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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 5359 | . . 3 ⊢ 〈𝑉, (𝐸 ∪ 𝐹)〉 ∈ V | |
9 | 8 | a1i 11 | . 2 ⊢ (𝜑 → 〈𝑉, (𝐸 ∪ 𝐹)〉 ∈ V) |
10 | 5 | fvexi 6687 | . . . 4 ⊢ 𝑉 ∈ V |
11 | 3 | fvexi 6687 | . . . . 5 ⊢ 𝐸 ∈ V |
12 | 4 | fvexi 6687 | . . . . 5 ⊢ 𝐹 ∈ V |
13 | 11, 12 | unex 7472 | . . . 4 ⊢ (𝐸 ∪ 𝐹) ∈ V |
14 | 10, 13 | pm3.2i 473 | . . 3 ⊢ (𝑉 ∈ V ∧ (𝐸 ∪ 𝐹) ∈ V) |
15 | opvtxfv 26792 | . . 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 26795 | . . 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 | upgrun 26906 | 1 ⊢ (𝜑 → 〈𝑉, (𝐸 ∪ 𝐹)〉 ∈ UPGraph) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1536 ∈ wcel 2113 Vcvv 3497 ∪ cun 3937 ∩ cin 3938 ∅c0 4294 〈cop 4576 dom cdm 5558 ‘cfv 6358 Vtxcvtx 26784 iEdgciedg 26785 UPGraphcupgr 26868 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-sbc 3776 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-fv 6366 df-1st 7692 df-2nd 7693 df-vtx 26786 df-iedg 26787 df-upgr 26870 |
This theorem is referenced by: uspgrunop 26974 |
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