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Theorem incistruhgr 28607
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 3445 . . . . . . . . 9 (𝑉 = 𝑃 → {𝑣𝑉𝑣𝐼𝑒} = {𝑣𝑃𝑣𝐼𝑒})
21mpteq2dv 5250 . . . . . . . 8 (𝑉 = 𝑃 → (𝑒𝐿 ↦ {𝑣𝑉𝑣𝐼𝑒}) = (𝑒𝐿 ↦ {𝑣𝑃𝑣𝐼𝑒}))
32eqeq2d 2742 . . . . . . 7 (𝑉 = 𝑃 → (𝐸 = (𝑒𝐿 ↦ {𝑣𝑉𝑣𝐼𝑒}) ↔ 𝐸 = (𝑒𝐿 ↦ {𝑣𝑃𝑣𝐼𝑒})))
4 xpeq1 5690 . . . . . . . . 9 (𝑉 = 𝑃 → (𝑉 × 𝐿) = (𝑃 × 𝐿))
54sseq2d 4014 . . . . . . . 8 (𝑉 = 𝑃 → (𝐼 ⊆ (𝑉 × 𝐿) ↔ 𝐼 ⊆ (𝑃 × 𝐿)))
653anbi2d 1440 . . . . . . 7 (𝑉 = 𝑃 → ((𝐺𝑊𝐼 ⊆ (𝑉 × 𝐿) ∧ ran 𝐼 = 𝐿) ↔ (𝐺𝑊𝐼 ⊆ (𝑃 × 𝐿) ∧ ran 𝐼 = 𝐿)))
73, 6anbi12d 630 . . . . . 6 (𝑉 = 𝑃 → ((𝐸 = (𝑒𝐿 ↦ {𝑣𝑉𝑣𝐼𝑒}) ∧ (𝐺𝑊𝐼 ⊆ (𝑉 × 𝐿) ∧ ran 𝐼 = 𝐿)) ↔ (𝐸 = (𝑒𝐿 ↦ {𝑣𝑃𝑣𝐼𝑒}) ∧ (𝐺𝑊𝐼 ⊆ (𝑃 × 𝐿) ∧ ran 𝐼 = 𝐿))))
8 dmeq 5903 . . . . . . . . 9 (𝐸 = (𝑒𝐿 ↦ {𝑣𝑉𝑣𝐼𝑒}) → dom 𝐸 = dom (𝑒𝐿 ↦ {𝑣𝑉𝑣𝐼𝑒}))
9 incistruhgr.v . . . . . . . . . . . 12 𝑉 = (Vtx‘𝐺)
109fvexi 6905 . . . . . . . . . . 11 𝑉 ∈ V
1110rabex 5332 . . . . . . . . . 10 {𝑣𝑉𝑣𝐼𝑒} ∈ V
12 eqid 2731 . . . . . . . . . 10 (𝑒𝐿 ↦ {𝑣𝑉𝑣𝐼𝑒}) = (𝑒𝐿 ↦ {𝑣𝑉𝑣𝐼𝑒})
1311, 12dmmpti 6694 . . . . . . . . 9 dom (𝑒𝐿 ↦ {𝑣𝑉𝑣𝐼𝑒}) = 𝐿
148, 13eqtrdi 2787 . . . . . . . 8 (𝐸 = (𝑒𝐿 ↦ {𝑣𝑉𝑣𝐼𝑒}) → dom 𝐸 = 𝐿)
15 ssrab2 4077 . . . . . . . . . . . . 13 {𝑣𝑉𝑣𝐼𝑒} ⊆ 𝑉
1615a1i 11 . . . . . . . . . . . 12 (((𝐺𝑊𝐼 ⊆ (𝑉 × 𝐿) ∧ ran 𝐼 = 𝐿) ∧ 𝑒𝐿) → {𝑣𝑉𝑣𝐼𝑒} ⊆ 𝑉)
1711elpw 4606 . . . . . . . . . . . 12 ({𝑣𝑉𝑣𝐼𝑒} ∈ 𝒫 𝑉 ↔ {𝑣𝑉𝑣𝐼𝑒} ⊆ 𝑉)
1816, 17sylibr 233 . . . . . . . . . . 11 (((𝐺𝑊𝐼 ⊆ (𝑉 × 𝐿) ∧ ran 𝐼 = 𝐿) ∧ 𝑒𝐿) → {𝑣𝑉𝑣𝐼𝑒} ∈ 𝒫 𝑉)
19 eleq2 2821 . . . . . . . . . . . . . . . 16 (ran 𝐼 = 𝐿 → (𝑒 ∈ ran 𝐼𝑒𝐿))
20193ad2ant3 1134 . . . . . . . . . . . . . . 15 ((𝐺𝑊𝐼 ⊆ (𝑉 × 𝐿) ∧ ran 𝐼 = 𝐿) → (𝑒 ∈ ran 𝐼𝑒𝐿))
21 ssrelrn 5894 . . . . . . . . . . . . . . . . 17 ((𝐼 ⊆ (𝑉 × 𝐿) ∧ 𝑒 ∈ ran 𝐼) → ∃𝑣𝑉 𝑣𝐼𝑒)
2221ex 412 . . . . . . . . . . . . . . . 16 (𝐼 ⊆ (𝑉 × 𝐿) → (𝑒 ∈ ran 𝐼 → ∃𝑣𝑉 𝑣𝐼𝑒))
23223ad2ant2 1133 . . . . . . . . . . . . . . 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 406 . . . . . . . . . . . . 13 (((𝐺𝑊𝐼 ⊆ (𝑉 × 𝐿) ∧ ran 𝐼 = 𝐿) ∧ 𝑒𝐿) → ∃𝑣𝑉 𝑣𝐼𝑒)
26 df-ne 2940 . . . . . . . . . . . . . 14 ({𝑣𝑉𝑣𝐼𝑒} ≠ ∅ ↔ ¬ {𝑣𝑉𝑣𝐼𝑒} = ∅)
