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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 5457 | . . 3 ⊢ ⟨𝑉, (𝐸 ∪ 𝐹)⟩ ∈ V | |
9 | 8 | a1i 11 | . 2 ⊢ (𝜑 → ⟨𝑉, (𝐸 ∪ 𝐹)⟩ ∈ V) |
10 | 5 | fvexi 6899 | . . . 4 ⊢ 𝑉 ∈ V |
11 | 3 | fvexi 6899 | . . . . 5 ⊢ 𝐸 ∈ V |
12 | 4 | fvexi 6899 | . . . . 5 ⊢ 𝐹 ∈ V |
13 | 11, 12 | unex 7730 | . . . 4 ⊢ (𝐸 ∪ 𝐹) ∈ V |
14 | 10, 13 | pm3.2i 470 | . . 3 ⊢ (𝑉 ∈ V ∧ (𝐸 ∪ 𝐹) ∈ V) |
15 | opvtxfv 28772 | . . 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 28775 | . . 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 28842 | 1 ⊢ (𝜑 → ⟨𝑉, (𝐸 ∪ 𝐹)⟩ ∈ UHGraph) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 Vcvv 3468 ∪ cun 3941 ∩ cin 3942 ∅c0 4317 ⟨cop 4629 dom cdm 5669 ‘cfv 6537 Vtxcvtx 28764 iEdgciedg 28765 UHGraphcuhgr 28824 |
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 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 ax-un 7722 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 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 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-sbc 3773 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-fv 6545 df-1st 7974 df-2nd 7975 df-vtx 28766 df-iedg 28767 df-uhgr 28826 |
This theorem is referenced by: ushgrunop 28845 |
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