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Mirrors > Home > MPE Home > Th. List > upgrunop | Structured version Visualization version GIF version |
Description: The union of two pseudographs (with the same vertex set): If 〈𝑉, 𝐸〉 and 〈𝑉, 𝐹〉 are pseudographs, then 〈𝑉, 𝐸 ∪ 𝐹〉 is a pseudograph (the vertex set stays the same, but the edges from both graphs are kept). (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 12-Oct-2020.) (Revised by AV, 24-Oct-2021.) |
Ref | Expression |
---|---|
upgrun.g | ⊢ (𝜑 → 𝐺 ∈ UPGraph) |
upgrun.h | ⊢ (𝜑 → 𝐻 ∈ UPGraph) |
upgrun.e | ⊢ 𝐸 = (iEdg‘𝐺) |
upgrun.f | ⊢ 𝐹 = (iEdg‘𝐻) |
upgrun.vg | ⊢ 𝑉 = (Vtx‘𝐺) |
upgrun.vh | ⊢ (𝜑 → (Vtx‘𝐻) = 𝑉) |
upgrun.i | ⊢ (𝜑 → (dom 𝐸 ∩ dom 𝐹) = ∅) |
Ref | Expression |
---|---|
upgrunop | ⊢ (𝜑 → 〈𝑉, (𝐸 ∪ 𝐹)〉 ∈ UPGraph) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | upgrun.g | . 2 ⊢ (𝜑 → 𝐺 ∈ UPGraph) | |
2 | upgrun.h | . 2 ⊢ (𝜑 → 𝐻 ∈ UPGraph) | |
3 | upgrun.e | . 2 ⊢ 𝐸 = (iEdg‘𝐺) | |
4 | upgrun.f | . 2 ⊢ 𝐹 = (iEdg‘𝐻) | |
5 | upgrun.vg | . 2 ⊢ 𝑉 = (Vtx‘𝐺) | |
6 | upgrun.vh | . 2 ⊢ (𝜑 → (Vtx‘𝐻) = 𝑉) | |
7 | upgrun.i | . 2 ⊢ (𝜑 → (dom 𝐸 ∩ dom 𝐹) = ∅) | |
8 | opex 4962 | . . 3 ⊢ 〈𝑉, (𝐸 ∪ 𝐹)〉 ∈ V | |
9 | 8 | a1i 11 | . 2 ⊢ (𝜑 → 〈𝑉, (𝐸 ∪ 𝐹)〉 ∈ V) |
10 | fvex 6239 | . . . . 5 ⊢ (Vtx‘𝐺) ∈ V | |
11 | 5, 10 | eqeltri 2726 | . . . 4 ⊢ 𝑉 ∈ V |
12 | fvex 6239 | . . . . . 6 ⊢ (iEdg‘𝐺) ∈ V | |
13 | 3, 12 | eqeltri 2726 | . . . . 5 ⊢ 𝐸 ∈ V |
14 | fvex 6239 | . . . . . 6 ⊢ (iEdg‘𝐻) ∈ V | |
15 | 4, 14 | eqeltri 2726 | . . . . 5 ⊢ 𝐹 ∈ V |
16 | 13, 15 | unex 6998 | . . . 4 ⊢ (𝐸 ∪ 𝐹) ∈ V |
17 | 11, 16 | pm3.2i 470 | . . 3 ⊢ (𝑉 ∈ V ∧ (𝐸 ∪ 𝐹) ∈ V) |
18 | opvtxfv 25929 | . . 3 ⊢ ((𝑉 ∈ V ∧ (𝐸 ∪ 𝐹) ∈ V) → (Vtx‘〈𝑉, (𝐸 ∪ 𝐹)〉) = 𝑉) | |
19 | 17, 18 | mp1i 13 | . 2 ⊢ (𝜑 → (Vtx‘〈𝑉, (𝐸 ∪ 𝐹)〉) = 𝑉) |
20 | opiedgfv 25932 | . . 3 ⊢ ((𝑉 ∈ V ∧ (𝐸 ∪ 𝐹) ∈ V) → (iEdg‘〈𝑉, (𝐸 ∪ 𝐹)〉) = (𝐸 ∪ 𝐹)) | |
21 | 17, 20 | mp1i 13 | . 2 ⊢ (𝜑 → (iEdg‘〈𝑉, (𝐸 ∪ 𝐹)〉) = (𝐸 ∪ 𝐹)) |
22 | 1, 2, 3, 4, 5, 6, 7, 9, 19, 21 | upgrun 26058 | 1 ⊢ (𝜑 → 〈𝑉, (𝐸 ∪ 𝐹)〉 ∈ UPGraph) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1523 ∈ wcel 2030 Vcvv 3231 ∪ cun 3605 ∩ cin 3606 ∅c0 3948 〈cop 4216 dom cdm 5143 ‘cfv 5926 Vtxcvtx 25919 iEdgciedg 25920 UPGraphcupgr 26020 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ral 2946 df-rex 2947 df-rab 2950 df-v 3233 df-sbc 3469 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-op 4217 df-uni 4469 df-br 4686 df-opab 4746 df-mpt 4763 df-id 5053 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-fv 5934 df-1st 7210 df-2nd 7211 df-vtx 25921 df-iedg 25922 df-upgr 26022 |
This theorem is referenced by: uspgrunop 26126 |
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