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Mirrors > Home > MPE Home > Th. List > uspgrunop | Structured version Visualization version GIF version |
Description: The union of two simple pseudographs (with the same vertex set): If 〈𝑉, 𝐸〉 and 〈𝑉, 𝐹〉 are simple pseudographs, then 〈𝑉, 𝐸 ∪ 𝐹〉 is a pseudograph (the vertex set stays the same, but the edges from both graphs are kept, maybe resulting incident two edges between two vertices). (Contributed by Alexander van der Vekens, 10-Aug-2017.) (Revised by AV, 16-Oct-2020.) (Revised by AV, 24-Oct-2021.) |
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) = ∅) |
Ref | Expression |
---|---|
uspgrunop | ⊢ (𝜑 → 〈𝑉, (𝐸 ∪ 𝐹)〉 ∈ UPGraph) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uspgrun.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ USPGraph) | |
2 | uspgrupgr 27121 | . . 3 ⊢ (𝐺 ∈ USPGraph → 𝐺 ∈ UPGraph) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝜑 → 𝐺 ∈ UPGraph) |
4 | uspgrun.h | . . 3 ⊢ (𝜑 → 𝐻 ∈ USPGraph) | |
5 | uspgrupgr 27121 | . . 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 | 3, 6, 7, 8, 9, 10, 11 | upgrunop 27064 | 1 ⊢ (𝜑 → 〈𝑉, (𝐸 ∪ 𝐹)〉 ∈ UPGraph) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2114 ∪ cun 3841 ∩ cin 3842 ∅c0 4211 〈cop 4522 dom cdm 5525 ‘cfv 6339 Vtxcvtx 26941 iEdgciedg 26942 UPGraphcupgr 27025 USPGraphcuspgr 27093 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-sep 5167 ax-nul 5174 ax-pr 5296 ax-un 7479 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-ral 3058 df-rex 3059 df-rab 3062 df-v 3400 df-sbc 3681 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-op 4523 df-uni 4797 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5429 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fv 6347 df-1st 7714 df-2nd 7715 df-vtx 26943 df-iedg 26944 df-upgr 27027 df-uspgr 27095 |
This theorem is referenced by: (None) |
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