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| 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 5469 | . . 3 ⊢ 〈𝑉, (𝐸 ∪ 𝐹)〉 ∈ V | |
| 9 | 8 | a1i 11 | . 2 ⊢ (𝜑 → 〈𝑉, (𝐸 ∪ 𝐹)〉 ∈ V) | 
| 10 | 5 | fvexi 6920 | . . . 4 ⊢ 𝑉 ∈ V | 
| 11 | 3 | fvexi 6920 | . . . . 5 ⊢ 𝐸 ∈ V | 
| 12 | 4 | fvexi 6920 | . . . . 5 ⊢ 𝐹 ∈ V | 
| 13 | 11, 12 | unex 7764 | . . . 4 ⊢ (𝐸 ∪ 𝐹) ∈ V | 
| 14 | 10, 13 | pm3.2i 470 | . . 3 ⊢ (𝑉 ∈ V ∧ (𝐸 ∪ 𝐹) ∈ V) | 
| 15 | opvtxfv 29021 | . . 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 29024 | . . 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 29135 | 1 ⊢ (𝜑 → 〈𝑉, (𝐸 ∪ 𝐹)〉 ∈ UPGraph) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ∪ cun 3949 ∩ cin 3950 ∅c0 4333 〈cop 4632 dom cdm 5685 ‘cfv 6561 Vtxcvtx 29013 iEdgciedg 29014 UPGraphcupgr 29097 | 
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fv 6569 df-1st 8014 df-2nd 8015 df-vtx 29015 df-iedg 29016 df-upgr 29099 | 
| This theorem is referenced by: uspgrunop 29206 | 
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