MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  incistruhgr Structured version   Visualization version   GIF version

Theorem incistruhgr 28072
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 3420 . . . . . . . . 9 (𝑉 = 𝑃 → {𝑣𝑉𝑣𝐼𝑒} = {𝑣𝑃𝑣𝐼𝑒})
21mpteq2dv 5208 . . . . . . . 8 (𝑉 = 𝑃 → (𝑒𝐿 ↦ {𝑣𝑉𝑣𝐼𝑒}) = (𝑒𝐿 ↦ {𝑣𝑃𝑣𝐼𝑒}))
32eqeq2d 2744 . . . . . . 7 (𝑉 = 𝑃 → (𝐸 = (𝑒𝐿 ↦ {𝑣𝑉𝑣𝐼𝑒}) ↔ 𝐸 = (𝑒𝐿 ↦ {𝑣𝑃𝑣𝐼𝑒})))
4 xpeq1 5648 . . . . . . . . 9 (𝑉 = 𝑃 → (𝑉 × 𝐿) = (𝑃 × 𝐿))
54sseq2d 3977 . . . . . . . 8 (𝑉 = 𝑃 → (𝐼 ⊆ (𝑉 × 𝐿) ↔ 𝐼 ⊆ (𝑃 × 𝐿)))
653anbi2d 1442 . . . . . . 7 (𝑉 = 𝑃 → ((𝐺𝑊𝐼 ⊆ (𝑉 × 𝐿) ∧ ran 𝐼 = 𝐿) ↔ (𝐺𝑊𝐼 ⊆ (𝑃 × 𝐿) ∧ ran 𝐼 = 𝐿)))
73, 6anbi12d 632 . . . . . 6 (𝑉 = 𝑃 → ((𝐸 = (𝑒𝐿 ↦ {𝑣𝑉𝑣𝐼𝑒}) ∧ (𝐺𝑊𝐼 ⊆ (𝑉 × 𝐿) ∧ ran 𝐼 = 𝐿)) ↔ (𝐸 = (𝑒𝐿 ↦ {𝑣𝑃𝑣𝐼𝑒}) ∧ (𝐺𝑊𝐼 ⊆ (𝑃 × 𝐿) ∧ ran 𝐼 = 𝐿))))
8 dmeq 5860 . . . . . . . . 9 (𝐸 = (𝑒𝐿 ↦ {𝑣𝑉𝑣𝐼𝑒}) → dom 𝐸 = dom (𝑒𝐿 ↦ {𝑣𝑉𝑣𝐼𝑒}))
9 incistruhgr.v . . . . . . . . . . . 12 𝑉 = (Vtx‘𝐺)
109fvexi 6857 . . . . . . . . . . 11 𝑉 ∈ V
1110rabex 5290 . . . . . . . . . 10 {𝑣𝑉𝑣𝐼𝑒} ∈ V
12 eqid 2733 . . . . . . . . . 10 (𝑒𝐿 ↦ {𝑣𝑉𝑣𝐼𝑒}) = (𝑒𝐿 ↦ {𝑣𝑉𝑣𝐼𝑒})
1311, 12dmmpti 6646 . . . . . . . . 9 dom (𝑒𝐿 ↦ {𝑣𝑉𝑣𝐼𝑒}) = 𝐿
148, 13eqtrdi 2789 . . . . . . . 8 (𝐸 = (𝑒𝐿 ↦ {𝑣𝑉𝑣𝐼𝑒}) → dom 𝐸 = 𝐿)
15 ssrab2 4038 . . . . . . . . . . . . 13 {𝑣𝑉𝑣𝐼𝑒} ⊆ 𝑉
1615a1i 11 . . . . . . . . . . . 12 (((𝐺𝑊𝐼 ⊆ (𝑉 × 𝐿) ∧ ran 𝐼 = 𝐿) ∧ 𝑒𝐿) → {𝑣𝑉𝑣𝐼𝑒} ⊆ 𝑉)
1711elpw 4565 . . . . . . . . . . . 12 ({𝑣𝑉𝑣𝐼𝑒} ∈ 𝒫 𝑉 ↔ {𝑣𝑉𝑣𝐼𝑒} ⊆ 𝑉)
1816, 17sylibr 233 . . . . . . . . . . 11 (((𝐺𝑊𝐼 ⊆ (𝑉 × 𝐿) ∧ ran 𝐼 = 𝐿) ∧ 𝑒𝐿) → {𝑣𝑉𝑣𝐼𝑒} ∈ 𝒫 𝑉)
19 eleq2 2823 . . . . . . . . . . . . . . . 16 (ran 𝐼 = 𝐿 → (𝑒 ∈ ran 𝐼𝑒𝐿))
20193ad2ant3 1136 . . . . . . . . . . . . . . 15 ((𝐺𝑊𝐼 ⊆ (𝑉 × 𝐿) ∧ ran 𝐼 = 𝐿) → (𝑒 ∈ ran 𝐼𝑒𝐿))
