Theorem incistruhgr 29096

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 ⊆ (𝑃 × 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 ∈ 𝑊 ∧ 𝐼 ⊆ (𝑃 × 𝐿) ∧ ran 𝐼 = 𝐿) → ((𝑉 = 𝑃 ∧ 𝐸 = (𝑒 ∈ 𝐿 ↦ {𝑣 ∈ 𝑃 ∣ 𝑣𝐼𝑒})) → 𝐺 ∈ 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		rabeq 3451	. . . . . . . . 9 ⊢ (𝑉 = 𝑃 → {𝑣 ∈ 𝑉 ∣ 𝑣𝐼𝑒} = {𝑣 ∈ 𝑃 ∣ 𝑣𝐼𝑒})
2	1	mpteq2dv 5244	. . . . . . . 8 ⊢ (𝑉 = 𝑃 → (𝑒 ∈ 𝐿 ↦ {𝑣 ∈ 𝑉 ∣ 𝑣𝐼𝑒}) = (𝑒 ∈ 𝐿 ↦ {𝑣 ∈ 𝑃 ∣ 𝑣𝐼𝑒}))
3	2	eqeq2d 2748	. . . . . . 7 ⊢ (𝑉 = 𝑃 → (𝐸 = (𝑒 ∈ 𝐿 ↦ {𝑣 ∈ 𝑉 ∣ 𝑣𝐼𝑒}) ↔ 𝐸 = (𝑒 ∈ 𝐿 ↦ {𝑣 ∈ 𝑃 ∣ 𝑣𝐼𝑒})))
4		xpeq1 5699	. . . . . . . . 9 ⊢ (𝑉 = 𝑃 → (𝑉 × 𝐿) = (𝑃 × 𝐿))
5	4	sseq2d 4016	. . . . . . . 8 ⊢ (𝑉 = 𝑃 → (𝐼 ⊆ (𝑉 × 𝐿) ↔ 𝐼 ⊆ (𝑃 × 𝐿)))
6	5	3anbi2d 1443	. . . . . . 7 ⊢ (𝑉 = 𝑃 → ((𝐺 ∈ 𝑊 ∧ 𝐼 ⊆ (𝑉 × 𝐿) ∧ ran 𝐼 = 𝐿) ↔ (𝐺 ∈ 𝑊 ∧ 𝐼 ⊆ (𝑃 × 𝐿) ∧ ran 𝐼 = 𝐿)))
7	3, 6	anbi12d 632	. . . . . 6 ⊢ (𝑉 = 𝑃 → ((𝐸 = (𝑒 ∈ 𝐿 ↦ {𝑣 ∈ 𝑉 ∣ 𝑣𝐼𝑒}) ∧ (𝐺 ∈ 𝑊 ∧ 𝐼 ⊆ (𝑉 × 𝐿) ∧ ran 𝐼 = 𝐿)) ↔ (𝐸 = (𝑒 ∈ 𝐿 ↦ {𝑣 ∈ 𝑃 ∣ 𝑣𝐼𝑒}) ∧ (𝐺 ∈ 𝑊 ∧ 𝐼 ⊆ (𝑃 × 𝐿) ∧ ran 𝐼 = 𝐿))))
8		dmeq 5914	. . . . . . . . 9 ⊢ (𝐸 = (𝑒 ∈ 𝐿 ↦ {𝑣 ∈ 𝑉 ∣ 𝑣𝐼𝑒}) → dom 𝐸 = dom (𝑒 ∈ 𝐿 ↦ {𝑣 ∈ 𝑉 ∣ 𝑣𝐼𝑒}))
9		incistruhgr.v	. . . . . . . . . . . 12 ⊢ 𝑉 = (Vtx‘𝐺)
10	9	fvexi 6920	. . . . . . . . . . 11 ⊢ 𝑉 ∈ V
11	10	rabex 5339	. . . . . . . . . 10 ⊢ {𝑣 ∈ 𝑉 ∣ 𝑣𝐼𝑒} ∈ V
12		eqid 2737	. . . . . . . . . 10 ⊢ (𝑒 ∈ 𝐿 ↦ {𝑣 ∈ 𝑉 ∣ 𝑣𝐼𝑒}) = (𝑒 ∈ 𝐿 ↦ {𝑣 ∈ 𝑉 ∣ 𝑣𝐼𝑒})
13	11, 12	dmmpti 6712	. . . . . . . . 9 ⊢ dom (𝑒 ∈ 𝐿 ↦ {𝑣 ∈ 𝑉 ∣ 𝑣𝐼𝑒}) = 𝐿
14	8, 13	eqtrdi 2793	. . . . . . . 8 ⊢ (𝐸 = (𝑒 ∈ 𝐿 ↦ {𝑣 ∈ 𝑉 ∣ 𝑣𝐼𝑒}) → dom 𝐸 = 𝐿)
15		ssrab2 4080	. . . . . . . . . . . . 13 ⊢ {𝑣 ∈ 𝑉 ∣ 𝑣𝐼𝑒} ⊆ 𝑉
