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Theorem incistruhgr 28328
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 ((𝐺𝑊𝐼 ⊆ (𝑃 × 𝐿) ∧ ran 𝐼 = 𝐿) → ((𝑉 = 𝑃𝐸 = (𝑒𝐿 ↦ {𝑣𝑃𝑣𝐼𝑒})) → 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
StepHypRef Expression
1 rabeq 3446 . . . . . . . . 9 (𝑉 = 𝑃 → {𝑣𝑉𝑣𝐼𝑒} = {𝑣𝑃𝑣𝐼𝑒})
21mpteq2dv 5249 . . . . . . . 8 (𝑉 = 𝑃 → (𝑒𝐿 ↦ {𝑣𝑉𝑣𝐼𝑒}) = (𝑒𝐿 ↦ {𝑣𝑃𝑣𝐼𝑒}))
32eqeq2d 2743 . . . . . . 7 (𝑉 = 𝑃 → (𝐸 = (𝑒𝐿 ↦ {𝑣𝑉𝑣𝐼𝑒}) ↔ 𝐸 = (𝑒𝐿 ↦ {𝑣𝑃𝑣𝐼𝑒})))
4 xpeq1 5689 . . . . . . . . 9 (𝑉 = 𝑃 → (𝑉 × 𝐿) = (𝑃 × 𝐿))
54sseq2d 4013 . . . . . . . 8 (𝑉 = 𝑃 → (𝐼 ⊆ (𝑉 × 𝐿) ↔ 𝐼 ⊆ (𝑃 × 𝐿)))
653anbi2d 1441 . . . . . . 7 (𝑉 = 𝑃 → ((𝐺𝑊𝐼 ⊆ (𝑉 × 𝐿) ∧ ran 𝐼 = 𝐿) ↔ (𝐺𝑊𝐼 ⊆ (𝑃 × 𝐿) ∧ ran 𝐼 = 𝐿)))
73, 6anbi12d 631 . . . . . 6 (𝑉 = 𝑃 → ((𝐸 = (𝑒𝐿 ↦ {𝑣𝑉𝑣𝐼𝑒}) ∧ (𝐺𝑊𝐼 ⊆ (𝑉 × 𝐿) ∧ ran 𝐼 = 𝐿)) ↔ (𝐸 = (𝑒𝐿 ↦ {𝑣𝑃𝑣𝐼𝑒}) ∧ (𝐺𝑊𝐼 ⊆ (𝑃 × 𝐿) ∧ ran 𝐼 = 𝐿))))
8 dmeq 5901 . . . . . . . . 9 (𝐸 = (𝑒𝐿 ↦ {𝑣𝑉𝑣𝐼𝑒}) → dom 𝐸 = dom (𝑒𝐿 ↦ {𝑣𝑉𝑣𝐼𝑒}))
9 incistruhgr.v . . . . . . . . . . . 12 𝑉 = (Vtx‘𝐺)
109fvexi 6902 . . . . . . . . . . 11 𝑉 ∈ V
1110rabex 5331 . . . . . . . . . 10 {𝑣𝑉𝑣𝐼𝑒} ∈ V
12 eqid 2732 . . . . . . . . . 10 (𝑒𝐿 ↦ {𝑣𝑉𝑣𝐼𝑒}) = (𝑒𝐿 ↦ {𝑣𝑉𝑣𝐼𝑒})
1311, 12dmmpti 6691 . . . . . . . . 9 dom (𝑒𝐿 ↦ {𝑣𝑉𝑣𝐼𝑒}) = 𝐿
148, 13eqtrdi 2788 . . . . . . . 8 (𝐸 = (𝑒𝐿 ↦ {𝑣𝑉𝑣𝐼𝑒}) → dom 𝐸 = 𝐿)
15 ssrab2 4076 . . . . . . . . . . . . 13 {𝑣𝑉𝑣𝐼𝑒} ⊆ 𝑉
1615a1i 11 . . . . . . . . . . . 12 (((𝐺𝑊𝐼 ⊆ (𝑉 × 𝐿) ∧ ran 𝐼 = 𝐿) ∧ 𝑒𝐿) → {𝑣𝑉𝑣𝐼𝑒} ⊆ 𝑉)
1711elpw 4605 . . . . . . . . . . . 12 ({𝑣𝑉𝑣𝐼𝑒} ∈ 𝒫 𝑉 ↔ {𝑣𝑉𝑣𝐼𝑒} ⊆ 𝑉)
1816, 17sylibr 233 . . . . . . . . . . 11 (((𝐺𝑊𝐼 ⊆ (𝑉 × 𝐿) ∧ ran 𝐼 = 𝐿) ∧ 𝑒𝐿) → {𝑣𝑉𝑣𝐼𝑒} ∈ 𝒫 𝑉)
19 eleq2 2822 . . . . . . . . . . . . . . . 16 (ran 𝐼 = 𝐿 → (𝑒 ∈ ran 𝐼𝑒𝐿))
20193ad2ant3 1135 . . . . . . . . . . . . . . 15 ((𝐺𝑊𝐼 ⊆ (𝑉 × 𝐿) ∧ ran 𝐼 = 𝐿) → (𝑒 ∈ ran 𝐼𝑒𝐿))