21 ssrelrn 5851 . . . . . . . . . . . . . . . . 17 ((𝐼 ⊆ (𝑉 × 𝐿) ∧ 𝑒 ∈ ran 𝐼) → ∃𝑣𝑉 𝑣𝐼𝑒)
2221ex 414 . . . . . . . . . . . . . . . 16 (𝐼 ⊆ (𝑉 × 𝐿) → (𝑒 ∈ ran 𝐼 → ∃𝑣𝑉 𝑣𝐼𝑒))
23223ad2ant2 1135 . . . . . . . . . . . . . . 15 ((𝐺𝑊𝐼 ⊆ (𝑉 × 𝐿) ∧ ran 𝐼 = 𝐿) → (𝑒 ∈ ran 𝐼 → ∃𝑣𝑉 𝑣𝐼𝑒))
2420, 23sylbird 260 . . . . . . . . . . . . . 14 ((𝐺𝑊𝐼 ⊆ (𝑉 × 𝐿) ∧ ran 𝐼 = 𝐿) → (𝑒𝐿 → ∃𝑣𝑉 𝑣𝐼𝑒))
2524imp 408 . . . . . . . . . . . . 13 (((𝐺𝑊𝐼 ⊆ (𝑉 × 𝐿) ∧ ran 𝐼 = 𝐿) ∧ 𝑒𝐿) → ∃𝑣𝑉 𝑣𝐼𝑒)
26 df-ne 2941 . . . . . . . . . . . . . 14 ({𝑣𝑉𝑣𝐼𝑒} ≠ ∅ ↔ ¬ {𝑣𝑉𝑣𝐼𝑒} = ∅)
27 rabn0 4346 . . . . . . . . . . . . . 14 ({𝑣𝑉𝑣𝐼𝑒} ≠ ∅ ↔ ∃𝑣𝑉 𝑣𝐼𝑒)
2826, 27bitr3i 277 . . . . . . . . . . . . 13 (¬ {𝑣𝑉𝑣𝐼𝑒} = ∅ ↔ ∃𝑣𝑉 𝑣𝐼𝑒)
2925, 28sylibr 233 . . . . . . . . . . . 12 (((𝐺𝑊𝐼 ⊆ (𝑉 × 𝐿) ∧ ran 𝐼 = 𝐿) ∧ 𝑒𝐿) → ¬ {𝑣𝑉𝑣𝐼𝑒} = ∅)
3011elsn 4602 . . . . . . . . . . . 12 ({𝑣𝑉𝑣𝐼𝑒} ∈ {∅} ↔ {𝑣𝑉𝑣𝐼𝑒} = ∅)
3129, 30sylnibr 329 . . . . . . . . . . 11 (((𝐺𝑊𝐼 ⊆ (𝑉 × 𝐿) ∧ ran 𝐼 = 𝐿) ∧ 𝑒𝐿) → ¬ {𝑣𝑉𝑣𝐼𝑒} ∈ {∅})
3218, 31eldifd 3922 . . . . . . . . . 10 (((𝐺𝑊𝐼 ⊆ (𝑉 × 𝐿) ∧ ran 𝐼 = 𝐿) ∧ 𝑒𝐿) → {𝑣𝑉𝑣𝐼𝑒} ∈ (𝒫 𝑉 ∖ {∅}))
3332fmpttd 7064 . . . . . . . . 9 ((𝐺𝑊𝐼 ⊆ (𝑉 × 𝐿) ∧ ran 𝐼 = 𝐿) → (𝑒𝐿 ↦ {𝑣𝑉𝑣𝐼𝑒}):𝐿⟶(𝒫 𝑉 ∖ {∅}))
34 simpl 484 . . . . . . . . . 10 ((𝐸 = (𝑒𝐿 ↦ {𝑣𝑉𝑣𝐼𝑒}) ∧ dom 𝐸 = 𝐿) → 𝐸 = (𝑒𝐿 ↦ {𝑣𝑉𝑣𝐼𝑒}))
35 simpr 486 . . . . . . . . . 10 ((𝐸 = (𝑒𝐿 ↦ {𝑣𝑉𝑣𝐼𝑒}) ∧ dom 𝐸 = 𝐿) → dom 𝐸 = 𝐿)
3634, 35feq12d 6657 . . . . . . . . 9 ((𝐸 = (𝑒𝐿 ↦ {𝑣𝑉𝑣𝐼𝑒}) ∧ dom 𝐸 = 𝐿) → (𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅}) ↔ (𝑒𝐿 ↦ {𝑣𝑉𝑣𝐼𝑒}):𝐿⟶(𝒫 𝑉 ∖ {∅})))
3733, 36syl5ibr 246 . . . . . . . 8 ((𝐸 = (𝑒𝐿 ↦ {𝑣𝑉𝑣𝐼𝑒}) ∧ dom 𝐸 = 𝐿) → ((𝐺𝑊𝐼 ⊆ (𝑉 × 𝐿) ∧ ran 𝐼 = 𝐿) → 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅})))
