Theorem incistruhgr 26858

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 3484	. . . . . . . . 9 ⊢ (𝑉 = 𝑃 → {𝑣 ∈ 𝑉 ∣ 𝑣𝐼𝑒} = {𝑣 ∈ 𝑃 ∣ 𝑣𝐼𝑒})
2	1	mpteq2dv 5155	. . . . . . . 8 ⊢ (𝑉 = 𝑃 → (𝑒 ∈ 𝐿 ↦ {𝑣 ∈ 𝑉 ∣ 𝑣𝐼𝑒}) = (𝑒 ∈ 𝐿 ↦ {𝑣 ∈ 𝑃 ∣ 𝑣𝐼𝑒}))
3	2	eqeq2d 2832	. . . . . . 7 ⊢ (𝑉 = 𝑃 → (𝐸 = (𝑒 ∈ 𝐿 ↦ {𝑣 ∈ 𝑉 ∣ 𝑣𝐼𝑒}) ↔ 𝐸 = (𝑒 ∈ 𝐿 ↦ {𝑣 ∈ 𝑃 ∣ 𝑣𝐼𝑒})))
4		xpeq1 5564	. . . . . . . . 9 ⊢ (𝑉 = 𝑃 → (𝑉 × 𝐿) = (𝑃 × 𝐿))
5	4	sseq2d 3999	. . . . . . . 8 ⊢ (𝑉 = 𝑃 → (𝐼 ⊆ (𝑉 × 𝐿) ↔ 𝐼 ⊆ (𝑃 × 𝐿)))
6	5	3anbi2d 1437	. . . . . . 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 5767	. . . . . . . . 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 6679	. . . . . . . . . . 11 ⊢ 𝑉 ∈ V
11	10	rabex 5228	. . . . . . . . . 10 ⊢ {𝑣 ∈ 𝑉 ∣ 𝑣𝐼𝑒} ∈ V
12		eqid 2821	. . . . . . . . . 10 ⊢ (𝑒 ∈ 𝐿 ↦ {𝑣 ∈ 𝑉 ∣ 𝑣𝐼𝑒}) = (𝑒 ∈ 𝐿 ↦ {𝑣 ∈ 𝑉 ∣ 𝑣𝐼𝑒})
13	11, 12	dmmpti 6487	. . . . . . . . 9 ⊢ dom (𝑒 ∈ 𝐿 ↦ {𝑣 ∈ 𝑉 ∣ 𝑣𝐼𝑒}) = 𝐿
14	8, 13	syl6eq 2872	. . . . . . . 8 ⊢ (𝐸 = (𝑒 ∈ 𝐿 ↦ {𝑣 ∈ 𝑉 ∣ 𝑣𝐼𝑒}) → dom 𝐸 = 𝐿)
15		ssrab2 4056	. . . . . . . . . . . . 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 4546	. . . . . . . . . . . 12 ⊢ ({𝑣 ∈ 𝑉 ∣ 𝑣𝐼𝑒} ∈ 𝒫 𝑉 ↔ {𝑣 ∈ 𝑉 ∣ 𝑣𝐼𝑒} ⊆ 𝑉)
18	16, 17	sylibr 236	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ 𝑊 ∧ 𝐼 ⊆ (𝑉 × 𝐿) ∧ ran 𝐼 = 𝐿) ∧ 𝑒 ∈ 𝐿) → {𝑣 ∈ 𝑉 ∣ 𝑣𝐼𝑒} ∈ 𝒫 𝑉)
19		eleq2 2901	. . . . . . . . . . . . . . . 16 ⊢ (ran 𝐼 = 𝐿 → (𝑒 ∈ ran 𝐼 ↔ 𝑒 ∈ 𝐿))
20	19	3ad2ant3 1131	. . . . . . . . . . . . . . 15 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝐼 ⊆ (𝑉 × 𝐿) ∧ ran 𝐼 = 𝐿) → (𝑒 ∈ ran 𝐼 ↔ 𝑒 ∈ 𝐿))
21		ssrelrn 5758	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐼 ⊆ (𝑉 × 𝐿) ∧ 𝑒 ∈ ran 𝐼) → ∃𝑣 ∈ 𝑉 𝑣𝐼𝑒)
22	21	ex 415	. . . . . . . . . . . . . . . 16 ⊢ (𝐼 ⊆ (𝑉 × 𝐿) → (𝑒 ∈ ran 𝐼 → ∃𝑣 ∈ 𝑉 𝑣𝐼𝑒))
23	22	3ad2ant2 1130	. . . . . . . . . . . . . . 15 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝐼 ⊆ (𝑉 × 𝐿) ∧ ran 𝐼 = 𝐿) → (𝑒 ∈ ran 𝐼 → ∃𝑣 ∈ 𝑉 𝑣𝐼𝑒))
24	20, 23	sylbird 262	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝐼 ⊆ (𝑉 × 𝐿) ∧ ran 𝐼 = 𝐿) → (𝑒 ∈ 𝐿 → ∃𝑣 ∈ 𝑉 𝑣𝐼𝑒))
25	24	imp 409	. . . . . . . . . . . . 13 ⊢ (((𝐺 ∈ 𝑊 ∧ 𝐼 ⊆ (𝑉 × 𝐿) ∧ ran 𝐼 = 𝐿) ∧ 𝑒 ∈ 𝐿) → ∃𝑣 ∈ 𝑉 𝑣𝐼𝑒)
26		df-ne 3017	. . . . . . . . . . . . . 14 ⊢ ({𝑣 ∈ 𝑉 ∣ 𝑣𝐼𝑒} ≠ ∅ ↔ ¬ {𝑣 ∈ 𝑉 ∣ 𝑣𝐼𝑒} = ∅)
27		rabn0 4339	. . . . . . . . . . . . . 14 ⊢ ({𝑣 ∈ 𝑉 ∣ 𝑣𝐼𝑒} ≠ ∅ ↔ ∃𝑣 ∈ 𝑉 𝑣𝐼𝑒)
28	26, 27	bitr3i 279	. . . . . . . . . . . . 13 ⊢ (¬ {𝑣 ∈ 𝑉 ∣ 𝑣𝐼𝑒} = ∅ ↔ ∃𝑣 ∈ 𝑉 𝑣𝐼𝑒)
29	25, 28	sylibr 236	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ 𝑊 ∧ 𝐼 ⊆ (𝑉 × 𝐿) ∧ ran 𝐼 = 𝐿) ∧ 𝑒 ∈ 𝐿) → ¬ {𝑣 ∈ 𝑉 ∣ 𝑣𝐼𝑒} = ∅)
30	11	elsn 4576	. . . . . . . . . . . 12 ⊢ ({𝑣 ∈ 𝑉 ∣ 𝑣𝐼𝑒} ∈ {∅} ↔ {𝑣 ∈ 𝑉 ∣ 𝑣𝐼𝑒} = ∅)
31	29, 30	sylnibr 331	. . . . . . . . . . 11 ⊢ (((𝐺 ∈ 𝑊 ∧ 𝐼 ⊆ (𝑉 × 𝐿) ∧ ran 𝐼 = 𝐿) ∧ 𝑒 ∈ 𝐿) → ¬ {𝑣 ∈ 𝑉 ∣ 𝑣𝐼𝑒} ∈ {∅})
32	18, 31	eldifd 3947	. . . . . . . . . 10 ⊢ (((𝐺 ∈ 𝑊 ∧ 𝐼 ⊆ (𝑉 × 𝐿) ∧ ran 𝐼 = 𝐿) ∧ 𝑒 ∈ 𝐿) → {𝑣 ∈ 𝑉 ∣ 𝑣𝐼𝑒} ∈ (𝒫 𝑉 ∖ {∅}))
33	32	fmpttd 6874	. . . . . . . . 9 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝐼 ⊆ (𝑉 × 𝐿) ∧ ran 𝐼 = 𝐿) → (𝑒 ∈ 𝐿 ↦ {𝑣 ∈ 𝑉 ∣ 𝑣𝐼𝑒}):𝐿⟶(𝒫 𝑉 ∖ {∅}))
34		simpl 485	. . . . . . . . . 10 ⊢ ((𝐸 = (𝑒 ∈ 𝐿 ↦ {𝑣 ∈ 𝑉 ∣ 𝑣𝐼𝑒}) ∧ dom 𝐸 = 𝐿) → 𝐸 = (𝑒 ∈ 𝐿 ↦ {𝑣 ∈ 𝑉 ∣ 𝑣𝐼𝑒}))
35		simpr 487	. . . . . . . . . 10 ⊢ ((𝐸 = (𝑒 ∈ 𝐿 ↦ {𝑣 ∈ 𝑉 ∣ 𝑣𝐼𝑒}) ∧ dom 𝐸 = 𝐿) → dom 𝐸 = 𝐿)
36	34, 35	feq12d 6497	. . . . . . . . 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	syl5ibr 248	. . . . . . . 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 685	. . . . . . 7 ⊢ (𝐸 = (𝑒 ∈ 𝐿 ↦ {𝑣 ∈ 𝑉 ∣ 𝑣𝐼𝑒}) → ((𝐺 ∈ 𝑊 ∧ 𝐼 ⊆ (𝑉 × 𝐿) ∧ ran 𝐼 = 𝐿) → 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅})))
