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Mirrors > Home > MPE Home > Th. List > pliguhgr | Structured version Visualization version GIF version |
Description: Any planar incidence geometry 𝐺 can be regarded as a hypergraph with its points as vertices and its lines as edges. See incistruhgr 26858 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.) |
Ref | Expression |
---|---|
pliguhgr | ⊢ (𝐺 ∈ Plig → 〈∪ 𝐺, ( I ↾ 𝐺)〉 ∈ UHGraph) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f1oi 6646 | . . . 4 ⊢ ( I ↾ 𝐺):𝐺–1-1-onto→𝐺 | |
2 | f1of 6609 | . . . 4 ⊢ (( I ↾ 𝐺):𝐺–1-1-onto→𝐺 → ( I ↾ 𝐺):𝐺⟶𝐺) | |
3 | pwuni 4867 | . . . . . . 7 ⊢ 𝐺 ⊆ 𝒫 ∪ 𝐺 | |
4 | n0lplig 28254 | . . . . . . . . . 10 ⊢ (𝐺 ∈ Plig → ¬ ∅ ∈ 𝐺) | |
5 | 4 | adantr 483 | . . . . . . . . 9 ⊢ ((𝐺 ∈ Plig ∧ 𝐺 ⊆ 𝒫 ∪ 𝐺) → ¬ ∅ ∈ 𝐺) |
6 | disjsn 4640 | . . . . . . . . 9 ⊢ ((𝐺 ∩ {∅}) = ∅ ↔ ¬ ∅ ∈ 𝐺) | |
7 | 5, 6 | sylibr 236 | . . . . . . . 8 ⊢ ((𝐺 ∈ Plig ∧ 𝐺 ⊆ 𝒫 ∪ 𝐺) → (𝐺 ∩ {∅}) = ∅) |
8 | reldisj 4401 | . . . . . . . . 9 ⊢ (𝐺 ⊆ 𝒫 ∪ 𝐺 → ((𝐺 ∩ {∅}) = ∅ ↔ 𝐺 ⊆ (𝒫 ∪ 𝐺 ∖ {∅}))) | |
9 | 8 | adantl 484 | . . . . . . . 8 ⊢ ((𝐺 ∈ Plig ∧ 𝐺 ⊆ 𝒫 ∪ 𝐺) → ((𝐺 ∩ {∅}) = ∅ ↔ 𝐺 ⊆ (𝒫 ∪ 𝐺 ∖ {∅}))) |
10 | 7, 9 | mpbid 234 | . . . . . . 7 ⊢ ((𝐺 ∈ Plig ∧ 𝐺 ⊆ 𝒫 ∪ 𝐺) → 𝐺 ⊆ (𝒫 ∪ 𝐺 ∖ {∅})) |
11 | 3, 10 | mpan2 689 | . . . . . 6 ⊢ (𝐺 ∈ Plig → 𝐺 ⊆ (𝒫 ∪ 𝐺 ∖ {∅})) |
12 | fss 6521 | . . . . . 6 ⊢ ((( I ↾ 𝐺):𝐺⟶𝐺 ∧ 𝐺 ⊆ (𝒫 ∪ 𝐺 ∖ {∅})) → ( I ↾ 𝐺):𝐺⟶(𝒫 ∪ 𝐺 ∖ {∅})) | |
13 | 11, 12 | sylan2 594 | . . . . 5 ⊢ ((( I ↾ 𝐺):𝐺⟶𝐺 ∧ 𝐺 ∈ Plig) → ( I ↾ 𝐺):𝐺⟶(𝒫 ∪ 𝐺 ∖ {∅})) |
14 | 13 | ex 415 | . . . 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 6531 | . 2 ⊢ (𝐺 ∈ Plig → ( I ↾ 𝐺):dom ( I ↾ 𝐺)⟶(𝒫 ∪ 𝐺 ∖ {∅})) |
17 | uniexg 7460 | . . 3 ⊢ (𝐺 ∈ Plig → ∪ 𝐺 ∈ V) | |
18 | resiexg 7613 | . . 3 ⊢ (𝐺 ∈ Plig → ( I ↾ 𝐺) ∈ V) | |
19 | isuhgrop 26849 | . . 3 ⊢ ((∪ 𝐺 ∈ V ∧ ( I ↾ 𝐺) ∈ V) → (〈∪ 𝐺, ( I ↾ 𝐺)〉 ∈ UHGraph ↔ ( I ↾ 𝐺):dom ( I ↾ 𝐺)⟶(𝒫 ∪ 𝐺 ∖ {∅}))) | |
20 | 17, 18, 19 | syl2anc 586 | . 2 ⊢ (𝐺 ∈ Plig → (〈∪ 𝐺, ( I ↾ 𝐺)〉 ∈ UHGraph ↔ ( I ↾ 𝐺):dom ( I ↾ 𝐺)⟶(𝒫 ∪ 𝐺 ∖ {∅}))) |
21 | 16, 20 | mpbird 259 | 1 ⊢ (𝐺 ∈ Plig → 〈∪ 𝐺, ( I ↾ 𝐺)〉 ∈ UHGraph) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 Vcvv 3494 ∖ cdif 3932 ∩ cin 3934 ⊆ wss 3935 ∅c0 4290 𝒫 cpw 4538 {csn 4560 〈cop 4566 ∪ cuni 4831 I cid 5453 dom cdm 5549 ↾ cres 5551 ⟶wf 6345 –1-1-onto→wf1o 6348 UHGraphcuhgr 26835 Pligcplig 28245 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-1st 7683 df-2nd 7684 df-vtx 26777 df-iedg 26778 df-uhgr 26837 df-plig 28246 |
This theorem is referenced by: (None) |
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