Theorem uhgrspan1 26195

Description: The induced subgraph S of a hypergraph G obtained by removing one vertex is actually a subgraph of G. A subgraph is called induced or spanned by a subset of vertices of a graph if it contains all edges of the original graph that join two vertices of the subgraph (see section I.1 in [Bollobas] p. 2 and section 1.1 in [Diestel] p. 4). (Contributed by AV, 19-Nov-2020.)

Hypotheses

Ref	Expression
uhgrspan1.v	|- V = (Vtx ` G )
uhgrspan1.i	|- I = (iEdg ` G )
uhgrspan1.f	|- F = { i e. dom I | N e/ ( I ` i ) }
uhgrspan1.s	|- S = <. ( V \ { N } ) , ( I |` F ) >.

Assertion

Ref	Expression
uhgrspan1	|- ( ( G e. UHGraph /\ N e. V ) -> S SubGraph G )

Distinct variable groups:   i, I   i, N

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

Proof of Theorem uhgrspan1

Dummy variables c j are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		difssd 3738	. 2 |- ( ( G e. UHGraph /\ N e. V ) -> ( V \ { N } ) C_ V )
2		uhgrspan1.v	. . . 4 |- V = (Vtx ` G )
3		uhgrspan1.i	. . . 4 |- I = (iEdg ` G )
4		uhgrspan1.f	. . . 4 |- F = { i e. dom I | N e/ ( I ` i ) }
5		uhgrspan1.s	. . . 4 |- S = <. ( V \ { N } ) , ( I |` F ) >.
6	2, 3, 4, 5	uhgrspan1lem3 26194	. . 3 |- (iEdg ` S ) = ( I |` F )
7		resresdm 5626	. . 3 |- ( (iEdg ` S ) = ( I |` F ) -> (iEdg ` S ) = ( I |` dom (iEdg ` S ) ) )
8	6, 7	mp1i 13	. 2 |- ( ( G e. UHGraph /\ N e. V ) -> (iEdg ` S ) = ( I |` dom (iEdg ` S ) ) )
9	3	uhgrfun 25961	. . . . . 6 |- ( G e. UHGraph -> Fun I )
10		fvelima 6248	. . . . . . 7 |- ( ( Fun I /\ c e. ( I " F ) ) -> E. j e. F ( I ` j ) = c )
11	10	ex 450	. . . . . 6 |- ( Fun I -> ( c e. ( I " F ) -> E. j e. F ( I ` j ) = c ) )
12	9, 11	syl 17	. . . . 5 |- ( G e. UHGraph -> ( c e. ( I " F ) -> E. j e. F ( I ` j ) = c ) )
13	12	adantr 481	. . . 4 |- ( ( G e. UHGraph /\ N e. V ) -> ( c e. ( I " F ) -> E. j e. F ( I ` j ) = c ) )
14		eqidd 2623	. . . . . . . 8 |- ( i = j -> N = N )
15		fveq2 6191	. . . . . . . 8 |- ( i = j -> ( I ` i ) = ( I ` j ) )
16	14, 15	neleq12d 2901	. . . . . . 7 |- ( i = j -> ( N e/ ( I ` i ) <-> N e/ ( I ` j ) ) )
17	16, 4	elrab2 3366	. . . . . 6 |- ( j e. F <-> ( j e. dom I /\ N e/ ( I ` j ) ) )
18		fvexd 6203	. . . . . . . . 9 |- ( ( ( G e. UHGraph /\ N e. V ) /\ ( j e. dom I /\ N e/ ( I ` j ) ) ) -> ( I ` j ) e. _V )
19	2, 3	uhgrss 25959	. . . . . . . . . 10 |- ( ( G e. UHGraph /\ j e. dom I ) -> ( I ` j ) C_ V )
20	19	ad2ant2r 783	. . . . . . . . 9 |- ( ( ( G e. UHGraph /\ N e. V ) /\ ( j e. dom I /\ N e/ ( I ` j ) ) ) -> ( I ` j ) C_ V )
21		simprr 796	. . . . . . . . 9 |- ( ( ( G e. UHGraph /\ N e. V ) /\ ( j e. dom I /\ N e/ ( I ` j ) ) ) -> N e/ ( I ` j ) )
22		elpwdifsn 4319	. . . . . . . . 9 |- ( ( ( I ` j ) e. _V /\ ( I ` j ) C_ V /\ N e/ ( I ` j ) ) -> ( I ` j ) e. ~P ( V \ { N } ) )
