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Theorem pliguhgr 30505
Description: Any planar incidence geometry 𝐺 can be regarded as a hypergraph with its points as vertices and its lines as edges. See incistruhgr 29096 for a generalization of this case for arbitrary incidence structures (planar incidence geometries are such incidence structures). (Proposed by Gerard Lang, 24-Nov-2021.) (Contributed by AV, 28-Nov-2021.)
Assertion
Ref Expression
pliguhgr (𝐺 ∈ Plig → ⟨ 𝐺, ( I ↾ 𝐺)⟩ ∈ UHGraph)

Proof of Theorem pliguhgr
StepHypRef Expression
1 f1oi 6886 . . . 4 ( I ↾ 𝐺):𝐺1-1-onto𝐺
2 f1of 6848 . . . 4 (( I ↾ 𝐺):𝐺1-1-onto𝐺 → ( I ↾ 𝐺):𝐺𝐺)
3 pwuni 4945 . . . . . . 7 𝐺 ⊆ 𝒫 𝐺
4 n0lplig 30502 . . . . . . . . . 10 (𝐺 ∈ Plig → ¬ ∅ ∈ 𝐺)
54adantr 480 . . . . . . . . 9 ((𝐺 ∈ Plig ∧ 𝐺 ⊆ 𝒫 𝐺) → ¬ ∅ ∈ 𝐺)
6 disjsn 4711 . . . . . . . . 9 ((𝐺 ∩ {∅}) = ∅ ↔ ¬ ∅ ∈ 𝐺)
75, 6sylibr 234 . . . . . . . 8 ((𝐺 ∈ Plig ∧ 𝐺 ⊆ 𝒫 𝐺) → (𝐺 ∩ {∅}) = ∅)
8 reldisj 4453 . . . . . . . . 9 (𝐺 ⊆ 𝒫 𝐺 → ((𝐺 ∩ {∅}) = ∅ ↔ 𝐺 ⊆ (𝒫 𝐺 ∖ {∅})))
98adantl 481 . . . . . . . 8 ((𝐺 ∈ Plig ∧ 𝐺 ⊆ 𝒫 𝐺) → ((𝐺 ∩ {∅}) = ∅ ↔ 𝐺 ⊆ (𝒫 𝐺 ∖ {∅})))
107, 9mpbid 232 . . . . . . 7 ((𝐺 ∈ Plig ∧ 𝐺 ⊆ 𝒫 𝐺) → 𝐺 ⊆ (𝒫 𝐺 ∖ {∅}))
113, 10mpan2 691 . . . . . 6 (𝐺 ∈ Plig → 𝐺 ⊆ (𝒫 𝐺 ∖ {∅}))
12 fss 6752 . . . . . 6 ((( I ↾ 𝐺):𝐺𝐺𝐺 ⊆ (𝒫 𝐺 ∖ {∅})) → ( I ↾ 𝐺):𝐺⟶(𝒫 𝐺 ∖ {∅}))
1311, 12sylan2 593 . . . . 5 ((( I ↾ 𝐺):𝐺𝐺𝐺 ∈ Plig) → ( I ↾ 𝐺):𝐺⟶(𝒫 𝐺 ∖ {∅}))
1413ex 412 . . . 4 (( I ↾ 𝐺):𝐺𝐺 → (𝐺 ∈ Plig → ( I ↾ 𝐺):𝐺⟶(𝒫 𝐺 ∖ {∅})))
151, 2, 14mp2b 10 . . 3 (𝐺 ∈ Plig → ( I ↾ 𝐺):𝐺⟶(𝒫 𝐺 ∖ {∅}))
1615ffdmd 6766 . 2 (𝐺 ∈ Plig → ( I ↾ 𝐺):dom ( I ↾ 𝐺)⟶(𝒫 𝐺 ∖ {∅}))
17 uniexg 7760 . . 3 (𝐺 ∈ Plig → 𝐺 ∈ V)
18 resiexg 7934 . . 3 (𝐺 ∈ Plig → ( I ↾ 𝐺) ∈ V)
19 isuhgrop 29087 . . 3 (( 𝐺 ∈ V ∧ ( I ↾ 𝐺) ∈ V) → (⟨ 𝐺, ( I ↾ 𝐺)⟩ ∈ UHGraph ↔ ( I ↾ 𝐺):dom ( I ↾ 𝐺)⟶(𝒫 𝐺 ∖ {∅})))
2017, 18, 19syl2anc 584 . 2 (𝐺 ∈ Plig → (⟨ 𝐺, ( I ↾ 𝐺)⟩ ∈ UHGraph ↔ ( I ↾ 𝐺):dom ( I ↾ 𝐺)⟶(𝒫 𝐺 ∖ {∅})))
2116, 20mpbird 257 1 (𝐺 ∈ Plig → ⟨ 𝐺, ( I ↾ 𝐺)⟩ ∈ UHGraph)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  Vcvv 3480  cdif 3948  cin 3950  wss 3951  c0 4333  𝒫 cpw 4600  {csn 4626  cop 4632   cuni 4907   I cid 5577  dom cdm 5685  cres 5687  wf 6557  1-1-ontowf1o 6560  UHGraphcuhgr 29073  Pligcplig 30493
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-pow 5365  ax-pr 5432  ax-un 7755
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-rmo 3380  df-reu 3381  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-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-1st 8014  df-2nd 8015  df-vtx 29015  df-iedg 29016  df-uhgr 29075  df-plig 30494
This theorem is referenced by: (None)
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