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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 29096 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 6886 | . . . 4 ⊢ ( I ↾ 𝐺):𝐺–1-1-onto→𝐺 | |
| 2 | f1of 6848 | . . . 4 ⊢ (( I ↾ 𝐺):𝐺–1-1-onto→𝐺 → ( I ↾ 𝐺):𝐺⟶𝐺) | |
| 3 | pwuni 4945 | . . . . . . 7 ⊢ 𝐺 ⊆ 𝒫 ∪ 𝐺 | |
| 4 | n0lplig 30502 | . . . . . . . . . 10 ⊢ (𝐺 ∈ Plig → ¬ ∅ ∈ 𝐺) | |
| 5 | 4 | adantr 480 | . . . . . . . . 9 ⊢ ((𝐺 ∈ Plig ∧ 𝐺 ⊆ 𝒫 ∪ 𝐺) → ¬ ∅ ∈ 𝐺) |
| 6 | disjsn 4711 | . . . . . . . . 9 ⊢ ((𝐺 ∩ {∅}) = ∅ ↔ ¬ ∅ ∈ 𝐺) | |
| 7 | 5, 6 | sylibr 234 | . . . . . . . 8 ⊢ ((𝐺 ∈ Plig ∧ 𝐺 ⊆ 𝒫 ∪ 𝐺) → (𝐺 ∩ {∅}) = ∅) |
| 8 | reldisj 4453 | . . . . . . . . 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 691 | . . . . . 6 ⊢ (𝐺 ∈ Plig → 𝐺 ⊆ (𝒫 ∪ 𝐺 ∖ {∅})) |
| 12 | fss 6752 | . . . . . 6 ⊢ ((( I ↾ 𝐺):𝐺⟶𝐺 ∧ 𝐺 ⊆ (𝒫 ∪ 𝐺 ∖ {∅})) → ( I ↾ 𝐺):𝐺⟶(𝒫 ∪ 𝐺 ∖ {∅})) | |
| 13 | 11, 12 | sylan2 593 | . . . . 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 6766 | . 2 ⊢ (𝐺 ∈ Plig → ( I ↾ 𝐺):dom ( I ↾ 𝐺)⟶(𝒫 ∪ 𝐺 ∖ {∅})) |
| 17 | uniexg 7760 | . . 3 ⊢ (𝐺 ∈ Plig → ∪ 𝐺 ∈ V) | |
| 18 | resiexg 7934 | . . 3 ⊢ (𝐺 ∈ Plig → ( I ↾ 𝐺) ∈ V) | |
| 19 | isuhgrop 29087 | . . 3 ⊢ ((∪ 𝐺 ∈ V ∧ ( I ↾ 𝐺) ∈ V) → (〈∪ 𝐺, ( I ↾ 𝐺)〉 ∈ UHGraph ↔ ( I ↾ 𝐺):dom ( I ↾ 𝐺)⟶(𝒫 ∪ 𝐺 ∖ {∅}))) | |
| 20 | 17, 18, 19 | syl2anc 584 | . 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 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ∖ cdif 3948 ∩ cin 3950 ⊆ wss 3951 ∅c0 4333 𝒫 cpw 4600 {csn 4626 〈cop 4632 ∪ cuni 4907 I cid 5577 dom cdm 5685 ↾ cres 5687 ⟶wf 6557 –1-1-onto→wf1o 6560 UHGraphcuhgr 29073 Pligcplig 30493 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-1st 8014 df-2nd 8015 df-vtx 29015 df-iedg 29016 df-uhgr 29075 df-plig 30494 |
| This theorem is referenced by: (None) |
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