16	15	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ 𝑊 ∧ 𝐼 ⊆ (𝑉 × 𝐿) ∧ ran 𝐼 = 𝐿) ∧ 𝑒 ∈ 𝐿) → {𝑣 ∈ 𝑉 ∣ 𝑣𝐼𝑒} ⊆ 𝑉)
17	11	elpw 4604	. . . . . . . . . . . 12 ⊢ ({𝑣 ∈ 𝑉 ∣ 𝑣𝐼𝑒} ∈ 𝒫 𝑉 ↔ {𝑣 ∈ 𝑉 ∣ 𝑣𝐼𝑒} ⊆ 𝑉)
18	16, 17	sylibr 234	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ 𝑊 ∧ 𝐼 ⊆ (𝑉 × 𝐿) ∧ ran 𝐼 = 𝐿) ∧ 𝑒 ∈ 𝐿) → {𝑣 ∈ 𝑉 ∣ 𝑣𝐼𝑒} ∈ 𝒫 𝑉)
19		eleq2 2830	. . . . . . . . . . . . . . . 16 ⊢ (ran 𝐼 = 𝐿 → (𝑒 ∈ ran 𝐼 ↔ 𝑒 ∈ 𝐿))
20	19	3ad2ant3 1136	. . . . . . . . . . . . . . 15 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝐼 ⊆ (𝑉 × 𝐿) ∧ ran 𝐼 = 𝐿) → (𝑒 ∈ ran 𝐼 ↔ 𝑒 ∈ 𝐿))
21		ssrelrn 5905	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐼 ⊆ (𝑉 × 𝐿) ∧ 𝑒 ∈ ran 𝐼) → ∃𝑣 ∈ 𝑉 𝑣𝐼𝑒)
22	21	ex 412	. . . . . . . . . . . . . . . 16 ⊢ (𝐼 ⊆ (𝑉 × 𝐿) → (𝑒 ∈ ran 𝐼 → ∃𝑣 ∈ 𝑉 𝑣𝐼𝑒))
23	22	3ad2ant2 1135	. . . . . . . . . . . . . . 15 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝐼 ⊆ (𝑉 × 𝐿) ∧ ran 𝐼 = 𝐿) → (𝑒 ∈ ran 𝐼 → ∃𝑣 ∈ 𝑉 𝑣𝐼𝑒))
24	20, 23	sylbird 260	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝐼 ⊆ (𝑉 × 𝐿) ∧ ran 𝐼 = 𝐿) → (𝑒 ∈ 𝐿 → ∃𝑣 ∈ 𝑉 𝑣𝐼𝑒))
25	24	imp 406	. . . . . . . . . . . . 13 ⊢ (((𝐺 ∈ 𝑊 ∧ 𝐼 ⊆ (𝑉 × 𝐿) ∧ ran 𝐼 = 𝐿) ∧ 𝑒 ∈ 𝐿) → ∃𝑣 ∈ 𝑉 𝑣𝐼𝑒)
26		df-ne 2941	. . . . . . . . . . . . . 14 ⊢ ({𝑣 ∈ 𝑉 ∣ 𝑣𝐼𝑒} ≠ ∅ ↔ ¬ {𝑣 ∈ 𝑉 ∣ 𝑣𝐼𝑒} = ∅)
27		rabn0 4389	. . . . . . . . . . . . . 14 ⊢ ({𝑣 ∈ 𝑉 ∣ 𝑣𝐼𝑒} ≠ ∅ ↔ ∃𝑣 ∈ 𝑉 𝑣𝐼𝑒)
28	26, 27	bitr3i 277	. . . . . . . . . . . . 13 ⊢ (¬ {𝑣 ∈ 𝑉 ∣ 𝑣𝐼𝑒} = ∅ ↔ ∃𝑣 ∈ 𝑉 𝑣𝐼𝑒)
29	25, 28	sylibr 234	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ 𝑊 ∧ 𝐼 ⊆ (𝑉 × 𝐿) ∧ ran 𝐼 = 𝐿) ∧ 𝑒 ∈ 𝐿) → ¬ {𝑣 ∈ 𝑉 ∣ 𝑣𝐼𝑒} = ∅)
