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Mirrors > Home > MPE Home > Th. List > uspgrun | Structured version Visualization version GIF version |
Description: The union 𝑈 of two simple pseudographs 𝐺 and 𝐻 with the same vertex set 𝑉 is a pseudograph with the vertex 𝑉 and the union (𝐸 ∪ 𝐹) of the (indexed) edges. (Contributed by AV, 16-Oct-2020.) |
Ref | Expression |
---|---|
uspgrun.g | ⊢ (𝜑 → 𝐺 ∈ USPGraph) |
uspgrun.h | ⊢ (𝜑 → 𝐻 ∈ USPGraph) |
uspgrun.e | ⊢ 𝐸 = (iEdg‘𝐺) |
uspgrun.f | ⊢ 𝐹 = (iEdg‘𝐻) |
uspgrun.vg | ⊢ 𝑉 = (Vtx‘𝐺) |
uspgrun.vh | ⊢ (𝜑 → (Vtx‘𝐻) = 𝑉) |
uspgrun.i | ⊢ (𝜑 → (dom 𝐸 ∩ dom 𝐹) = ∅) |
uspgrun.u | ⊢ (𝜑 → 𝑈 ∈ 𝑊) |
uspgrun.v | ⊢ (𝜑 → (Vtx‘𝑈) = 𝑉) |
uspgrun.un | ⊢ (𝜑 → (iEdg‘𝑈) = (𝐸 ∪ 𝐹)) |
Ref | Expression |
---|---|
uspgrun | ⊢ (𝜑 → 𝑈 ∈ UPGraph) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uspgrun.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ USPGraph) | |
2 | uspgrupgr 27291 | . . 3 ⊢ (𝐺 ∈ USPGraph → 𝐺 ∈ UPGraph) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝜑 → 𝐺 ∈ UPGraph) |
4 | uspgrun.h | . . 3 ⊢ (𝜑 → 𝐻 ∈ USPGraph) | |
5 | uspgrupgr 27291 | . . 3 ⊢ (𝐻 ∈ USPGraph → 𝐻 ∈ UPGraph) | |
6 | 4, 5 | syl 17 | . 2 ⊢ (𝜑 → 𝐻 ∈ UPGraph) |
7 | uspgrun.e | . 2 ⊢ 𝐸 = (iEdg‘𝐺) | |
8 | uspgrun.f | . 2 ⊢ 𝐹 = (iEdg‘𝐻) | |
9 | uspgrun.vg | . 2 ⊢ 𝑉 = (Vtx‘𝐺) | |
10 | uspgrun.vh | . 2 ⊢ (𝜑 → (Vtx‘𝐻) = 𝑉) | |
11 | uspgrun.i | . 2 ⊢ (𝜑 → (dom 𝐸 ∩ dom 𝐹) = ∅) | |
12 | uspgrun.u | . 2 ⊢ (𝜑 → 𝑈 ∈ 𝑊) | |
13 | uspgrun.v | . 2 ⊢ (𝜑 → (Vtx‘𝑈) = 𝑉) | |
14 | uspgrun.un | . 2 ⊢ (𝜑 → (iEdg‘𝑈) = (𝐸 ∪ 𝐹)) | |
15 | 3, 6, 7, 8, 9, 10, 11, 12, 13, 14 | upgrun 27233 | 1 ⊢ (𝜑 → 𝑈 ∈ UPGraph) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2111 ∪ cun 3879 ∩ cin 3880 ∅c0 4252 dom cdm 5566 ‘cfv 6398 Vtxcvtx 27111 iEdgciedg 27112 UPGraphcupgr 27195 USPGraphcuspgr 27263 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2159 ax-12 2176 ax-ext 2709 ax-sep 5207 ax-nul 5214 ax-pr 5337 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2072 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2887 df-ral 3067 df-rex 3068 df-rab 3071 df-v 3423 df-sbc 3710 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4253 df-if 4455 df-pw 4530 df-sn 4557 df-pr 4559 df-op 4563 df-uni 4835 df-br 5069 df-opab 5131 df-id 5470 df-rel 5573 df-cnv 5574 df-co 5575 df-dm 5576 df-rn 5577 df-iota 6356 df-fun 6400 df-fn 6401 df-f 6402 df-f1 6403 df-fv 6406 df-upgr 27197 df-uspgr 27265 |
This theorem is referenced by: (None) |
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