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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 28072 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 6823 | . . . 4 ⊢ ( I ↾ 𝐺):𝐺–1-1-onto→𝐺 | |
2 | f1of 6785 | . . . 4 ⊢ (( I ↾ 𝐺):𝐺–1-1-onto→𝐺 → ( I ↾ 𝐺):𝐺⟶𝐺) | |
3 | pwuni 4907 | . . . . . . 7 ⊢ 𝐺 ⊆ 𝒫 ∪ 𝐺 | |
4 | n0lplig 29467 | . . . . . . . . . 10 ⊢ (𝐺 ∈ Plig → ¬ ∅ ∈ 𝐺) | |
5 | 4 | adantr 482 | . . . . . . . . 9 ⊢ ((𝐺 ∈ Plig ∧ 𝐺 ⊆ 𝒫 ∪ 𝐺) → ¬ ∅ ∈ 𝐺) |
6 | disjsn 4673 | . . . . . . . . 9 ⊢ ((𝐺 ∩ {∅}) = ∅ ↔ ¬ ∅ ∈ 𝐺) | |
7 | 5, 6 | sylibr 233 | . . . . . . . 8 ⊢ ((𝐺 ∈ Plig ∧ 𝐺 ⊆ 𝒫 ∪ 𝐺) → (𝐺 ∩ {∅}) = ∅) |
8 | reldisj 4412 | . . . . . . . . 9 ⊢ (𝐺 ⊆ 𝒫 ∪ 𝐺 → ((𝐺 ∩ {∅}) = ∅ ↔ 𝐺 ⊆ (𝒫 ∪ 𝐺 ∖ {∅}))) | |
9 | 8 | adantl 483 | . . . . . . . 8 ⊢ ((𝐺 ∈ Plig ∧ 𝐺 ⊆ 𝒫 ∪ 𝐺) → ((𝐺 ∩ {∅}) = ∅ ↔ 𝐺 ⊆ (𝒫 ∪ 𝐺 ∖ {∅}))) |
10 | 7, 9 | mpbid 231 | . . . . . . 7 ⊢ ((𝐺 ∈ Plig ∧ 𝐺 ⊆ 𝒫 ∪ 𝐺) → 𝐺 ⊆ (𝒫 ∪ 𝐺 ∖ {∅})) |
11 | 3, 10 | mpan2 690 | . . . . . 6 ⊢ (𝐺 ∈ Plig → 𝐺 ⊆ (𝒫 ∪ 𝐺 ∖ {∅})) |
12 | fss 6686 | . . . . . 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 414 | . . . 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 6700 | . 2 ⊢ (𝐺 ∈ Plig → ( I ↾ 𝐺):dom ( I ↾ 𝐺)⟶(𝒫 ∪ 𝐺 ∖ {∅})) |
17 | uniexg 7678 | . . 3 ⊢ (𝐺 ∈ Plig → ∪ 𝐺 ∈ V) | |
18 | resiexg 7852 | . . 3 ⊢ (𝐺 ∈ Plig → ( I ↾ 𝐺) ∈ V) | |
19 | isuhgrop 28063 | . . 3 ⊢ ((∪ 𝐺 ∈ V ∧ ( I ↾ 𝐺) ∈ V) → (⟨∪ 𝐺, ( I ↾ 𝐺)⟩ ∈ UHGraph ↔ ( I ↾ 𝐺):dom ( I ↾ 𝐺)⟶(𝒫 ∪ 𝐺 ∖ {∅}))) | |
20 | 17, 18, 19 | syl2anc 585 | . 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) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 Vcvv 3444 ∖ cdif 3908 ∩ cin 3910 ⊆ wss 3911 ∅c0 4283 𝒫 cpw 4561 {csn 4587 ⟨cop 4593 ∪ cuni 4866 I cid 5531 dom cdm 5634 ↾ cres 5636 ⟶wf 6493 –1-1-onto→wf1o 6496 UHGraphcuhgr 28049 Pligcplig 29458 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-1st 7922 df-2nd 7923 df-vtx 27991 df-iedg 27992 df-uhgr 28051 df-plig 29459 |
This theorem is referenced by: (None) |
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