Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 5358 | . . 3 ⊢ 〈𝑉, (𝐸 ∪ 𝐹)〉 ∈ V | |
9 | 8 | a1i 11 | . 2 ⊢ (𝜑 → 〈𝑉, (𝐸 ∪ 𝐹)〉 ∈ V) |
10 | 5 | fvexi 6686 | . . . 4 ⊢ 𝑉 ∈ V |
11 | 3 | fvexi 6686 | . . . . 5 ⊢ 𝐸 ∈ V |
12 | 4 | fvexi 6686 | . . . . 5 ⊢ 𝐹 ∈ V |
13 | 11, 12 | unex 7471 | . . . 4 ⊢ (𝐸 ∪ 𝐹) ∈ V |
14 | 10, 13 | pm3.2i 473 | . . 3 ⊢ (𝑉 ∈ V ∧ (𝐸 ∪ 𝐹) ∈ V) |
15 | opvtxfv 26791 | . . 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 26794 | . . 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 26861 | 1 ⊢ (𝜑 → 〈𝑉, (𝐸 ∪ 𝐹)〉 ∈ UHGraph) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 Vcvv 3496 ∪ cun 3936 ∩ cin 3937 ∅c0 4293 〈cop 4575 dom cdm 5557 ‘cfv 6357 Vtxcvtx 26783 iEdgciedg 26784 UHGraphcuhgr 26843 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-fv 6365 df-1st 7691 df-2nd 7692 df-vtx 26785 df-iedg 26786 df-uhgr 26845 |
This theorem is referenced by: ushgrunop 26864 |
Copyright terms: Public domain | W3C validator |