Theorem upgrres 26198

Description: A subgraph obtained by removing one vertex and all edges incident with this vertex from a pseudograph (see uhgrspan1 26195) is a pseudograph. (Contributed by AV, 8-Nov-2020.) (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
upgrres	|- ( ( G e. UPGraph /\ N e. V ) -> S e. UPGraph )

Distinct variable groups:   i, E   i, N

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

Proof of Theorem upgrres

Dummy variable p is distinct from all other variables.

Step	Hyp	Ref	Expression
1		upgruhgr 25997	. . . . . 6 |- ( G e. UPGraph -> G e. UHGraph )
2		upgrres.e	. . . . . . 7 |- E = (iEdg ` G )
3	2	uhgrfun 25961	. . . . . 6 |- ( G e. UHGraph -> Fun E )
4		funres 5929	. . . . . 6 |- ( Fun E -> Fun ( E |` F ) )
5	1, 3, 4	3syl 18	. . . . 5 |- ( G e. UPGraph -> Fun ( E |` F ) )
6	5	funfnd 5919	. . . 4 |- ( G e. UPGraph -> ( E |` F ) Fn dom ( E |` F ) )
7	6	adantr 481	. . 3 |- ( ( G e. UPGraph /\ N e. V ) -> ( E |` F ) Fn dom ( E |` F ) )
8		upgrres.v	. . . 4 |- V = (Vtx ` G )
9		upgrres.f	. . . 4 |- F = { i e. dom E | N e/ ( E ` i ) }
10	8, 2, 9	upgrreslem 26196	. . 3 |- ( ( G e. UPGraph /\ N e. V ) -> ran ( E |` F ) C_ { p e. ( ~P ( V \ { N } ) \ { (/) } ) | ( # ` p ) <_ 2 } )
11		df-f 5892	. . 3 |- ( ( E |` F ) : dom ( E |` F ) --> { p e. ( ~P ( V \ { N } ) \ { (/) } ) | ( # ` p ) <_ 2 } <-> ( ( E |` F ) Fn dom ( E |` F ) /\ ran ( E |` F ) C_ { p e. ( ~P ( V \ { N } ) \ { (/) } ) | ( # ` p ) <_ 2 } ) )
12	7, 10, 11	sylanbrc 698	. 2 |- ( ( G e. UPGraph /\ N e. V ) -> ( E |` F ) : dom ( E |` F ) --> { p e. ( ~P ( V \ { N } ) \ { (/) } ) | ( # ` p ) <_ 2 } )
13		upgrres.s	. . . 4 |- S = <. ( V \ { N } ) , ( E |` F ) >.
14		opex 4932	. . . 4 |- <. ( V \ { N } ) , ( E |` F ) >. e. _V
15	13, 14	eqeltri 2697	. . 3 |- S e. _V
16	8, 2, 9, 13	uhgrspan1lem2 26193	. . . . 5 |- (Vtx ` S ) = ( V \ { N } )
17	16	eqcomi 2631	. . . 4 |- ( V \ { N } ) = (Vtx ` S )
18	8, 2, 9, 13	uhgrspan1lem3 26194	. . . . 5 |- (iEdg ` S ) = ( E |` F )
19	18	eqcomi 2631	. . . 4 |- ( E |` F ) = (iEdg ` S )
20	17, 19	isupgr 25979	. . 3 |- ( S e. _V -> ( S e. UPGraph <-> ( E |` F ) : dom ( E |` F ) --> { p e. ( ~P ( V \ { N } ) \ { (/) } ) | ( # ` p ) <_ 2 } ) )
21	15, 20	mp1i 13	. 2 |- ( ( G e. UPGraph /\ N e. V ) -> ( S e. UPGraph <-> ( E |` F ) : dom ( E |` F ) --> { p e. ( ~P ( V \ { N } ) \ { (/) } ) | ( # ` p ) <_ 2 } ) )
22	12, 21	mpbird 247	1 |- ( ( G e. UPGraph /\ N e. V ) -> S e. UPGraph )

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   class class class wbr 4653   dom cdm 5114   ran crn 5115   |` cres 5116   Fun wfun 5882   Fn wfn 5883   -->wf 5884   ` cfv 5888   <_ cle 10075   2c2 11070   #chash 13117  Vtxcvtx 25874  iEdgciedg 25875   UHGraph cuhgr 25951   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-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-fv 5896  df-1st 7168  df-2nd 7169  df-vtx 25876  df-iedg 25877  df-uhgr 25953  df-upgr 25977

This theorem is referenced by:  finsumvtxdg2size  26446