Theorem incistruhgr 15730

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 2765	. . . . . . . . 9 |- ( V = P -> { v e. V | v I e } = { v e. P | v I e } )
2	1	mpteq2dv 4139	. . . . . . . 8 |- ( V = P -> ( e e. L |-> { v e. V | v I e } ) = ( e e. L |-> { v e. P | v I e } ) )
3	2	eqeq2d 2218	. . . . . . 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 4693	. . . . . . . . 9 |- ( V = P -> ( V X. L ) = ( P X. L ) )
5	4	sseq2d 3224	. . . . . . . 8 |- ( V = P -> ( I C_ ( V X. L ) <-> I C_ ( P X. L ) ) )
6	5	3anbi2d 1330	. . . . . . 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 4883	. . . . . . . . 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 2206	. . . . . . . . . 10 |- ( e e. L |-> { v e. V | v I e } ) = ( e e. L |-> { v e. V | v I e } )
11		eqid 2206	. . . . . . . . . . 11 |- { v e. V | v I e } = { v e. V | v I e }
12		incistruhgr.v	. . . . . . . . . . . 12 |- V = (Vtx ` G )
13		simpl1 1003	. . . . . . . . . . . . 13 |- ( ( ( G e. W /\ I C_ ( V X. L ) /\ ran I = L ) /\ e e. L ) -> G e. W )
14		vtxex 15661	. . . . . . . . . . . . 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 2293	. . . . . . . . . . 11 |- ( ( ( G e. W /\ I C_ ( V X. L ) /\ ran I = L ) /\ e e. L ) -> V e. _V )
17	11, 16	rabexd 4193	. . . . . . . . . 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 5412	. . . . . . . . 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 2259	. . . . . . . 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 2270	. . . . . . . . . . 11 |- ( s = { v e. V | v I e } -> ( j e. s <-> j e. { v e. V | v I e } ) )
23	22	exbidv 1849	. . . . . . . . . 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 3279	. . . . . . . . . . 11 |- { v e. V | v I e } C_ V
25		elpwg 3625	. . . . . . . . . . . 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 2270	. . . . . . . . . . . . . 14 |- ( ran I = L -> ( e e. ran I <-> e e. L ) )
29	28	3ad2ant3 1023	. . . . . . . . . . . . 13 |- ( ( G e. W /\ I C_ ( V X. L ) /\ ran I = L ) -> ( e e. ran I <-> e e. L ) )
30		ssrelrn 4874	. . . . . . . . . . . . . . 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 1022	. . . . . . . . . . . . 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 3489	. . . . . . . . . . 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 2932	. . . . . . . . 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 5742	. . . . . . . 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 5421	. . . . . . . 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 15711	. . . . 5 |- ( G e. W -> ( G e. UHGraph <-> E : dom E --> { s e. ~P V | E. j j e. s } ) )
49	48	3ad2ant1 1021	. . . 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 981   = wceq 1373   E.wex 1516   e. wcel 2177   E.wrex 2486   {crab 2489   _Vcvv 2773   C_ wss 3167   ~Pcpw 3617   class class class wbr 4047   |-> cmpt 4109   X. cxp 4677   dom cdm 4679   ran crn 4680   -->wf 5272   ` cfv 5276  Vtxcvtx 15655  iEdgciedg 15656  UHGraphcuhgr 15707

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-sep 4166  ax-pow 4222  ax-pr 4257  ax-un 4484  ax-setind 4589  ax-cnex 8023  ax-resscn 8024  ax-1cn 8025  ax-1re 8026  ax-icn 8027  ax-addcl 8028  ax-addrcl 8029  ax-mulcl 8030  ax-addcom 8032  ax-mulcom 8033  ax-addass 8034  ax-mulass 8035  ax-distr 8036  ax-i2m1 8037  ax-1rid 8039  ax-0id 8040  ax-rnegex 8041  ax-cnre 8043

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-ral 2490  df-rex 2491  df-reu 2492  df-rab 2494  df-v 2775  df-sbc 3000  df-csb 3095  df-dif 3169  df-un 3171  df-in 3173  df-ss 3180  df-if 3573  df-pw 3619  df-sn 3640  df-pr 3641  df-op 3643  df-uni 3853  df-int 3888  df-br 4048  df-opab 4110  df-mpt 4111  df-id 4344  df-xp 4685  df-rel 4686  df-cnv 4687  df-co 4688  df-dm 4689  df-rn 4690  df-res 4691  df-ima 4692  df-iota 5237  df-fun 5278  df-fn 5279  df-f 5280  df-fo 5282  df-fv 5284  df-riota 5906  df-ov 5954  df-oprab 5955  df-mpo 5956  df-1st 6233  df-2nd 6234  df-sub 8252  df-inn 9044  df-2 9102  df-3 9103  df-4 9104  df-5 9105  df-6 9106  df-7 9107  df-8 9108  df-9 9109  df-n0 9303  df-dec 9512  df-ndx 12879  df-slot 12880  df-base 12882  df-edgf 15648  df-vtx 15657  df-iedg 15658  df-uhgrm 15709

This theorem is referenced by: (None)