Theorem incistruhgr 25974

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 (analogous to LineG and Itv) 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

Step	Hyp	Ref	Expression
1		id 22	. . . . . . . . . 10 |- ( V = P -> V = P )
2	1	rabeqdv 3194	. . . . . . . . 9 |- ( V = P -> { v e. V | v I e } = { v e. P | v I e } )
3	2	mpteq2dv 4745	. . . . . . . 8 |- ( V = P -> ( e e. L |-> { v e. V | v I e } ) = ( e e. L |-> { v e. P | v I e } ) )
4	3	eqeq2d 2632	. . . . . . 7 |- ( V = P -> ( E = ( e e. L |-> { v e. V | v I e } ) <-> E = ( e e. L |-> { v e. P | v I e } ) ) )
5		xpeq1 5128	. . . . . . . . 9 |- ( V = P -> ( V X. L ) = ( P X. L ) )
6	5	sseq2d 3633	. . . . . . . 8 |- ( V = P -> ( I C_ ( V X. L ) <-> I C_ ( P X. L ) ) )
7	6	3anbi2d 1404	. . . . . . 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 ) ) )
8	4, 7	anbi12d 747	. . . . . 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 ) ) ) )
9		dmeq 5324	. . . . . . . . 9 |- ( E = ( e e. L |-> { v e. V | v I e } ) -> dom E = dom ( e e. L |-> { v e. V | v I e } ) )
10		incistruhgr.v	. . . . . . . . . . . . 13 |- V = (Vtx ` G )
11		fvex 6201	. . . . . . . . . . . . 13 |- (Vtx ` G ) e. _V
12	10, 11	eqeltri 2697	. . . . . . . . . . . 12 |- V e. _V
13	12	rabex 4813	. . . . . . . . . . 11 |- { v e. V | v I e } e. _V
14		eqid 2622	. . . . . . . . . . 11 |- ( e e. L |-> { v e. V | v I e } ) = ( e e. L |-> { v e. V | v I e } )
15	13, 14	dmmpti 6023	. . . . . . . . . 10 |- dom ( e e. L |-> { v e. V | v I e } ) = L
16	15	a1i 11	. . . . . . . . 9 |- ( E = ( e e. L |-> { v e. V | v I e } ) -> dom ( e e. L |-> { v e. V | v I e } ) = L )
17	9, 16	eqtrd 2656	. . . . . . . 8 |- ( E = ( e e. L |-> { v e. V | v I e } ) -> dom E = L )
18		ssrab2 3687	. . . . . . . . . . . . 13 |- { v e. V | v I e } C_ V
19	18	a1i 11	. . . . . . . . . . . 12 |- ( ( ( G e. W /\ I C_ ( V X. L ) /\ ran I = L ) /\ e e. L ) -> { v e. V | v I e } C_ V )
20	13	elpw 4164	. . . . . . . . . . . 12 |- ( { v e. V | v I e } e. ~P V <-> { v e. V | v I e } C_ V )
21	19, 20	sylibr 224	. . . . . . . . . . 11 |- ( ( ( G e. W /\ I C_ ( V X. L ) /\ ran I = L ) /\ e e. L ) -> { v e. V | v I e } e. ~P V )
22		eleq2 2690	. . . . . . . . . . . . . . . 16 |- ( ran I = L -> ( e e. ran I <-> e e. L ) )
23	22	3ad2ant3 1084	. . . . . . . . . . . . . . 15 |- ( ( G e. W /\ I C_ ( V X. L ) /\ ran I = L ) -> ( e e. ran I <-> e e. L ) )
24		ssrelrn 5315	. . . . . . . . . . . . . . . . 17 |- ( ( I C_ ( V X. L ) /\ e e. ran I ) -> E. v e. V v I e )
25	24	ex 450	. . . . . . . . . . . . . . . 16 |- ( I C_ ( V X. L ) -> ( e e. ran I -> E. v e. V v I e ) )
26	25	3ad2ant2 1083	. . . . . . . . . . . . . . 15 |- ( ( G e. W /\ I C_ ( V X. L ) /\ ran I = L ) -> ( e e. ran I -> E. v e. V v I e ) )
27	23, 26	sylbird 250	. . . . . . . . . . . . . 14 |- ( ( G e. W /\ I C_ ( V X. L ) /\ ran I = L ) -> ( e e. L -> E. v e. V v I e ) )
28	27	imp 445	. . . . . . . . . . . . 13 |- ( ( ( G e. W /\ I C_ ( V X. L ) /\ ran I = L ) /\ e e. L ) -> E. v e. V v I e )
29		df-ne 2795	. . . . . . . . . . . . . 14 |- ( { v e. V | v I e } =/= (/) <-> -. { v e. V | v I e } = (/) )
30		rabn0 3958	. . . . . . . . . . . . . 14 |- ( { v e. V | v I e } =/= (/) <-> E. v e. V v I e )
