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Theorem pliguhgr 27570
Description: Any planar incidence geometry 𝐺 can be regarded as a hypergraph with its points as vertices and its lines as edges. See incistruhgr 26094 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 6287 . . . 4 ( I ↾ 𝐺):𝐺1-1-onto𝐺
2 f1of 6250 . . . 4 (( I ↾ 𝐺):𝐺1-1-onto𝐺 → ( I ↾ 𝐺):𝐺𝐺)
3 pwuni 4582 . . . . . . 7 𝐺 ⊆ 𝒫 𝐺
4 n0lplig 27567 . . . . . . . . . 10 (𝐺 ∈ Plig → ¬ ∅ ∈ 𝐺)
54adantr 472 . . . . . . . . 9 ((𝐺 ∈ Plig ∧ 𝐺 ⊆ 𝒫 𝐺) → ¬ ∅ ∈ 𝐺)
6 disjsn 4353 . . . . . . . . 9 ((𝐺 ∩ {∅}) = ∅ ↔ ¬ ∅ ∈ 𝐺)
75, 6sylibr 224 . . . . . . . 8 ((𝐺 ∈ Plig ∧ 𝐺 ⊆ 𝒫 𝐺) → (𝐺 ∩ {∅}) = ∅)
8 reldisj 4128 . . . . . . . . 9 (𝐺 ⊆ 𝒫 𝐺 → ((𝐺 ∩ {∅}) = ∅ ↔ 𝐺 ⊆ (𝒫 𝐺 ∖ {∅})))
98adantl 473 . . . . . . . 8 ((𝐺 ∈ Plig ∧ 𝐺 ⊆ 𝒫 𝐺) → ((𝐺 ∩ {∅}) = ∅ ↔ 𝐺 ⊆ (𝒫 𝐺 ∖ {∅})))
107, 9mpbid 222 . . . . . . 7 ((𝐺 ∈ Plig ∧ 𝐺 ⊆ 𝒫 𝐺) → 𝐺 ⊆ (𝒫 𝐺 ∖ {∅}))
113, 10mpan2 709 . . . . . 6 (𝐺 ∈ Plig → 𝐺 ⊆ (𝒫 𝐺 ∖ {∅}))
12 fss 6169 . . . . . 6 ((( I ↾ 𝐺):𝐺𝐺𝐺 ⊆ (𝒫 𝐺 ∖ {∅})) → ( I ↾ 𝐺):𝐺⟶(𝒫 𝐺 ∖ {∅}))
1311, 12sylan2 492 . . . . 5 ((( I ↾ 𝐺):𝐺𝐺𝐺 ∈ Plig) → ( I ↾ 𝐺):𝐺⟶(𝒫 𝐺 ∖ {∅}))
1413ex 449 . . . 4 (( I ↾ 𝐺):𝐺𝐺 → (𝐺 ∈ Plig → ( I ↾ 𝐺):𝐺⟶(𝒫 𝐺 ∖ {∅})))
151, 2, 14mp2b 10 . . 3 (𝐺 ∈ Plig → ( I ↾ 𝐺):𝐺⟶(𝒫 𝐺 ∖ {∅}))
1615ffdmd 6176 . 2 (𝐺 ∈ Plig → ( I ↾ 𝐺):dom ( I ↾ 𝐺)⟶(𝒫 𝐺 ∖ {∅}))
17 uniexg 7072 . . 3 (𝐺 ∈ Plig → 𝐺 ∈ V)
18 resiexg 7219 . . 3 (𝐺 ∈ Plig → ( I ↾ 𝐺) ∈ V)
19 isuhgrop 26085 . . 3 (( 𝐺 ∈ V ∧ ( I ↾ 𝐺) ∈ V) → (⟨ 𝐺, ( I ↾ 𝐺)⟩ ∈ UHGraph ↔ ( I ↾ 𝐺):dom ( I ↾ 𝐺)⟶(𝒫 𝐺 ∖ {∅})))
2017, 18, 19syl2anc 696 . 2 (𝐺 ∈ Plig → (⟨ 𝐺, ( I ↾ 𝐺)⟩ ∈ UHGraph ↔ ( I ↾ 𝐺):dom ( I ↾ 𝐺)⟶(𝒫 𝐺 ∖ {∅})))
2116, 20mpbird 247 1 (𝐺 ∈ Plig → ⟨ 𝐺, ( I ↾ 𝐺)⟩ ∈ UHGraph)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383   = wceq 1596  wcel 2103  Vcvv 3304  cdif 3677  cin 3679  wss 3680  c0 4023  𝒫 cpw 4266  {csn 4285  cop 4291   cuni 4544   I cid 5127  dom cdm 5218  cres 5220  wf 5997  1-1-ontowf1o 6000  UHGraphcuhgr 26071  Pligcplig 27558
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-sep 4889  ax-nul 4897  ax-pow 4948  ax-pr 5011  ax-un 7066  ax-reg 8613
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-ral 3019  df-rex 3020  df-reu 3021  df-rab 3023  df-v 3306  df-sbc 3542  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-nul 4024  df-if 4195  df-pw 4268  df-sn 4286  df-pr 4288  df-op 4292  df-uni 4545  df-br 4761  df-opab 4821  df-mpt 4838  df-id 5128  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-res 5230  df-ima 5231  df-iota 5964  df-fun 6003  df-fn 6004  df-f 6005  df-f1 6006  df-fo 6007  df-f1o 6008  df-fv 6009  df-1st 7285  df-2nd 7286  df-vtx 25996  df-iedg 25997  df-uhgr 26073  df-plig 27559
This theorem is referenced by: (None)
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