30	11	elsn 4641	. . . . . . . . . . . 12 ⊢ ({𝑣 ∈ 𝑉 ∣ 𝑣𝐼𝑒} ∈ {∅} ↔ {𝑣 ∈ 𝑉 ∣ 𝑣𝐼𝑒} = ∅)
31	29, 30	sylnibr 329	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ 𝑊 ∧ 𝐼 ⊆ (𝑉 × 𝐿) ∧ ran 𝐼 = 𝐿) ∧ 𝑒 ∈ 𝐿) → ¬ {𝑣 ∈ 𝑉 ∣ 𝑣𝐼𝑒} ∈ {∅})
32	18, 31	eldifd 3962	. . . . . . . . . 10 ⊢ (((𝐺 ∈ 𝑊 ∧ 𝐼 ⊆ (𝑉 × 𝐿) ∧ ran 𝐼 = 𝐿) ∧ 𝑒 ∈ 𝐿) → {𝑣 ∈ 𝑉 ∣ 𝑣𝐼𝑒} ∈ (𝒫 𝑉 ∖ {∅}))
33	32	fmpttd 7135	. . . . . . . . 9 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝐼 ⊆ (𝑉 × 𝐿) ∧ ran 𝐼 = 𝐿) → (𝑒 ∈ 𝐿 ↦ {𝑣 ∈ 𝑉 ∣ 𝑣𝐼𝑒}):𝐿⟶(𝒫 𝑉 ∖ {∅}))
34		simpl 482	. . . . . . . . . 10 ⊢ ((𝐸 = (𝑒 ∈ 𝐿 ↦ {𝑣 ∈ 𝑉 ∣ 𝑣𝐼𝑒}) ∧ dom 𝐸 = 𝐿) → 𝐸 = (𝑒 ∈ 𝐿 ↦ {𝑣 ∈ 𝑉 ∣ 𝑣𝐼𝑒}))
35		simpr 484	. . . . . . . . . 10 ⊢ ((𝐸 = (𝑒 ∈ 𝐿 ↦ {𝑣 ∈ 𝑉 ∣ 𝑣𝐼𝑒}) ∧ dom 𝐸 = 𝐿) → dom 𝐸 = 𝐿)
36	34, 35	feq12d 6724	. . . . . . . . 9 ⊢ ((𝐸 = (𝑒 ∈ 𝐿 ↦ {𝑣 ∈ 𝑉 ∣ 𝑣𝐼𝑒}) ∧ dom 𝐸 = 𝐿) → (𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅}) ↔ (𝑒 ∈ 𝐿 ↦ {𝑣 ∈ 𝑉 ∣ 𝑣𝐼𝑒}):𝐿⟶(𝒫 𝑉 ∖ {∅})))
37	33, 36	imbitrrid 246	. . . . . . . 8 ⊢ ((𝐸 = (𝑒 ∈ 𝐿 ↦ {𝑣 ∈ 𝑉 ∣ 𝑣𝐼𝑒}) ∧ dom 𝐸 = 𝐿) → ((𝐺 ∈ 𝑊 ∧ 𝐼 ⊆ (𝑉 × 𝐿) ∧ ran 𝐼 = 𝐿) → 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅})))
38	14, 37	mpdan 687	. . . . . . 7 ⊢ (𝐸 = (𝑒 ∈ 𝐿 ↦ {𝑣 ∈ 𝑉 ∣ 𝑣𝐼𝑒}) → ((𝐺 ∈ 𝑊 ∧ 𝐼 ⊆ (𝑉 × 𝐿) ∧ ran 𝐼 = 𝐿) → 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅})))
39	38	imp 406	. . . . . 6 ⊢ ((𝐸 = (𝑒 ∈ 𝐿 ↦ {𝑣 ∈ 𝑉 ∣ 𝑣𝐼𝑒}) ∧ (𝐺 ∈ 𝑊 ∧ 𝐼 ⊆ (𝑉 × 𝐿) ∧ ran 𝐼 = 𝐿)) → 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅}))
40	7, 39	biimtrrdi 254	. . . . 5 ⊢ (𝑉 = 𝑃 → ((𝐸 = (𝑒 ∈ 𝐿 ↦ {𝑣 ∈ 𝑃 ∣ 𝑣𝐼𝑒}) ∧ (𝐺 ∈ 𝑊 ∧ 𝐼 ⊆ (𝑃 × 𝐿) ∧ ran 𝐼 = 𝐿)) → 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅})))
41	40	expdimp 452	. . . 4 ⊢ ((𝑉 = 𝑃 ∧ 𝐸 = (𝑒 ∈ 𝐿 ↦ {𝑣 ∈ 𝑃 ∣ 𝑣𝐼𝑒})) → ((𝐺 ∈ 𝑊 ∧ 𝐼 ⊆ (𝑃 × 𝐿) ∧ ran 𝐼 = 𝐿) → 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅})))
42	41	impcom 407	. . 3 ⊢ (((𝐺 ∈ 𝑊 ∧ 𝐼 ⊆ (𝑃 × 𝐿) ∧ ran 𝐼 = 𝐿) ∧ (𝑉 = 𝑃 ∧ 𝐸 = (𝑒 ∈ 𝐿 ↦ {𝑣 ∈ 𝑃 ∣ 𝑣𝐼𝑒}))) → 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅}))
43		incistruhgr.e	. . . . . 6 ⊢ 𝐸 = (iEdg‘𝐺)
44	9, 43	isuhgr 29077	. . . . 5 ⊢ (𝐺 ∈ 𝑊 → (𝐺 ∈ UHGraph ↔ 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅})))
45	44	3ad2ant1 1134	. . . 4 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝐼 ⊆ (𝑃 × 𝐿) ∧ ran 𝐼 = 𝐿) → (𝐺 ∈ UHGraph ↔ 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅})))
46	45	adantr 480	. . 3 ⊢ (((𝐺 ∈ 𝑊 ∧ 𝐼 ⊆ (𝑃 × 𝐿) ∧ ran 𝐼 = 𝐿) ∧ (𝑉 = 𝑃 ∧ 𝐸 = (𝑒 ∈ 𝐿 ↦ {𝑣 ∈ 𝑃 ∣ 𝑣𝐼𝑒}))) → (𝐺 ∈ UHGraph ↔ 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅})))
47	42, 46	mpbird 257	. 2 ⊢ (((𝐺 ∈ 𝑊 ∧ 𝐼 ⊆ (𝑃 × 𝐿) ∧ ran 𝐼 = 𝐿) ∧ (𝑉 = 𝑃 ∧ 𝐸 = (𝑒 ∈ 𝐿 ↦ {𝑣 ∈ 𝑃 ∣ 𝑣𝐼𝑒}))) → 𝐺 ∈ UHGraph)
48	47	ex 412	1 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝐼 ⊆ (𝑃 × 𝐿) ∧ ran 𝐼 = 𝐿) → ((𝑉 = 𝑃 ∧ 𝐸 = (𝑒 ∈ 𝐿 ↦ {𝑣 ∈ 𝑃 ∣ 𝑣𝐼𝑒})) → 𝐺 ∈ UHGraph))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1540   ∈ wcel 2108   ≠ wne 2940  ∃wrex 3070  {crab 3436   ∖ cdif 3948   ⊆ wss 3951  ∅c0 4333  𝒫 cpw 4600  {csn 4626   class class class wbr 5143   ↦ cmpt 5225   × cxp 5683  dom cdm 5685  ran crn 5686  ⟶wf 6557  ‘cfv 6561  Vtxcvtx 29013  iEdgciedg 29014  UHGraphcuhgr 29073

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-uhgr 29075

This theorem is referenced by: (None)