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Theorem pliguhgr 27468
Description: Any planar incidence geometry 𝐺 can be regarded as a hypergraph with its points as vertices and its lines as edges. See incistruhgr 26019 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 6212 . . . 4 ( I ↾ 𝐺):𝐺1-1-onto𝐺
2 f1of 6175 . . . 4 (( I ↾ 𝐺):𝐺1-1-onto𝐺 → ( I ↾ 𝐺):𝐺𝐺)
3 pwuni 4506 . . . . . . 7 𝐺 ⊆ 𝒫 𝐺
4 n0lplig 27465 . . . . . . . . . 10 (𝐺 ∈ Plig → ¬ ∅ ∈ 𝐺)
54adantr 480 . . . . . . . . 9 ((𝐺 ∈ Plig ∧ 𝐺 ⊆ 𝒫 𝐺) → ¬ ∅ ∈ 𝐺)
6 disjsn 4278 . . . . . . . . 9 ((𝐺 ∩ {∅}) = ∅ ↔ ¬ ∅ ∈ 𝐺)
75, 6sylibr 224 . . . . . . . 8 ((𝐺 ∈ Plig ∧ 𝐺 ⊆ 𝒫 𝐺) → (𝐺 ∩ {∅}) = ∅)
8 reldisj 4053 . . . . . . . . 9 (𝐺 ⊆ 𝒫 𝐺 → ((𝐺 ∩ {∅}) = ∅ ↔ 𝐺 ⊆ (𝒫 𝐺 ∖ {∅})))
98adantl 481 . . . . . . . 8 ((𝐺 ∈ Plig ∧ 𝐺 ⊆ 𝒫 𝐺) → ((𝐺 ∩ {∅}) = ∅ ↔ 𝐺 ⊆ (𝒫 𝐺 ∖ {∅})))
107, 9mpbid 222 . . . . . . 7 ((𝐺 ∈ Plig ∧ 𝐺 ⊆ 𝒫 𝐺) → 𝐺 ⊆ (𝒫 𝐺 ∖ {∅}))
113, 10mpan2 707 . . . . . 6 (𝐺 ∈ Plig → 𝐺 ⊆ (𝒫 𝐺 ∖ {∅}))
12 fss 6094 . . . . . 6 ((( I ↾ 𝐺):𝐺𝐺𝐺 ⊆ (𝒫 𝐺 ∖ {∅})) → ( I ↾ 𝐺):𝐺⟶(𝒫 𝐺 ∖ {∅}))
1311, 12sylan2 490 . . . . 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 6101 . 2 (𝐺 ∈ Plig → ( I ↾ 𝐺):dom ( I ↾ 𝐺)⟶(𝒫 𝐺 ∖ {∅}))
17 uniexg 6997 . . 3 (𝐺 ∈ Plig → 𝐺 ∈ V)
18 resiexg 7144 . . 3 (𝐺 ∈ Plig → ( I ↾ 𝐺) ∈ V)
19 isuhgrop 26010 . . 3 (( 𝐺 ∈ V ∧ ( I ↾ 𝐺) ∈ V) → (⟨ 𝐺, ( I ↾ 𝐺)⟩ ∈ UHGraph ↔ ( I ↾ 𝐺):dom ( I ↾ 𝐺)⟶(𝒫 𝐺 ∖ {∅})))
2017, 18, 19syl2anc 694 . 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 1523  wcel 2030  Vcvv 3231  cdif 3604  cin 3606  wss 3607  c0 3948  𝒫 cpw 4191  {csn 4210  cop 4216   cuni 4468   I cid 5052  dom cdm 5143  cres 5145  wf 5922  1-1-ontowf1o 5925  UHGraphcuhgr 25996  Pligcplig 27456
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-reg 8538
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-1st 7210  df-2nd 7211  df-vtx 25921  df-iedg 25922  df-uhgr 25998  df-plig 27457
This theorem is referenced by: (None)
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