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Mirrors > Home > MPE Home > Th. List > uhgrunop | Structured version Visualization version GIF version |
Description: The union of two (undirected) hypergraphs (with the same vertex set) represented as ordered pair: If 〈𝑉, 𝐸〉 and 〈𝑉, 𝐹〉 are hypergraphs, then 〈𝑉, 𝐸 ∪ 𝐹〉 is a hypergraph (the vertex set stays the same, but the edges from both graphs are kept, possibly resulting in two edges between two vertices). (Contributed by Alexander van der Vekens, 27-Dec-2017.) (Revised by AV, 11-Oct-2020.) (Revised by AV, 24-Oct-2021.) |
Ref | Expression |
---|---|
uhgrun.g | ⊢ (𝜑 → 𝐺 ∈ UHGraph) |
uhgrun.h | ⊢ (𝜑 → 𝐻 ∈ UHGraph) |
uhgrun.e | ⊢ 𝐸 = (iEdg‘𝐺) |
uhgrun.f | ⊢ 𝐹 = (iEdg‘𝐻) |
uhgrun.vg | ⊢ 𝑉 = (Vtx‘𝐺) |
uhgrun.vh | ⊢ (𝜑 → (Vtx‘𝐻) = 𝑉) |
uhgrun.i | ⊢ (𝜑 → (dom 𝐸 ∩ dom 𝐹) = ∅) |
Ref | Expression |
---|---|
uhgrunop | ⊢ (𝜑 → 〈𝑉, (𝐸 ∪ 𝐹)〉 ∈ UHGraph) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uhgrun.g | . 2 ⊢ (𝜑 → 𝐺 ∈ UHGraph) | |
2 | uhgrun.h | . 2 ⊢ (𝜑 → 𝐻 ∈ UHGraph) | |
3 | uhgrun.e | . 2 ⊢ 𝐸 = (iEdg‘𝐺) | |
4 | uhgrun.f | . 2 ⊢ 𝐹 = (iEdg‘𝐻) | |
5 | uhgrun.vg | . 2 ⊢ 𝑉 = (Vtx‘𝐺) | |
6 | uhgrun.vh | . 2 ⊢ (𝜑 → (Vtx‘𝐻) = 𝑉) | |
7 | uhgrun.i | . 2 ⊢ (𝜑 → (dom 𝐸 ∩ dom 𝐹) = ∅) | |
8 | opex 5321 | . . 3 ⊢ 〈𝑉, (𝐸 ∪ 𝐹)〉 ∈ V | |
9 | 8 | a1i 11 | . 2 ⊢ (𝜑 → 〈𝑉, (𝐸 ∪ 𝐹)〉 ∈ V) |
10 | 5 | fvexi 6659 | . . . 4 ⊢ 𝑉 ∈ V |
11 | 3 | fvexi 6659 | . . . . 5 ⊢ 𝐸 ∈ V |
12 | 4 | fvexi 6659 | . . . . 5 ⊢ 𝐹 ∈ V |
13 | 11, 12 | unex 7449 | . . . 4 ⊢ (𝐸 ∪ 𝐹) ∈ V |
14 | 10, 13 | pm3.2i 474 | . . 3 ⊢ (𝑉 ∈ V ∧ (𝐸 ∪ 𝐹) ∈ V) |
15 | opvtxfv 26797 | . . 3 ⊢ ((𝑉 ∈ V ∧ (𝐸 ∪ 𝐹) ∈ V) → (Vtx‘〈𝑉, (𝐸 ∪ 𝐹)〉) = 𝑉) | |
16 | 14, 15 | mp1i 13 | . 2 ⊢ (𝜑 → (Vtx‘〈𝑉, (𝐸 ∪ 𝐹)〉) = 𝑉) |
17 | opiedgfv 26800 | . . 3 ⊢ ((𝑉 ∈ V ∧ (𝐸 ∪ 𝐹) ∈ V) → (iEdg‘〈𝑉, (𝐸 ∪ 𝐹)〉) = (𝐸 ∪ 𝐹)) | |
18 | 14, 17 | mp1i 13 | . 2 ⊢ (𝜑 → (iEdg‘〈𝑉, (𝐸 ∪ 𝐹)〉) = (𝐸 ∪ 𝐹)) |
19 | 1, 2, 3, 4, 5, 6, 7, 9, 16, 18 | uhgrun 26867 | 1 ⊢ (𝜑 → 〈𝑉, (𝐸 ∪ 𝐹)〉 ∈ UHGraph) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 Vcvv 3441 ∪ cun 3879 ∩ cin 3880 ∅c0 4243 〈cop 4531 dom cdm 5519 ‘cfv 6324 Vtxcvtx 26789 iEdgciedg 26790 UHGraphcuhgr 26849 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-fv 6332 df-1st 7671 df-2nd 7672 df-vtx 26791 df-iedg 26792 df-uhgr 26851 |
This theorem is referenced by: ushgrunop 26870 |
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