Theorem incistruhgr 15934

Description: An incidence structure <. P , L , I >. "where P is a set whose elements are called points, L is a distinct set whose elements are called lines and I C_ ( P X. L ) is the incidence relation" (see Wikipedia "Incidence structure" (24-Oct-2020), https://en.wikipedia.org/wiki/Incidence_structure) implies an undirected hypergraph, if the incidence relation is right-total (to exclude empty edges). The points become the vertices, and the edge function is derived from the incidence relation by mapping each line ("edge") to the set of vertices incident to the line/edge. With P = ( Base ` S ) and by defining two new slots for lines and incidence relations and enhancing the definition of iEdg accordingly, it would even be possible to express that a corresponding incidence structure is an undirected hypergraph. By choosing the incident relation appropriately, other kinds of undirected graphs (pseudographs, multigraphs, simple graphs, etc.) could be defined. (Contributed by AV, 24-Oct-2020.)

Hypotheses

Ref	Expression
incistruhgr.v	|- V = (Vtx ` G )
incistruhgr.e	|- E = (iEdg ` G )

Assertion

Ref	Expression
incistruhgr	|- ( ( G e. W /\ I C_ ( P X. L ) /\ ran I = L ) -> ( ( V = P /\ E = ( e e. L |-> { v e. P | v I e } ) ) -> G e. UHGraph ) )

Distinct variable groups:   e, E   e, G   e, I, v   e, L, v   P, e, v   e, V, v   e, W

Allowed substitution hints:   E( v)   G( v)   W( v)

