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

Theorem pliguhgr 30007
Description: Any planar incidence geometry 𝐺 can be regarded as a hypergraph with its points as vertices and its lines as edges. See incistruhgr 28607 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 6871 . . . 4 ( I ↾ 𝐺):𝐺1-1-onto𝐺
2 f1of 6833 . . . 4 (( I ↾ 𝐺):𝐺1-1-onto𝐺 → ( I ↾ 𝐺):𝐺𝐺)
3 pwuni 4949 . . . . . . 7 𝐺 ⊆ 𝒫 𝐺
4 n0lplig 30004 . . . . . . . . . 10 (𝐺 ∈ Plig → ¬ ∅ ∈ 𝐺)
54adantr 480 . . . . . . . . 9 ((𝐺 ∈ Plig ∧ 𝐺 ⊆ 𝒫 𝐺) → ¬ ∅ ∈ 𝐺)
6 disjsn 4715 . . . . . . . . 9 ((𝐺 ∩ {∅}) = ∅ ↔ ¬ ∅ ∈ 𝐺)
75, 6sylibr 233 . . . . . . . 8 ((𝐺 ∈ Plig ∧ 𝐺 ⊆ 𝒫 𝐺) → (𝐺 ∩ {∅}) = ∅)
8 reldisj 4451 . . . . . . . . 9 (𝐺 ⊆ 𝒫 𝐺 → ((𝐺 ∩ {∅}) = ∅ ↔ 𝐺 ⊆ (𝒫 𝐺 ∖ {∅})))
98adantl 481 . . . . . . . 8 ((𝐺 ∈ Plig ∧ 𝐺 ⊆ 𝒫 𝐺) → ((𝐺 ∩ {∅}) = ∅ ↔ 𝐺 ⊆ (𝒫 𝐺 ∖ {∅})))
107, 9mpbid 231 . . . . . . 7 ((𝐺 ∈ Plig ∧ 𝐺 ⊆ 𝒫 𝐺) → 𝐺 ⊆ (𝒫 𝐺 ∖ {∅}))
113, 10mpan2 688 . . . . . 6 (𝐺 ∈ Plig → 𝐺 ⊆ (𝒫 𝐺 ∖ {∅}))
12 fss 6734 . . . . . 6 ((( I ↾ 𝐺):𝐺𝐺𝐺 ⊆ (𝒫 𝐺 ∖ {∅})) → ( I ↾ 𝐺):𝐺⟶(𝒫 𝐺 ∖ {∅}))
1311, 12sylan2 592 . . . . 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 6748 . 2 (𝐺 ∈ Plig → ( I ↾ 𝐺):dom ( I ↾ 𝐺)⟶(𝒫 𝐺 ∖ {∅}))
17 uniexg 7734 . . 3 (𝐺 ∈ Plig → 𝐺 ∈ V)
18 resiexg 7909 . . 3 (𝐺 ∈ Plig → ( I ↾ 𝐺) ∈ V)
19 isuhgrop 28598 . . 3 (( 𝐺 ∈ V ∧ ( I ↾ 𝐺) ∈ V) → (⟨ 𝐺, ( I ↾ 𝐺)⟩ ∈ UHGraph ↔ ( I ↾ 𝐺):dom ( I ↾ 𝐺)⟶(𝒫 𝐺 ∖ {∅})))
2017, 18, 19syl2anc 583 . 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 205  wa 395   = wceq 1540  wcel 2105  Vcvv 3473  cdif 3945  cin 3947  wss 3948  c0 4322  𝒫 cpw 4602  {csn 4628  cop 4634   cuni 4908   I cid 5573  dom cdm 5676  cres 5678  wf 6539  1-1-ontowf1o 6542  UHGraphcuhgr 28584  Pligcplig 29995
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-pow 5363  ax-pr 5427  ax-un 7729
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-rmo 3375  df-reu 3376  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-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-1st 7979  df-2nd 7980  df-vtx 28526  df-iedg 28527  df-uhgr 28586  df-plig 29996
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator