Theorem incistruhgr 16314

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 2813	. . . . . . . . 9 |- ( V = P -> { v e. V | v I e } = { v e. P | v I e } )
2	1	mpteq2dv 4220	. . . . . . . 8 |- ( V = P -> ( e e. L |-> { v e. V | v I e } ) = ( e e. L |-> { v e. P | v I e } ) )
3	2	eqeq2d 2250	. . . . . . 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 4786	. . . . . . . . 9 |- ( V = P -> ( V X. L ) = ( P X. L ) )
5	4	sseq2d 3278	. . . . . . . 8 |- ( V = P -> ( I C_ ( V X. L ) <-> I C_ ( P X. L ) ) )
6	5	3anbi2d 1358	. . . . . . 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 477	. . . . . 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 4979	. . . . . . . . 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 2238	. . . . . . . . . 10 |- ( e e. L |-> { v e. V | v I e } ) = ( e e. L |-> { v e. V | v I e } )
11		eqid 2238	. . . . . . . . . . 11 |- { v e. V | v I e } = { v e. V | v I e }
12		incistruhgr.v	. . . . . . . . . . . 12 |- V = (Vtx ` G )
13		simpl1 1031	. . . . . . . . . . . . 13 |- ( ( ( G e. W /\ I C_ ( V X. L ) /\ ran I = L ) /\ e e. L ) -> G e. W )
14		vtxex 16242	. . . . . . . . . . . . 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 2325	. . . . . . . . . . 11 |- ( ( ( G e. W /\ I C_ ( V X. L ) /\ ran I = L ) /\ e e. L ) -> V e. _V )
17	11, 16	rabexd 4279	. . . . . . . . . 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 5512	. . . . . . . . 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 2291	. . . . . . . 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 2302	. . . . . . . . . . 11 |- ( s = { v e. V | v I e } -> ( j e. s <-> j e. { v e. V | v I e } ) )
23	22	exbidv 1878	. . . . . . . . . 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 3333	. . . . . . . . . . 11 |- { v e. V | v I e } C_ V
25		elpwg 3696	. . . . . . . . . . . 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 2302	. . . . . . . . . . . . . 14 |- ( ran I = L -> ( e e. ran I <-> e e. L ) )
29	28	3ad2ant3 1051	. . . . . . . . . . . . 13 |- ( ( G e. W /\ I C_ ( V X. L ) /\ ran I = L ) -> ( e e. ran I <-> e e. L ) )
30		ssrelrn 4970	. . . . . . . . . . . . . . 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 1050	. . . . . . . . . . . . 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 3549	. . . . . . . . . . 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 2984	. . . . . . . . 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 5857	. . . . . . . 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 5521	. . . . . . . 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 16295	. . . . 5 |- ( G e. W -> ( G e. UHGraph <-> E : dom E --> { s e. ~P V | E. j j e. s } ) )
49	48	3ad2ant1 1049	. . . 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 1009   = wceq 1402   E.wex 1545   e. wcel 2209   E.wrex 2529   {crab 2532   _Vcvv 2821   C_ wss 3220   ~Pcpw 3688   class class class wbr 4128   |-> cmpt 4190   X. cxp 4770   dom cdm 4772   ran crn 4773   -->wf 5371   ` cfv 5375  Vtxcvtx 16236  iEdgciedg 16237  UHGraphcuhgr 16291

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4247  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-cnex 8264  ax-resscn 8265  ax-1cn 8266  ax-1re 8267  ax-icn 8268  ax-addcl 8269  ax-addrcl 8270  ax-mulcl 8271  ax-addcom 8273  ax-mulcom 8274  ax-addass 8275  ax-mulass 8276  ax-distr 8277  ax-i2m1 8278  ax-1rid 8280  ax-0id 8281  ax-rnegex 8282  ax-cnre 8284

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-if 3639  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-br 4129  df-opab 4191  df-mpt 4192  df-id 4436  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-fo 5381  df-fv 5383  df-riota 6032  df-ov 6082  df-oprab 6083  df-mpo 6084  df-1st 6368  df-2nd 6369  df-sub 8493  df-inn 9288  df-2 9346  df-3 9347  df-4 9348  df-5 9349  df-6 9350  df-7 9351  df-8 9352  df-9 9353  df-n0 9547  df-dec 9761  df-ndx 13338  df-slot 13339  df-base 13341  df-edgf 16229  df-vtx 16238  df-iedg 16239  df-uhgrm 16293

This theorem is referenced by: (None)