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Theorem pliguhgr 27954
Description: Any planar incidence geometry 𝐺 can be regarded as a hypergraph with its points as vertices and its lines as edges. See incistruhgr 26547 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 6520 . . . 4 ( I ↾ 𝐺):𝐺1-1-onto𝐺
2 f1of 6483 . . . 4 (( I ↾ 𝐺):𝐺1-1-onto𝐺 → ( I ↾ 𝐺):𝐺𝐺)
3 pwuni 4781 . . . . . . 7 𝐺 ⊆ 𝒫 𝐺
4 n0lplig 27951 . . . . . . . . . 10 (𝐺 ∈ Plig → ¬ ∅ ∈ 𝐺)
54adantr 481 . . . . . . . . 9 ((𝐺 ∈ Plig ∧ 𝐺 ⊆ 𝒫 𝐺) → ¬ ∅ ∈ 𝐺)
6 disjsn 4554 . . . . . . . . 9 ((𝐺 ∩ {∅}) = ∅ ↔ ¬ ∅ ∈ 𝐺)
75, 6sylibr 235 . . . . . . . 8 ((𝐺 ∈ Plig ∧ 𝐺 ⊆ 𝒫 𝐺) → (𝐺 ∩ {∅}) = ∅)
8 reldisj 4316 . . . . . . . . 9 (𝐺 ⊆ 𝒫 𝐺 → ((𝐺 ∩ {∅}) = ∅ ↔ 𝐺 ⊆ (𝒫 𝐺 ∖ {∅})))
98adantl 482 . . . . . . . 8 ((𝐺 ∈ Plig ∧ 𝐺 ⊆ 𝒫 𝐺) → ((𝐺 ∩ {∅}) = ∅ ↔ 𝐺 ⊆ (𝒫 𝐺 ∖ {∅})))
107, 9mpbid 233 . . . . . . 7 ((𝐺 ∈ Plig ∧ 𝐺 ⊆ 𝒫 𝐺) → 𝐺 ⊆ (𝒫 𝐺 ∖ {∅}))
113, 10mpan2 687 . . . . . 6 (𝐺 ∈ Plig → 𝐺 ⊆ (𝒫 𝐺 ∖ {∅}))
12 fss 6395 . . . . . 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 413 . . . 4 (( I ↾ 𝐺):𝐺𝐺 → (𝐺 ∈ Plig → ( I ↾ 𝐺):𝐺⟶(𝒫 𝐺 ∖ {∅})))
151, 2, 14mp2b 10 . . 3 (𝐺 ∈ Plig → ( I ↾ 𝐺):𝐺⟶(𝒫 𝐺 ∖ {∅}))
1615ffdmd 6405 . 2 (𝐺 ∈ Plig → ( I ↾ 𝐺):dom ( I ↾ 𝐺)⟶(𝒫 𝐺 ∖ {∅}))
17 uniexg 7325 . . 3 (𝐺 ∈ Plig → 𝐺 ∈ V)
18 resiexg 7475 . . 3 (𝐺 ∈ Plig → ( I ↾ 𝐺) ∈ V)
19 isuhgrop 26538 . . 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 258 1 (𝐺 ∈ Plig → ⟨ 𝐺, ( I ↾ 𝐺)⟩ ∈ UHGraph)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396   = wceq 1522  wcel 2081  Vcvv 3437  cdif 3856  cin 3858  wss 3859  c0 4211  𝒫 cpw 4453  {csn 4472  cop 4478   cuni 4745   I cid 5347  dom cdm 5443  cres 5445  wf 6221  1-1-ontowf1o 6224  UHGraphcuhgr 26524  Pligcplig 27942
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-ral 3110  df-rex 3111  df-reu 3112  df-rab 3114  df-v 3439  df-sbc 3707  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-op 4479  df-uni 4746  df-br 4963  df-opab 5025  df-mpt 5042  df-id 5348  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-f1 6230  df-fo 6231  df-f1o 6232  df-fv 6233  df-1st 7545  df-2nd 7546  df-vtx 26466  df-iedg 26467  df-uhgr 26526  df-plig 27943
This theorem is referenced by: (None)
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