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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 5403 | . . 3 ⊢ ⟨𝑉, (𝐸 ∪ 𝐹)⟩ ∈ V | |
9 | 8 | a1i 11 | . 2 ⊢ (𝜑 → ⟨𝑉, (𝐸 ∪ 𝐹)⟩ ∈ V) |
10 | 5 | fvexi 6833 | . . . 4 ⊢ 𝑉 ∈ V |
11 | 3 | fvexi 6833 | . . . . 5 ⊢ 𝐸 ∈ V |
12 | 4 | fvexi 6833 | . . . . 5 ⊢ 𝐹 ∈ V |
13 | 11, 12 | unex 7650 | . . . 4 ⊢ (𝐸 ∪ 𝐹) ∈ V |
14 | 10, 13 | pm3.2i 471 | . . 3 ⊢ (𝑉 ∈ V ∧ (𝐸 ∪ 𝐹) ∈ V) |
15 | opvtxfv 27604 | . . 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 27607 | . . 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 27718 | 1 ⊢ (𝜑 → ⟨𝑉, (𝐸 ∪ 𝐹)⟩ ∈ UPGraph) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1540 ∈ wcel 2105 Vcvv 3441 ∪ cun 3895 ∩ cin 3896 ∅c0 4268 ⟨cop 4578 dom cdm 5614 ‘cfv 6473 Vtxcvtx 27596 iEdgciedg 27597 UPGraphcupgr 27680 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-sep 5240 ax-nul 5247 ax-pr 5369 ax-un 7642 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3404 df-v 3443 df-sbc 3727 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-nul 4269 df-if 4473 df-pw 4548 df-sn 4573 df-pr 4575 df-op 4579 df-uni 4852 df-br 5090 df-opab 5152 df-mpt 5173 df-id 5512 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-iota 6425 df-fun 6475 df-fn 6476 df-f 6477 df-fv 6481 df-1st 7891 df-2nd 7892 df-vtx 27598 df-iedg 27599 df-upgr 27682 |
This theorem is referenced by: uspgrunop 27786 |
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