![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 26417 | . . 3 ⊢ (𝐺 ∈ USHGraph → 𝐺 ∈ UHGraph) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝜑 → 𝐺 ∈ UHGraph) |
4 | ushgrun.h | . . 3 ⊢ (𝜑 → 𝐻 ∈ USHGraph) | |
5 | ushgruhgr 26417 | . . 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 26422 | 1 ⊢ (𝜑 → 𝑈 ∈ UHGraph) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1601 ∈ wcel 2107 ∪ cun 3790 ∩ cin 3791 ∅c0 4141 dom cdm 5355 ‘cfv 6135 Vtxcvtx 26344 iEdgciedg 26345 UHGraphcuhgr 26404 USHGraphcushgr 26405 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5017 ax-nul 5025 ax-pr 5138 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3400 df-sbc 3653 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4672 df-br 4887 df-opab 4949 df-id 5261 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fv 6143 df-uhgr 26406 df-ushgr 26407 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |