Theorem pliguhgr 30518

Description: Any planar incidence geometry 𝐺 can be regarded as a hypergraph with its points as vertices and its lines as edges. See incistruhgr 29114 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

Step	Hyp	Ref	Expression
1		f1oi 6900	. . . 4 ⊢ ( I ↾ 𝐺):𝐺–1-1-onto→𝐺
2		f1of 6862	. . . 4 ⊢ (( I ↾ 𝐺):𝐺–1-1-onto→𝐺 → ( I ↾ 𝐺):𝐺⟶𝐺)
3		pwuni 4969	. . . . . . 7 ⊢ 𝐺 ⊆ 𝒫 ∪ 𝐺
4		n0lplig 30515	. . . . . . . . . 10 ⊢ (𝐺 ∈ Plig → ¬ ∅ ∈ 𝐺)
5	4	adantr 480	. . . . . . . . 9 ⊢ ((𝐺 ∈ Plig ∧ 𝐺 ⊆ 𝒫 ∪ 𝐺) → ¬ ∅ ∈ 𝐺)
6		disjsn 4736	. . . . . . . . 9 ⊢ ((𝐺 ∩ {∅}) = ∅ ↔ ¬ ∅ ∈ 𝐺)
7	5, 6	sylibr 234	. . . . . . . 8 ⊢ ((𝐺 ∈ Plig ∧ 𝐺 ⊆ 𝒫 ∪ 𝐺) → (𝐺 ∩ {∅}) = ∅)
8		reldisj 4476	. . . . . . . . 9 ⊢ (𝐺 ⊆ 𝒫 ∪ 𝐺 → ((𝐺 ∩ {∅}) = ∅ ↔ 𝐺 ⊆ (𝒫 ∪ 𝐺 ∖ {∅})))
9	8	adantl 481	. . . . . . . 8 ⊢ ((𝐺 ∈ Plig ∧ 𝐺 ⊆ 𝒫 ∪ 𝐺) → ((𝐺 ∩ {∅}) = ∅ ↔ 𝐺 ⊆ (𝒫 ∪ 𝐺 ∖ {∅})))
10	7, 9	mpbid 232	. . . . . . 7 ⊢ ((𝐺 ∈ Plig ∧ 𝐺 ⊆ 𝒫 ∪ 𝐺) → 𝐺 ⊆ (𝒫 ∪ 𝐺 ∖ {∅}))
11	3, 10	mpan2 690	. . . . . 6 ⊢ (𝐺 ∈ Plig → 𝐺 ⊆ (𝒫 ∪ 𝐺 ∖ {∅}))
12		fss 6763	. . . . . 6 ⊢ ((( I ↾ 𝐺):𝐺⟶𝐺 ∧ 𝐺 ⊆ (𝒫 ∪ 𝐺 ∖ {∅})) → ( I ↾ 𝐺):𝐺⟶(𝒫 ∪ 𝐺 ∖ {∅}))
13	11, 12	sylan2 592	. . . . 5 ⊢ ((( I ↾ 𝐺):𝐺⟶𝐺 ∧ 𝐺 ∈ Plig) → ( I ↾ 𝐺):𝐺⟶(𝒫 ∪ 𝐺 ∖ {∅}))
14	13	ex 412	. . . 4 ⊢ (( I ↾ 𝐺):𝐺⟶𝐺 → (𝐺 ∈ Plig → ( I ↾ 𝐺):𝐺⟶(𝒫 ∪ 𝐺 ∖ {∅})))
15	1, 2, 14	mp2b 10	. . 3 ⊢ (𝐺 ∈ Plig → ( I ↾ 𝐺):𝐺⟶(𝒫 ∪ 𝐺 ∖ {∅}))
16	15	ffdmd 6778	. 2 ⊢ (𝐺 ∈ Plig → ( I ↾ 𝐺):dom ( I ↾ 𝐺)⟶(𝒫 ∪ 𝐺 ∖ {∅}))
17		uniexg 7775	. . 3 ⊢ (𝐺 ∈ Plig → ∪ 𝐺 ∈ V)
18		resiexg 7952	. . 3 ⊢ (𝐺 ∈ Plig → ( I ↾ 𝐺) ∈ V)
19		isuhgrop 29105	. . 3 ⊢ ((∪ 𝐺 ∈ V ∧ ( I ↾ 𝐺) ∈ V) → (⟨∪ 𝐺, ( I ↾ 𝐺)⟩ ∈ UHGraph ↔ ( I ↾ 𝐺):dom ( I ↾ 𝐺)⟶(𝒫 ∪ 𝐺 ∖ {∅})))
20	17, 18, 19	syl2anc 583	. 2 ⊢ (𝐺 ∈ Plig → (⟨∪ 𝐺, ( I ↾ 𝐺)⟩ ∈ UHGraph ↔ ( I ↾ 𝐺):dom ( I ↾ 𝐺)⟶(𝒫 ∪ 𝐺 ∖ {∅})))
21	16, 20	mpbird 257	1 ⊢ (𝐺 ∈ Plig → ⟨∪ 𝐺, ( I ↾ 𝐺)⟩ ∈ UHGraph)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2108  Vcvv 3488   ∖ cdif 3973   ∩ cin 3975   ⊆ wss 3976  ∅c0 4352  𝒫 cpw 4622  {csn 4648  ⟨cop 4654  ∪ cuni 4931   I cid 5592  dom cdm 5700   ↾ cres 5702  ⟶wf 6569  –1-1-onto→wf1o 6572  UHGraphcuhgr 29091  Pligcplig 30506

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-1st 8030  df-2nd 8031  df-vtx 29033  df-iedg 29034  df-uhgr 29093  df-plig 30507

This theorem is referenced by: (None)