39	38	imp 409	. . . . . 6 ⊢ ((𝐸 = (𝑒 ∈ 𝐿 ↦ {𝑣 ∈ 𝑉 ∣ 𝑣𝐼𝑒}) ∧ (𝐺 ∈ 𝑊 ∧ 𝐼 ⊆ (𝑉 × 𝐿) ∧ ran 𝐼 = 𝐿)) → 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅}))
40	7, 39	syl6bir 256	. . . . 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 455	. . . 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 410	. . 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 26839	. . . . 5 ⊢ (𝐺 ∈ 𝑊 → (𝐺 ∈ UHGraph ↔ 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅})))
45	44	3ad2ant1 1129	. . . 4 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝐼 ⊆ (𝑃 × 𝐿) ∧ ran 𝐼 = 𝐿) → (𝐺 ∈ UHGraph ↔ 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅})))
46	45	adantr 483	. . 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 259	. 2 ⊢ (((𝐺 ∈ 𝑊 ∧ 𝐼 ⊆ (𝑃 × 𝐿) ∧ ran 𝐼 = 𝐿) ∧ (𝑉 = 𝑃 ∧ 𝐸 = (𝑒 ∈ 𝐿 ↦ {𝑣 ∈ 𝑃 ∣ 𝑣𝐼𝑒}))) → 𝐺 ∈ UHGraph)
48	47	ex 415	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 208   ∧ wa 398   ∧ w3a 1083   = wceq 1533   ∈ wcel 2110   ≠ wne 3016  ∃wrex 3139  {crab 3142   ∖ cdif 3933   ⊆ wss 3936  ∅c0 4291  𝒫 cpw 4539  {csn 4561   class class class wbr 5059   ↦ cmpt 5139   × cxp 5548  dom cdm 5550  ran crn 5551  ⟶wf 6346  ‘cfv 6350  Vtxcvtx 26775  iEdgciedg 26776  UHGraphcuhgr 26835

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-sep 5196  ax-nul 5203  ax-pr 5322

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3497  df-sbc 3773  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4833  df-br 5060  df-opab 5122  df-mpt 5140  df-id 5455  df-xp 5556  df-rel 5557  df-cnv 5558  df-co 5559  df-dm 5560  df-rn 5561  df-res 5562  df-ima 5563  df-iota 6309  df-fun 6352  df-fn 6353  df-f 6354  df-fv 6358  df-uhgr 26837

This theorem is referenced by: (None)