21 ssrelrn 5892 . . . . . . . . . . . . . . . . 17 ((𝐼 ⊆ (𝑉 × 𝐿) ∧ 𝑒 ∈ ran 𝐼) → ∃𝑣𝑉 𝑣𝐼𝑒)
2221ex 413 . . . . . . . . . . . . . . . 16 (𝐼 ⊆ (𝑉 × 𝐿) → (𝑒 ∈ ran 𝐼 → ∃𝑣𝑉 𝑣𝐼𝑒))
23223ad2ant2 1134 . . . . . . . . . . . . . . 15 ((𝐺𝑊𝐼 ⊆ (𝑉 × 𝐿) ∧ ran 𝐼 = 𝐿) → (𝑒 ∈ ran 𝐼 → ∃𝑣𝑉 𝑣𝐼𝑒))
2420, 23sylbird 259 . . . . . . . . . . . . . 14 ((𝐺𝑊𝐼 ⊆ (𝑉 × 𝐿) ∧ ran 𝐼 = 𝐿) → (𝑒𝐿 → ∃𝑣𝑉 𝑣𝐼𝑒))
2524imp 407 . . . . . . . . . . . . 13 (((𝐺𝑊𝐼 ⊆ (𝑉 × 𝐿) ∧ ran 𝐼 = 𝐿) ∧ 𝑒𝐿) → ∃𝑣𝑉 𝑣𝐼𝑒)
26 df-ne 2941 . . . . . . . . . . . . . 14 ({𝑣𝑉𝑣𝐼𝑒} ≠ ∅ ↔ ¬ {𝑣𝑉𝑣𝐼𝑒} = ∅)
27 rabn0 4384 . . . . . . . . . . . . . 14 ({𝑣𝑉𝑣𝐼𝑒} ≠ ∅ ↔ ∃𝑣𝑉 𝑣𝐼𝑒)
2826, 27bitr3i 276 . . . . . . . . . . . . 13 (¬ {𝑣𝑉𝑣𝐼𝑒} = ∅ ↔ ∃𝑣𝑉 𝑣𝐼𝑒)
2925, 28sylibr 233 . . . . . . . . . . . 12 (((𝐺𝑊𝐼 ⊆ (𝑉 × 𝐿) ∧ ran 𝐼 = 𝐿) ∧ 𝑒𝐿) → ¬ {𝑣𝑉𝑣𝐼𝑒} = ∅)
3011elsn 4642 . . . . . . . . . . . 12 ({𝑣𝑉𝑣𝐼𝑒} ∈ {∅} ↔ {𝑣𝑉𝑣𝐼𝑒} = ∅)
3129, 30sylnibr 328 . . . . . . . . . . 11 (((𝐺𝑊𝐼 ⊆ (𝑉 × 𝐿) ∧ ran 𝐼 = 𝐿) ∧ 𝑒𝐿) → ¬ {𝑣𝑉𝑣𝐼𝑒} ∈ {∅})
3218, 31eldifd 3958 . . . . . . . . . 10 (((𝐺𝑊𝐼 ⊆ (𝑉 × 𝐿) ∧ ran 𝐼 = 𝐿) ∧ 𝑒𝐿) → {𝑣𝑉𝑣𝐼𝑒} ∈ (𝒫 𝑉 ∖ {∅}))
3332fmpttd 7111 . . . . . . . . 9 ((𝐺𝑊𝐼 ⊆ (𝑉 × 𝐿) ∧ ran 𝐼 = 𝐿) → (𝑒𝐿 ↦ {𝑣𝑉𝑣𝐼𝑒}):𝐿⟶(𝒫 𝑉 ∖ {∅}))
34 simpl 483 . . . . . . . . . 10 ((𝐸 = (𝑒𝐿 ↦ {𝑣𝑉𝑣𝐼𝑒}) ∧ dom 𝐸 = 𝐿) → 𝐸 = (𝑒𝐿 ↦ {𝑣𝑉𝑣𝐼𝑒}))
35 simpr 485 . . . . . . . . . 10 ((𝐸 = (𝑒𝐿 ↦ {𝑣𝑉𝑣𝐼𝑒}) ∧ dom 𝐸 = 𝐿) → dom 𝐸 = 𝐿)
3634, 35feq12d 6702 . . . . . . . . 9 ((𝐸 = (𝑒𝐿 ↦ {𝑣𝑉𝑣𝐼𝑒}) ∧ dom 𝐸 = 𝐿) → (𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅}) ↔ (𝑒𝐿 ↦ {𝑣𝑉𝑣𝐼𝑒}):𝐿⟶(𝒫 𝑉 ∖ {∅})))
3733, 36imbitrrid 245 . . . . . . . 8 ((𝐸 = (𝑒𝐿 ↦ {𝑣𝑉𝑣𝐼𝑒}) ∧ dom 𝐸 = 𝐿) → ((𝐺𝑊𝐼 ⊆ (𝑉 × 𝐿) ∧ ran 𝐼 = 𝐿) → 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅})))
