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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 26547 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 6520 | . . . 4 ⊢ ( I ↾ 𝐺):𝐺–1-1-onto→𝐺 | |
2 | f1of 6483 | . . . 4 ⊢ (( I ↾ 𝐺):𝐺–1-1-onto→𝐺 → ( I ↾ 𝐺):𝐺⟶𝐺) | |
3 | pwuni 4781 | . . . . . . 7 ⊢ 𝐺 ⊆ 𝒫 ∪ 𝐺 | |
4 | n0lplig 27951 | . . . . . . . . . 10 ⊢ (𝐺 ∈ Plig → ¬ ∅ ∈ 𝐺) | |
5 | 4 | adantr 481 | . . . . . . . . 9 ⊢ ((𝐺 ∈ Plig ∧ 𝐺 ⊆ 𝒫 ∪ 𝐺) → ¬ ∅ ∈ 𝐺) |
6 | disjsn 4554 | . . . . . . . . 9 ⊢ ((𝐺 ∩ {∅}) = ∅ ↔ ¬ ∅ ∈ 𝐺) | |
7 | 5, 6 | sylibr 235 | . . . . . . . 8 ⊢ ((𝐺 ∈ Plig ∧ 𝐺 ⊆ 𝒫 ∪ 𝐺) → (𝐺 ∩ {∅}) = ∅) |
8 | reldisj 4316 | . . . . . . . . 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 687 | . . . . . 6 ⊢ (𝐺 ∈ Plig → 𝐺 ⊆ (𝒫 ∪ 𝐺 ∖ {∅})) |
12 | fss 6395 | . . . . . 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 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 6405 | . 2 ⊢ (𝐺 ∈ Plig → ( I ↾ 𝐺):dom ( I ↾ 𝐺)⟶(𝒫 ∪ 𝐺 ∖ {∅})) |
17 | uniexg 7325 | . . 3 ⊢ (𝐺 ∈ Plig → ∪ 𝐺 ∈ V) | |
18 | resiexg 7475 | . . 3 ⊢ (𝐺 ∈ Plig → ( I ↾ 𝐺) ∈ V) | |
19 | isuhgrop 26538 | . . 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 258 | 1 ⊢ (𝐺 ∈ Plig → 〈∪ 𝐺, ( I ↾ 𝐺)〉 ∈ UHGraph) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1522 ∈ wcel 2081 Vcvv 3437 ∖ cdif 3856 ∩ cin 3858 ⊆ wss 3859 ∅c0 4211 𝒫 cpw 4453 {csn 4472 〈cop 4478 ∪ cuni 4745 I cid 5347 dom cdm 5443 ↾ cres 5445 ⟶wf 6221 –1-1-onto→wf1o 6224 UHGraphcuhgr 26524 Pligcplig 27942 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-sep 5094 ax-nul 5101 ax-pow 5157 ax-pr 5221 ax-un 7319 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-ral 3110 df-rex 3111 df-reu 3112 df-rab 3114 df-v 3439 df-sbc 3707 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-nul 4212 df-if 4382 df-pw 4455 df-sn 4473 df-pr 4475 df-op 4479 df-uni 4746 df-br 4963 df-opab 5025 df-mpt 5042 df-id 5348 df-xp 5449 df-rel 5450 df-cnv 5451 df-co 5452 df-dm 5453 df-rn 5454 df-res 5455 df-ima 5456 df-iota 6189 df-fun 6227 df-fn 6228 df-f 6229 df-f1 6230 df-fo 6231 df-f1o 6232 df-fv 6233 df-1st 7545 df-2nd 7546 df-vtx 26466 df-iedg 26467 df-uhgr 26526 df-plig 27943 |
This theorem is referenced by: (None) |
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