iedg 25877  df-edg 25940  df-upgr 25977

This theorem is referenced by:  nbupgrres  26266