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Mirrors > Home > MPE Home > Th. List > ushgrun | Structured version Visualization version GIF version |
Description: The union 𝑈 of two (undirected) simple hypergraphs 𝐺 and 𝐻 with the same vertex set 𝑉 is a (not necessarily simple) hypergraph with the vertex 𝑉 and the union (𝐸 ∪ 𝐹) of the (indexed) edges. (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) = ∅) |
ushgrun.u | ⊢ (𝜑 → 𝑈 ∈ 𝑊) |
ushgrun.v | ⊢ (𝜑 → (Vtx‘𝑈) = 𝑉) |
ushgrun.un | ⊢ (𝜑 → (iEdg‘𝑈) = (𝐸 ∪ 𝐹)) |
Ref | Expression |
---|---|
ushgrun | ⊢ (𝜑 → 𝑈 ∈ UHGraph) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ushgrun.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ USHGraph) | |
2 | ushgruhgr 28902 | . . 3 ⊢ (𝐺 ∈ USHGraph → 𝐺 ∈ UHGraph) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝜑 → 𝐺 ∈ UHGraph) |
4 | ushgrun.h | . . 3 ⊢ (𝜑 → 𝐻 ∈ USHGraph) | |
5 | ushgruhgr 28902 | . . 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 | ushgrun.u | . 2 ⊢ (𝜑 → 𝑈 ∈ 𝑊) | |
13 | ushgrun.v | . 2 ⊢ (𝜑 → (Vtx‘𝑈) = 𝑉) | |
14 | ushgrun.un | . 2 ⊢ (𝜑 → (iEdg‘𝑈) = (𝐸 ∪ 𝐹)) | |
15 | 3, 6, 7, 8, 9, 10, 11, 12, 13, 14 | uhgrun 28907 | 1 ⊢ (𝜑 → 𝑈 ∈ UHGraph) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ∪ cun 3947 ∩ cin 3948 ∅c0 4326 dom cdm 5682 ‘cfv 6553 Vtxcvtx 28829 iEdgciedg 28830 UHGraphcuhgr 28889 USHGraphcushgr 28890 |
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-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pr 5433 |
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-clab 2706 df-cleq 2720 df-clel 2806 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-sbc 3779 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-opab 5215 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fv 6561 df-uhgr 28891 df-ushgr 28892 |
This theorem is referenced by: (None) |
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