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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 5465 | . . 3 ⊢ ⟨𝑉, (𝐸 ∪ 𝐹)⟩ ∈ V | |
9 | 8 | a1i 11 | . 2 ⊢ (𝜑 → ⟨𝑉, (𝐸 ∪ 𝐹)⟩ ∈ V) |
10 | 5 | fvexi 6906 | . . . 4 ⊢ 𝑉 ∈ V |
11 | 3 | fvexi 6906 | . . . . 5 ⊢ 𝐸 ∈ V |
12 | 4 | fvexi 6906 | . . . . 5 ⊢ 𝐹 ∈ V |
13 | 11, 12 | unex 7733 | . . . 4 ⊢ (𝐸 ∪ 𝐹) ∈ V |
14 | 10, 13 | pm3.2i 472 | . . 3 ⊢ (𝑉 ∈ V ∧ (𝐸 ∪ 𝐹) ∈ V) |
15 | opvtxfv 28264 | . . 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 28267 | . . 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 28378 | 1 ⊢ (𝜑 → ⟨𝑉, (𝐸 ∪ 𝐹)⟩ ∈ UPGraph) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 Vcvv 3475 ∪ cun 3947 ∩ cin 3948 ∅c0 4323 ⟨cop 4635 dom cdm 5677 ‘cfv 6544 Vtxcvtx 28256 iEdgciedg 28257 UPGraphcupgr 28340 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pr 5428 ax-un 7725 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-sbc 3779 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-fv 6552 df-1st 7975 df-2nd 7976 df-vtx 28258 df-iedg 28259 df-upgr 28342 |
This theorem is referenced by: uspgrunop 28446 |
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