3814, 37mpdan 685 . . . . . . 7 (𝐸 = (𝑒𝐿 ↦ {𝑣𝑉𝑣𝐼𝑒}) → ((𝐺𝑊𝐼 ⊆ (𝑉 × 𝐿) ∧ ran 𝐼 = 𝐿) → 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅})))
3938imp 407 . . . . . 6 ((𝐸 = (𝑒𝐿 ↦ {𝑣𝑉𝑣𝐼𝑒}) ∧ (𝐺𝑊𝐼 ⊆ (𝑉 × 𝐿) ∧ ran 𝐼 = 𝐿)) → 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅}))
407, 39syl6bir 253 . . . . 5 (𝑉 = 𝑃 → ((𝐸 = (𝑒𝐿 ↦ {𝑣𝑃𝑣𝐼𝑒}) ∧ (𝐺𝑊𝐼 ⊆ (𝑃 × 𝐿) ∧ ran 𝐼 = 𝐿)) → 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅})))
4140expdimp 453 . . . 4 ((𝑉 = 𝑃𝐸 = (𝑒𝐿 ↦ {𝑣𝑃𝑣𝐼𝑒})) → ((𝐺𝑊𝐼 ⊆ (𝑃 × 𝐿) ∧ ran 𝐼 = 𝐿) → 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅})))
4241impcom 408 . . 3 (((𝐺𝑊𝐼 ⊆ (𝑃 × 𝐿) ∧ ran 𝐼 = 𝐿) ∧ (𝑉 = 𝑃𝐸 = (𝑒𝐿 ↦ {𝑣𝑃𝑣𝐼𝑒}))) → 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅}))
43 incistruhgr.e . . . . . 6 𝐸 = (iEdg‘𝐺)
449, 43isuhgr 28309 . . . . 5 (𝐺𝑊 → (𝐺 ∈ UHGraph ↔ 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅})))
45443ad2ant1 1133 . . . 4 ((𝐺𝑊𝐼 ⊆ (𝑃 × 𝐿) ∧ ran 𝐼 = 𝐿) → (𝐺 ∈ UHGraph ↔ 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅})))
4645adantr 481 . . 3 (((𝐺𝑊𝐼 ⊆ (𝑃 × 𝐿) ∧ ran 𝐼 = 𝐿) ∧ (𝑉 = 𝑃𝐸 = (𝑒𝐿 ↦ {𝑣𝑃𝑣𝐼𝑒}))) → (𝐺 ∈ UHGraph ↔ 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅})))
4742, 46mpbird 256 . 2 (((𝐺𝑊𝐼 ⊆ (𝑃 × 𝐿) ∧ ran 𝐼 = 𝐿) ∧ (𝑉 = 𝑃𝐸 = (𝑒𝐿 ↦ {𝑣𝑃𝑣𝐼𝑒}))) → 𝐺 ∈ UHGraph)
4847ex 413 1 ((𝐺𝑊𝐼 ⊆ (𝑃 × 𝐿) ∧ ran 𝐼 = 𝐿) → ((𝑉 = 𝑃𝐸 = (𝑒𝐿 ↦ {𝑣𝑃𝑣𝐼𝑒})) → 𝐺 ∈ UHGraph))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wne 2940  wrex 3070  {crab 3432  cdif 3944  wss 3947  c0 4321  𝒫 cpw 4601  {csn 4627   class class class wbr 5147  cmpt 5230   × cxp 5673  dom cdm 5675  ran crn 5676  wf 6536  cfv 6540  Vtxcvtx 28245  iEdgciedg 28246  UHGraphcuhgr 28305
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-fv 6548  df-uhgr 28307
This theorem is referenced by: (None)
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