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Mirrors > Home > MPE Home > Th. List > ushgrunop | Structured version Visualization version GIF version |
Description: The union of two (undirected) simple hypergraphs (with the same vertex set) represented as ordered pair: If ⟨𝑉, 𝐸⟩ and ⟨𝑉, 𝐹⟩ are simple hypergraphs, then ⟨𝑉, 𝐸 ∪ 𝐹⟩ is a (not necessarily simple) 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 AV, 29-Nov-2020.) (Revised by AV, 24-Oct-2021.) |
Ref | Expression |
---|---|
ushgrun.g | ⊢ (𝜑 → 𝐺 ∈ USHGraph) |
ushgrun.h | ⊢ (𝜑 → 𝐻 ∈ USHGraph) |
ushgrun.e | ⊢ 𝐸 = (iEdg‘𝐺) |
ushgrun.f | ⊢ 𝐹 = (iEdg‘𝐻) |
ushgrun.vg | ⊢ 𝑉 = (Vtx‘𝐺) |
ushgrun.vh | ⊢ (𝜑 → (Vtx‘𝐻) = 𝑉) |
ushgrun.i | ⊢ (𝜑 → (dom 𝐸 ∩ dom 𝐹) = ∅) |
Ref | Expression |
---|---|
ushgrunop | ⊢ (𝜑 → ⟨𝑉, (𝐸 ∪ 𝐹)⟩ ∈ UHGraph) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ushgrun.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ USHGraph) | |
2 | ushgruhgr 28926 | . . 3 ⊢ (𝐺 ∈ USHGraph → 𝐺 ∈ UHGraph) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝜑 → 𝐺 ∈ UHGraph) |
4 | ushgrun.h | . . 3 ⊢ (𝜑 → 𝐻 ∈ USHGraph) | |
5 | ushgruhgr 28926 | . . 3 ⊢ (𝐻 ∈ USHGraph → 𝐻 ∈ UHGraph) | |
6 | 4, 5 | syl 17 | . 2 ⊢ (𝜑 → 𝐻 ∈ UHGraph) |
7 | ushgrun.e | . 2 ⊢ 𝐸 = (iEdg‘𝐺) | |
8 | ushgrun.f | . 2 ⊢ 𝐹 = (iEdg‘𝐻) | |
9 | ushgrun.vg | . 2 ⊢ 𝑉 = (Vtx‘𝐺) | |
10 | ushgrun.vh | . 2 ⊢ (𝜑 → (Vtx‘𝐻) = 𝑉) | |
11 | ushgrun.i | . 2 ⊢ (𝜑 → (dom 𝐸 ∩ dom 𝐹) = ∅) | |
12 | 3, 6, 7, 8, 9, 10, 11 | uhgrunop 28932 | 1 ⊢ (𝜑 → ⟨𝑉, (𝐸 ∪ 𝐹)⟩ ∈ UHGraph) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ∪ cun 3937 ∩ cin 3938 ∅c0 4318 ⟨cop 4630 dom cdm 5672 ‘cfv 6543 Vtxcvtx 28853 iEdgciedg 28854 UHGraphcuhgr 28913 USHGraphcushgr 28914 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5294 ax-nul 5301 ax-pr 5423 ax-un 7738 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3465 df-sbc 3769 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5144 df-opab 5206 df-mpt 5227 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fv 6551 df-1st 7991 df-2nd 7992 df-vtx 28855 df-iedg 28856 df-uhgr 28915 df-ushgr 28916 |
This theorem is referenced by: (None) |
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