23	18, 20, 21, 22	syl3anc 1326	. . . . . . . 8 |- ( ( ( G e. UHGraph /\ N e. V ) /\ ( j e. dom I /\ N e/ ( I ` j ) ) ) -> ( I ` j ) e. ~P ( V \ { N } ) )
24		eleq1 2689	. . . . . . . . 9 |- ( c = ( I ` j ) -> ( c e. ~P ( V \ { N } ) <-> ( I ` j ) e. ~P ( V \ { N } ) ) )
25	24	eqcoms 2630	. . . . . . . 8 |- ( ( I ` j ) = c -> ( c e. ~P ( V \ { N } ) <-> ( I ` j ) e. ~P ( V \ { N } ) ) )
26	23, 25	syl5ibrcom 237	. . . . . . 7 |- ( ( ( G e. UHGraph /\ N e. V ) /\ ( j e. dom I /\ N e/ ( I ` j ) ) ) -> ( ( I ` j ) = c -> c e. ~P ( V \ { N } ) ) )
27	26	ex 450	. . . . . 6 |- ( ( G e. UHGraph /\ N e. V ) -> ( ( j e. dom I /\ N e/ ( I ` j ) ) -> ( ( I ` j ) = c -> c e. ~P ( V \ { N } ) ) ) )
28	17, 27	syl5bi 232	. . . . 5 |- ( ( G e. UHGraph /\ N e. V ) -> ( j e. F -> ( ( I ` j ) = c -> c e. ~P ( V \ { N } ) ) ) )
29	28	rexlimdv 3030	. . . 4 |- ( ( G e. UHGraph /\ N e. V ) -> ( E. j e. F ( I ` j ) = c -> c e. ~P ( V \ { N } ) ) )
30	13, 29	syld 47	. . 3 |- ( ( G e. UHGraph /\ N e. V ) -> ( c e. ( I " F ) -> c e. ~P ( V \ { N } ) ) )
31	30	ssrdv 3609	. 2 |- ( ( G e. UHGraph /\ N e. V ) -> ( I " F ) C_ ~P ( V \ { N } ) )
32		opex 4932	. . . . 5 |- <. ( V \ { N } ) , ( I |` F ) >. e. _V
33	5, 32	eqeltri 2697	. . . 4 |- S e. _V
34	33	a1i 11	. . 3 |- ( N e. V -> S e. _V )
35	2, 3, 4, 5	uhgrspan1lem2 26193	. . . . 5 |- (Vtx ` S ) = ( V \ { N } )
36	35	eqcomi 2631	. . . 4 |- ( V \ { N } ) = (Vtx ` S )
37		eqid 2622	. . . 4 |- (iEdg ` S ) = (iEdg ` S )
38	6	rneqi 5352	. . . . 5 |- ran (iEdg ` S ) = ran ( I |` F )
39		edgval 25941	. . . . 5 |- (Edg ` S ) = ran (iEdg ` S )
40		df-ima 5127	. . . . 5 |- ( I " F ) = ran ( I |` F )
41	38, 39, 40	3eqtr4ri 2655	. . . 4 |- ( I " F ) = (Edg ` S )
42	36, 2, 37, 3, 41	issubgr 26163	. . 3 |- ( ( G e. UHGraph /\ S e. _V ) -> ( S SubGraph G <-> ( ( V \ { N } ) C_ V /\ (iEdg ` S ) = ( I |` dom (iEdg ` S ) ) /\ ( I " F ) C_ ~P ( V \ { N } ) ) ) )
43	34, 42	sylan2 491	. 2 |- ( ( G e. UHGraph /\ N e. V ) -> ( S SubGraph G <-> ( ( V \ { N } ) C_ V /\ (iEdg ` S ) = ( I |` dom (iEdg ` S ) ) /\ ( I " F ) C_ ~P ( V \ { N } ) ) ) )
44	1, 8, 31, 43	mpbir3and 1245	1 |- ( ( G e. UHGraph /\ N e. V ) -> S SubGraph G )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   e/ wnel 2897   E.wrex 2913   {crab 2916   _Vcvv 3200   \ cdif 3571   C_ wss 3574   ~Pcpw 4158   {csn 4177   <.cop 4183   class class class wbr 4653   dom cdm 5114   ran crn 5115   |` cres 5116   "cima 5117   Fun wfun 5882   ` cfv 5888  Vtxcvtx 25874  iEdgciedg 25875  Edgcedg 25939   UHGraph cuhgr 25951   SubGraph csubgr 26159

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-fv 5896  df-1st 7168  df-2nd 7169  df-vtx 25876  df-iedg 25877  df-edg 25940  df-uhgr 25953  df-subgr 26160

This theorem is referenced by: (None)