Proof of Theorem incistruhgr

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

Step	Hyp	Ref	Expression
1		rabeq 2792	. . . . . . . . 9 |- ( V = P -> { v e. V | v I e } = { v e. P | v I e } )
2	1	mpteq2dv 4178	. . . . . . . 8 |- ( V = P -> ( e e. L |-> { v e. V | v I e } ) = ( e e. L |-> { v e. P | v I e } ) )
3	2	eqeq2d 2241	. . . . . . 7 |- ( V = P -> ( E = ( e e. L |-> { v e. V | v I e } ) <-> E = ( e e. L |-> { v e. P | v I e } ) ) )
4		xpeq1 4737	. . . . . . . . 9 |- ( V = P -> ( V X. L ) = ( P X. L ) )
5	4	sseq2d 3255	. . . . . . . 8 |- ( V = P -> ( I C_ ( V X. L ) <-> I C_ ( P X. L ) ) )
6	5	3anbi2d 1351	. . . . . . 7 |- ( V = P -> ( ( G e. W /\ I C_ ( V X. L ) /\ ran I = L ) <-> ( G e. W /\ I C_ ( P X. L ) /\ ran I = L ) ) )
7	3, 6	anbi12d 473	. . . . . 6 |- ( V = P -> ( ( E = ( e e. L |-> { v e. V | v I e } ) /\ ( G e. W /\ I C_ ( V X. L ) /\ ran I = L ) ) <-> ( E = ( e e. L |-> { v e. P | v I e } ) /\ ( G e. W /\ I C_ ( P X. L ) /\ ran I = L ) ) ) )
8		simpl 109	. . . . . . . 8 |- ( ( E = ( e e. L |-> { v e. V | v I e } ) /\ ( G e. W /\ I C_ ( V X. L ) /\ ran I = L ) ) -> E = ( e e. L |-> { v e. V | v I e } ) )
9		dmeq 4929	. . . . . . . . 9 |- ( E = ( e e. L |-> { v e. V | v I e } ) -> dom E = dom ( e e. L |-> { v e. V | v I e } ) )
10		eqid 2229	. . . . . . . . . 10 |- ( e e. L |-> { v e. V | v I e } ) = ( e e. L |-> { v e. V | v I e } )
11		eqid 2229	. . . . . . . . . . 11 |- { v e. V | v I e } = { v e. V | v I e }
12		incistruhgr.v	. . . . . . . . . . . 12 |- V = (Vtx ` G )
13		simpl1 1024	. . . . . . . . . . . . 13 |- ( ( ( G e. W /\ I C_ ( V X. L ) /\ ran I = L ) /\ e e. L ) -> G e. W )
14		vtxex 15862	. . . . . . . . . . . . 13 |- ( G e. W -> (Vtx ` G ) e. _V )
15	13, 14	syl 14	. . . . . . . . . . . 12 |- ( ( ( G e. W /\ I C_ ( V X. L ) /\ ran I = L ) /\ e e. L ) -> (Vtx ` G ) e. _V )
16	12, 15	eqeltrid 2316	. . . . . . . . . . 11 |- ( ( ( G e. W /\ I C_ ( V X. L ) /\ ran I = L ) /\ e e. L ) -> V e. _V )
17	11, 16	rabexd 4233	. . . . . . . . . 10 |- ( ( ( G e. W /\ I C_ ( V X. L ) /\ ran I = L ) /\ e e. L ) -> { v e. V | v I e } e. _V )
18	10, 17	dmmptd 5460	. . . . . . . . 9 |- ( ( G e. W /\ I C_ ( V X. L ) /\ ran I = L ) -> dom ( e e. L |-> { v e. V | v I e } ) = L )
19	9, 18	sylan9eq 2282	. . . . . . . 8 |- ( ( E = ( e e. L |-> { v e. V | v I e } ) /\ ( G e. W /\ I C_ ( V X. L ) /\ ran I = L ) ) -> dom E = L )
20	8, 19	jca 306	. . . . . . 7 |- ( ( E = ( e e. L |-> { v e. V | v I e } ) /\ ( G e. W /\ I C_ ( V X. L ) /\ ran I = L ) ) -> ( E = ( e e. L |-> { v e. V | v I e } ) /\ dom E = L ) )
21		simpr 110	. . . . . . 7 |- ( ( E = ( e e. L |-> { v e. V | v I e } ) /\ ( G e. W /\ I C_ ( V X. L ) /\ ran I = L ) ) -> ( G e. W /\ I C_ ( V X. L ) /\ ran I = L ) )
22		eleq2 2293	. . . . . . . . . . 11 |- ( s = { v e. V | v I e } -> ( j e. s <-> j e. { v e. V | v I e } ) )
23	22	exbidv 1871	. . . . . . . . . 10 |- ( s = { v e. V | v I e } -> ( E. j j e. s <-> E. j j e. { v e. V | v I e } ) )
24		ssrab2 3310	. . . . . . . . . . 11 |- { v e. V | v I e } C_ V
25		elpwg 3658	. . . . . . . . . . . 12 |- ( { v e. V | v I e } e. _V -> ( { v e. V | v I e } e. ~P V <-> { v e. V | v I e } C_ V ) )
26	17, 25	syl 14	. . . . . . . . . . 11 |- ( ( ( G e. W /\ I C_ ( V X. L ) /\ ran I = L ) /\ e e. L ) -> ( { v e. V | v I e } e. ~P V <-> { v e. V | v I e } C_ V ) )
27	24, 26	mpbiri 168	. . . . . . . . . 10 |- ( ( ( G e. W /\ I C_ ( V X. L ) /\ ran I = L ) /\ e e. L ) -> { v e. V | v I e } e. ~P V )
28		eleq2 2293	. . . . . . . . . . . . . 14 |- ( ran I = L -> ( e e. ran I <-> e e. L ) )
29	28	3ad2ant3 1044	. . . . . . . . . . . . 13 |- ( ( G e. W /\ I C_ ( V X. L ) /\ ran I = L ) -> ( e e. ran I <-> e e. L ) )
30		ssrelrn 4920	. . . . . . . . . . . . . . 15 |- ( ( I C_ ( V X. L ) /\ e e. ran I ) -> E. v e. V v I e )
31	30	ex 115	. . . . . . . . . . . . . 14 |- ( I C_ ( V X. L ) -> ( e e. ran I -> E. v e. V v I e ) )
32	31	3ad2ant2 1043	. . . . . . . . . . . . 13 |- ( ( G e. W /\ I C_ ( V X. L ) /\ ran I = L ) -> ( e e. ran I -> E. v e. V v I e ) )
33	29, 32	sylbird 170	. . . . . . . . . . . 12 |- ( ( G e. W /\ I C_ ( V X. L ) /\ ran I = L ) -> ( e e. L -> E. v e. V v I e ) )