31	29, 30	bitr3i 266	. . . . . . . . . . . . 13 |- ( -. { v e. V | v I e } = (/) <-> E. v e. V v I e )
32	28, 31	sylibr 224	. . . . . . . . . . . 12 |- ( ( ( G e. W /\ I C_ ( V X. L ) /\ ran I = L ) /\ e e. L ) -> -. { v e. V | v I e } = (/) )
33	13	elsn 4192	. . . . . . . . . . . 12 |- ( { v e. V | v I e } e. { (/) } <-> { v e. V | v I e } = (/) )
34	32, 33	sylnibr 319	. . . . . . . . . . 11 |- ( ( ( G e. W /\ I C_ ( V X. L ) /\ ran I = L ) /\ e e. L ) -> -. { v e. V | v I e } e. { (/) } )
35	21, 34	eldifd 3585	. . . . . . . . . 10 |- ( ( ( G e. W /\ I C_ ( V X. L ) /\ ran I = L ) /\ e e. L ) -> { v e. V | v I e } e. ( ~P V \ { (/) } ) )
36	35, 14	fmptd 6385	. . . . . . . . 9 |- ( ( G e. W /\ I C_ ( V X. L ) /\ ran I = L ) -> ( e e. L |-> { v e. V | v I e } ) : L --> ( ~P V \ { (/) } ) )
37		simpl 473	. . . . . . . . . 10 |- ( ( E = ( e e. L |-> { v e. V | v I e } ) /\ dom E = L ) -> E = ( e e. L |-> { v e. V | v I e } ) )
38		simpr 477	. . . . . . . . . 10 |- ( ( E = ( e e. L |-> { v e. V | v I e } ) /\ dom E = L ) -> dom E = L )
39	37, 38	feq12d 6033	. . . . . . . . 9 |- ( ( E = ( e e. L |-> { v e. V | v I e } ) /\ dom E = L ) -> ( E : dom E --> ( ~P V \ { (/) } ) <-> ( e e. L |-> { v e. V | v I e } ) : L --> ( ~P V \ { (/) } ) ) )
40	36, 39	syl5ibr 236	. . . . . . . 8 |- ( ( 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 --> ( ~P V \ { (/) } ) ) )
41	17, 40	mpdan 702	. . . . . . 7 |- ( E = ( e e. L |-> { v e. V | v I e } ) -> ( ( G e. W /\ I C_ ( V X. L ) /\ ran I = L ) -> E : dom E --> ( ~P V \ { (/) } ) ) )
42	41	imp 445	. . . . . 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 --> ( ~P V \ { (/) } ) )
43	8, 42	syl6bir 244	. . . . 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 --> ( ~P V \ { (/) } ) ) )
44	43	expdimp 453	. . . 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 --> ( ~P V \ { (/) } ) ) )
45	44	impcom 446	. . 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 --> ( ~P V \ { (/) } ) )
46		incistruhgr.e	. . . . . 6 |- E = (iEdg ` G )
47	10, 46	isuhgr 25955	. . . . 5 |- ( G e. W -> ( G e. UHGraph <-> E : dom E --> ( ~P V \ { (/) } ) ) )
48	47	3ad2ant1 1082	. . . 4 |- ( ( G e. W /\ I C_ ( P X. L ) /\ ran I = L ) -> ( G e. UHGraph <-> E : dom E --> ( ~P V \ { (/) } ) ) )
49	48	adantr 481	. . 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 --> ( ~P V \ { (/) } ) ) )
50	45, 49	mpbird 247	. 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 )
51	50	ex 450	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:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913   {crab 2916   _Vcvv 3200   \ cdif 3571   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   {csn 4177   class class class wbr 4653   |-> cmpt 4729   X. cxp 5112   dom cdm 5114   ran crn 5115   -->wf 5884   ` cfv 5888  Vtxcvtx 25874  iEdgciedg 25875   UHGraph cuhgr 25951

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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

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-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-uhgr 25953

This theorem is referenced by: (None)