3814, 37mpdan 686 . . . . . . 7 (𝐸 = (𝑒𝐿 ↦ {𝑣𝑉𝑣𝐼𝑒}) → ((𝐺𝑊𝐼 ⊆ (𝑉 × 𝐿) ∧ ran 𝐼 = 𝐿) → 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅})))
3938imp 408 . . . . . 6 ((𝐸 = (𝑒𝐿 ↦ {𝑣𝑉𝑣𝐼𝑒}) ∧ (𝐺𝑊𝐼 ⊆ (𝑉 × 𝐿) ∧ ran 𝐼 = 𝐿)) → 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅}))
407, 39syl6bir 254 . . . . 5 (𝑉 = 𝑃 → ((𝐸 = (𝑒𝐿 ↦ {𝑣𝑃𝑣𝐼𝑒}) ∧ (𝐺𝑊𝐼 ⊆ (𝑃 × 𝐿) ∧ ran 𝐼 = 𝐿)) → 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅})))
4140expdimp 454 . . . 4 ((𝑉 = 𝑃𝐸 = (𝑒𝐿 ↦ {𝑣𝑃𝑣𝐼𝑒})) → ((𝐺𝑊𝐼 ⊆ (𝑃 × 𝐿) ∧ ran 𝐼 = 𝐿) → 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅})))
4241impcom 409 . . 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 28053 . . . . 5 (𝐺𝑊 → (𝐺 ∈ UHGraph ↔ 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅})))
45443ad2ant1 1134 . . . 4 ((𝐺𝑊𝐼 ⊆ (𝑃 × 𝐿) ∧ ran 𝐼 = 𝐿) → (𝐺 ∈ UHGraph ↔ 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅})))
4645adantr 482 . . 3 (((𝐺𝑊𝐼 ⊆ (𝑃 × 𝐿) ∧ ran 𝐼 = 𝐿) ∧ (𝑉 = 𝑃𝐸 = (𝑒𝐿 ↦ {𝑣𝑃𝑣𝐼𝑒}))) → (𝐺 ∈ UHGraph ↔ 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅})))
4742, 46mpbird 257 . 2 (((𝐺𝑊𝐼 ⊆ (𝑃 × 𝐿) ∧ ran 𝐼 = 𝐿) ∧ (𝑉 = 𝑃𝐸 = (𝑒𝐿 ↦ {𝑣𝑃𝑣𝐼𝑒}))) → 𝐺 ∈ UHGraph)
4847ex 414 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 397  w3a 1088   = wceq 1542  wcel 2107  wne 2940  wrex 3070  {crab 3406  cdif 3908  wss 3911  c0 4283  𝒫 cpw 4561  {csn 4587   class class class wbr 5106  cmpt 5189   × cxp 5632  dom cdm 5634  ran crn 5635  wf 6493  cfv 6497  Vtxcvtx 27989  iEdgciedg 27990  UHGraphcuhgr 28049
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-sbc 3741  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-fv 6505  df-uhgr 28051
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator