Theorem pliguhgr 30575

Description: Any planar incidence geometry 𝐺 can be regarded as a hypergraph with its points as vertices and its lines as edges. See incistruhgr 29166 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 6805	. . . 4 ⊢ ( I ↾ 𝐺):𝐺–1-1-onto→𝐺
2		f1of 6767	. . . 4 ⊢ (( I ↾ 𝐺):𝐺–1-1-onto→𝐺 → ( I ↾ 𝐺):𝐺⟶𝐺)
3		pwuni 4876	. . . . . . 7 ⊢ 𝐺 ⊆ 𝒫 ∪ 𝐺
4		n0lplig 30572	. . . . . . . . . 10 ⊢ (𝐺 ∈ Plig → ¬ ∅ ∈ 𝐺)
5	4	adantr 481	. . . . . . . . 9 ⊢ ((𝐺 ∈ Plig ∧ 𝐺 ⊆ 𝒫 ∪ 𝐺) → ¬ ∅ ∈ 𝐺)
6		disjsn 4643	. . . . . . . . 9 ⊢ ((𝐺 ∩ {∅}) = ∅ ↔ ¬ ∅ ∈ 𝐺)
7	5, 6	sylibr 235	. . . . . . . 8 ⊢ ((𝐺 ∈ Plig ∧ 𝐺 ⊆ 𝒫 ∪ 𝐺) → (𝐺 ∩ {∅}) = ∅)
8		reldisj 4381	. . . . . . . . 9 ⊢ (𝐺 ⊆ 𝒫 ∪ 𝐺 → ((𝐺 ∩ {∅}) = ∅ ↔ 𝐺 ⊆ (𝒫 ∪ 𝐺 ∖ {∅})))
9	8	adantl 482	. . . . . . . 8 ⊢ ((𝐺 ∈ Plig ∧ 𝐺 ⊆ 𝒫 ∪ 𝐺) → ((𝐺 ∩ {∅}) = ∅ ↔ 𝐺 ⊆ (𝒫 ∪ 𝐺 ∖ {∅})))
10	7, 9	mpbid 233	. . . . . . 7 ⊢ ((𝐺 ∈ Plig ∧ 𝐺 ⊆ 𝒫 ∪ 𝐺) → 𝐺 ⊆ (𝒫 ∪ 𝐺 ∖ {∅}))
11	3, 10	mpan2 697	. . . . . 6 ⊢ (𝐺 ∈ Plig → 𝐺 ⊆ (𝒫 ∪ 𝐺 ∖ {∅}))
12		fss 6671	. . . . . 6 ⊢ ((( I ↾ 𝐺):𝐺⟶𝐺 ∧ 𝐺 ⊆ (𝒫 ∪ 𝐺 ∖ {∅})) → ( I ↾ 𝐺):𝐺⟶(𝒫 ∪ 𝐺 ∖ {∅}))
13	11, 12	sylan2 599	. . . . 5 ⊢ ((( I ↾ 𝐺):𝐺⟶𝐺 ∧ 𝐺 ∈ Plig) → ( I ↾ 𝐺):𝐺⟶(𝒫 ∪ 𝐺 ∖ {∅}))
14	13	ex 413	. . . 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 6685	. 2 ⊢ (𝐺 ∈ Plig → ( I ↾ 𝐺):dom ( I ↾ 𝐺)⟶(𝒫 ∪ 𝐺 ∖ {∅}))
17		uniexg 7683	. . 3 ⊢ (𝐺 ∈ Plig → ∪ 𝐺 ∈ V)
18		resiexg 7852	. . 3 ⊢ (𝐺 ∈ Plig → ( I ↾ 𝐺) ∈ V)
19		isuhgrop 29157	. . 3 ⊢ ((∪ 𝐺 ∈ V ∧ ( I ↾ 𝐺) ∈ V) → (⟨∪ 𝐺, ( I ↾ 𝐺)⟩ ∈ UHGraph ↔ ( I ↾ 𝐺):dom ( I ↾ 𝐺)⟶(𝒫 ∪ 𝐺 ∖ {∅})))
20	17, 18, 19	syl2anc 590	. 2 ⊢ (𝐺 ∈ Plig → (⟨∪ 𝐺, ( I ↾ 𝐺)⟩ ∈ UHGraph ↔ ( I ↾ 𝐺):dom ( I ↾ 𝐺)⟶(𝒫 ∪ 𝐺 ∖ {∅})))
21	16, 20	mpbird 258	1 ⊢ (𝐺 ∈ Plig → ⟨∪ 𝐺, ( I ↾ 𝐺)⟩ ∈ UHGraph)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1547   ∈ wcel 2119  Vcvv 3431   ∖ cdif 3880   ∩ cin 3882   ⊆ wss 3883  ∅c0 4261  𝒫 cpw 4529  {csn 4555  ⟨cop 4561  ∪ cuni 4838   I cid 5512  dom cdm 5618   ↾ cres 5620  ⟶wf 6481  –1-1-onto→wf1o 6484  UHGraphcuhgr 29143  Pligcplig 30563

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-rmo 3344  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-1st 7931  df-2nd 7932  df-vtx 29085  df-iedg 29086  df-uhgr 29145  df-plig 30564

This theorem is referenced by: (None)