34	33	imp 124	. . . . . . . . . . 11 |- ( ( ( G e. W /\ I C_ ( V X. L ) /\ ran I = L ) /\ e e. L ) -> E. v e. V v I e )
35		rabn0m 3520	. . . . . . . . . . 11 |- ( E. j j e. { v e. V | v I e } <-> E. v e. V v I e )
36	34, 35	sylibr 134	. . . . . . . . . 10 |- ( ( ( G e. W /\ I C_ ( V X. L ) /\ ran I = L ) /\ e e. L ) -> E. j j e. { v e. V | v I e } )
37	23, 27, 36	elrabd 2962	. . . . . . . . 9 |- ( ( ( G e. W /\ I C_ ( V X. L ) /\ ran I = L ) /\ e e. L ) -> { v e. V | v I e } e. { s e. ~P V | E. j j e. s } )
38	37	fmpttd 5798	. . . . . . . 8 |- ( ( G e. W /\ I C_ ( V X. L ) /\ ran I = L ) -> ( e e. L |-> { v e. V | v I e } ) : L --> { s e. ~P V | E. j j e. s } )
39		simpl 109	. . . . . . . . 9 |- ( ( E = ( e e. L |-> { v e. V | v I e } ) /\ dom E = L ) -> E = ( e e. L |-> { v e. V | v I e } ) )
40		simpr 110	. . . . . . . . 9 |- ( ( E = ( e e. L |-> { v e. V | v I e } ) /\ dom E = L ) -> dom E = L )
41	39, 40	feq12d 5469	. . . . . . . 8 |- ( ( E = ( e e. L |-> { v e. V | v I e } ) /\ dom E = L ) -> ( E : dom E --> { s e. ~P V | E. j j e. s } <-> ( e e. L |-> { v e. V | v I e } ) : L --> { s e. ~P V | E. j j e. s } ) )
42	38, 41	imbitrrid 156	. . . . . . 7 |- ( ( E = ( e e. L |-> { v e. V | v I e } ) /\ dom E = L ) -> ( ( G e. W /\ I C_ ( V X. L ) /\ ran I = L ) -> E : dom E --> { s e. ~P V | E. j j e. s } ) )
43	20, 21, 42	sylc 62	. . . . . 6 |- ( ( E = ( e e. L |-> { v e. V | v I e } ) /\ ( G e. W /\ I C_ ( V X. L ) /\ ran I = L ) ) -> E : dom E --> { s e. ~P V | E. j j e. s } )
44	7, 43	biimtrrdi 164	. . . . 5 |- ( V = P -> ( ( E = ( e e. L |-> { v e. P | v I e } ) /\ ( G e. W /\ I C_ ( P X. L ) /\ ran I = L ) ) -> E : dom E --> { s e. ~P V | E. j j e. s } ) )
45	44	expdimp 259	. . . 4 |- ( ( V = P /\ E = ( e e. L |-> { v e. P | v I e } ) ) -> ( ( G e. W /\ I C_ ( P X. L ) /\ ran I = L ) -> E : dom E --> { s e. ~P V | E. j j e. s } ) )
46	45	impcom 125	. . 3 |- ( ( ( G e. W /\ I C_ ( P X. L ) /\ ran I = L ) /\ ( V = P /\ E = ( e e. L |-> { v e. P | v I e } ) ) ) -> E : dom E --> { s e. ~P V | E. j j e. s } )
47		incistruhgr.e	. . . . . 6 |- E = (iEdg ` G )
48	12, 47	isuhgrm 15915	. . . . 5 |- ( G e. W -> ( G e. UHGraph <-> E : dom E --> { s e. ~P V | E. j j e. s } ) )
49	48	3ad2ant1 1042	. . . 4 |- ( ( G e. W /\ I C_ ( P X. L ) /\ ran I = L ) -> ( G e. UHGraph <-> E : dom E --> { s e. ~P V | E. j j e. s } ) )
50	49	adantr 276	. . 3 |- ( ( ( G e. W /\ I C_ ( P X. L ) /\ ran I = L ) /\ ( V = P /\ E = ( e e. L |-> { v e. P | v I e } ) ) ) -> ( G e. UHGraph <-> E : dom E --> { s e. ~P V | E. j j e. s } ) )
51	46, 50	mpbird 167	. 2 |- ( ( ( G e. W /\ I C_ ( P X. L ) /\ ran I = L ) /\ ( V = P /\ E = ( e e. L |-> { v e. P | v I e } ) ) ) -> G e. UHGraph )
52	51	ex 115	1 |- ( ( G e. W /\ I C_ ( P X. L ) /\ ran I = L ) -> ( ( V = P /\ E = ( e e. L |-> { v e. P | v I e } ) ) -> G e. UHGraph ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1002   = wceq 1395   E.wex 1538   e. wcel 2200   E.wrex 2509   {crab 2512   _Vcvv 2800   C_ wss 3198   ~Pcpw 3650   class class class wbr 4086   |-> cmpt 4148   X. cxp 4721   dom cdm 4723   ran crn 4724   -->wf 5320   ` cfv 5324  Vtxcvtx 15856  iEdgciedg 15857  UHGraphcuhgr 15911

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-cnex 8116  ax-resscn 8117  ax-1cn 8118  ax-1re 8119  ax-icn 8120  ax-addcl 8121  ax-addrcl 8122  ax-mulcl 8123  ax-addcom 8125  ax-mulcom 8126  ax-addass 8127  ax-mulass 8128  ax-distr 8129  ax-i2m1 8130  ax-1rid 8132  ax-0id 8133  ax-rnegex 8134  ax-cnre 8136

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-if 3604  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-br 4087  df-opab 4149  df-mpt 4150  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-fo 5330  df-fv 5332  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1st 6298  df-2nd 6299  df-sub 8345  df-inn 9137  df-2 9195  df-3 9196  df-4 9197  df-5 9198  df-6 9199  df-7 9200  df-8 9201  df-9 9202  df-n0 9396  df-dec 9605  df-ndx 13078  df-slot 13079  df-base 13081  df-edgf 15849  df-vtx 15858  df-iedg 15859  df-uhgrm 15913

This theorem is referenced by: (None)