27 rabn0 4385 . . . . . . . . . . . . . 14 ({𝑣𝑉𝑣𝐼𝑒} ≠ ∅ ↔ ∃𝑣𝑉 𝑣𝐼𝑒)
2826, 27bitr3i 277 . . . . . . . . . . . . 13 (¬ {𝑣𝑉𝑣𝐼𝑒} = ∅ ↔ ∃𝑣𝑉 𝑣𝐼𝑒)
2925, 28sylibr 233 . . . . . . . . . . . 12 (((𝐺𝑊𝐼 ⊆ (𝑉 × 𝐿) ∧ ran 𝐼 = 𝐿) ∧ 𝑒𝐿) → ¬ {𝑣𝑉𝑣𝐼𝑒} = ∅)
3011elsn 4643 . . . . . . . . . . . 12 ({𝑣𝑉𝑣𝐼𝑒} ∈ {∅} ↔ {𝑣𝑉𝑣𝐼𝑒} = ∅)
3129, 30sylnibr 329 . . . . . . . . . . 11 (((𝐺𝑊𝐼 ⊆ (𝑉 × 𝐿) ∧ ran 𝐼 = 𝐿) ∧ 𝑒𝐿) → ¬ {𝑣𝑉𝑣𝐼𝑒} ∈ {∅})
3218, 31eldifd 3959 . . . . . . . . . 10 (((𝐺𝑊𝐼 ⊆ (𝑉 × 𝐿) ∧ ran 𝐼 = 𝐿) ∧ 𝑒𝐿) → {𝑣𝑉𝑣𝐼𝑒} ∈ (𝒫 𝑉 ∖ {∅}))
3332fmpttd 7116 . . . . . . . . 9 ((𝐺𝑊𝐼 ⊆ (𝑉 × 𝐿) ∧ ran 𝐼 = 𝐿) → (𝑒𝐿 ↦ {𝑣𝑉𝑣𝐼𝑒}):𝐿⟶(𝒫 𝑉 ∖ {∅}))
34 simpl 482 . . . . . . . . . 10 ((𝐸 = (𝑒𝐿 ↦ {𝑣𝑉𝑣𝐼𝑒}) ∧ dom 𝐸 = 𝐿) → 𝐸 = (𝑒𝐿 ↦ {𝑣𝑉𝑣𝐼𝑒}))
35 simpr 484 . . . . . . . . . 10 ((𝐸 = (𝑒𝐿 ↦ {𝑣𝑉𝑣𝐼𝑒}) ∧ dom 𝐸 = 𝐿) → dom 𝐸 = 𝐿)
3634, 35feq12d 6705 . . . . . . . . 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 684 . . . . . . 7 (𝐸 = (𝑒𝐿 ↦ {𝑣𝑉𝑣𝐼𝑒}) → ((𝐺𝑊𝐼 ⊆ (𝑉 × 𝐿) ∧ ran 𝐼 = 𝐿) → 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅})))
3938imp 406 . . . . . 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 452 . . . 4 ((𝑉 = 𝑃𝐸 = (𝑒𝐿 ↦ {𝑣𝑃𝑣𝐼𝑒})) → ((𝐺𝑊𝐼 ⊆ (𝑃 × 𝐿) ∧ ran 𝐼 = 𝐿) → 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅})))
4241impcom 407 . . 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 28588 . . . . 5 (𝐺𝑊 → (𝐺 ∈ UHGraph ↔ 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅})))
45443ad2ant1 1132 . . . 4 ((𝐺𝑊𝐼 ⊆ (𝑃 × 𝐿) ∧ ran 𝐼 = 𝐿) → (𝐺 ∈ UHGraph ↔ 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅})))
4645adantr 480 . . 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 412 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 395  w3a 1086   = wceq 1540  wcel 2105  wne 2939  wrex 3069  {crab 3431  cdif 3945  wss 3948  c0 4322  𝒫 cpw 4602  {csn 4628   class class class wbr 5148  cmpt 5231   × cxp 5674  dom cdm 5676  ran crn 5677  wf 6539  cfv 6543  Vtxcvtx 28524  iEdgciedg 28525  UHGraphcuhgr 28584
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551  df-uhgr 28586
This theorem is referenced by: (None)
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