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| Description: The union of two simple pseudographs (with the same vertex set): If 〈𝑉, 𝐸〉 and 〈𝑉, 𝐹〉 are simple pseudographs, then 〈𝑉, 𝐸 ∪ 𝐹〉 is a pseudograph (the vertex set stays the same, but the edges from both graphs are kept, maybe resulting incident two edges between two vertices). (Contributed by Alexander van der Vekens, 10-Aug-2017.) (Revised by AV, 16-Oct-2020.) (Revised by AV, 24-Oct-2021.) | 
| Ref | Expression | 
|---|---|
| uspgrun.g | ⊢ (𝜑 → 𝐺 ∈ USPGraph) | 
| uspgrun.h | ⊢ (𝜑 → 𝐻 ∈ USPGraph) | 
| uspgrun.e | ⊢ 𝐸 = (iEdg‘𝐺) | 
| uspgrun.f | ⊢ 𝐹 = (iEdg‘𝐻) | 
| uspgrun.vg | ⊢ 𝑉 = (Vtx‘𝐺) | 
| uspgrun.vh | ⊢ (𝜑 → (Vtx‘𝐻) = 𝑉) | 
| uspgrun.i | ⊢ (𝜑 → (dom 𝐸 ∩ dom 𝐹) = ∅) | 
| Ref | Expression | 
|---|---|
| uspgrunop | ⊢ (𝜑 → 〈𝑉, (𝐸 ∪ 𝐹)〉 ∈ UPGraph) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | uspgrun.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ USPGraph) | |
| 2 | uspgrupgr 29196 | . . 3 ⊢ (𝐺 ∈ USPGraph → 𝐺 ∈ UPGraph) | |
| 3 | 1, 2 | syl 17 | . 2 ⊢ (𝜑 → 𝐺 ∈ UPGraph) | 
| 4 | uspgrun.h | . . 3 ⊢ (𝜑 → 𝐻 ∈ USPGraph) | |
| 5 | uspgrupgr 29196 | . . 3 ⊢ (𝐻 ∈ USPGraph → 𝐻 ∈ UPGraph) | |
| 6 | 4, 5 | syl 17 | . 2 ⊢ (𝜑 → 𝐻 ∈ UPGraph) | 
| 7 | uspgrun.e | . 2 ⊢ 𝐸 = (iEdg‘𝐺) | |
| 8 | uspgrun.f | . 2 ⊢ 𝐹 = (iEdg‘𝐻) | |
| 9 | uspgrun.vg | . 2 ⊢ 𝑉 = (Vtx‘𝐺) | |
| 10 | uspgrun.vh | . 2 ⊢ (𝜑 → (Vtx‘𝐻) = 𝑉) | |
| 11 | uspgrun.i | . 2 ⊢ (𝜑 → (dom 𝐸 ∩ dom 𝐹) = ∅) | |
| 12 | 3, 6, 7, 8, 9, 10, 11 | upgrunop 29137 | 1 ⊢ (𝜑 → 〈𝑉, (𝐸 ∪ 𝐹)〉 ∈ UPGraph) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2107 ∪ cun 3948 ∩ cin 3949 ∅c0 4332 〈cop 4631 dom cdm 5684 ‘cfv 6560 Vtxcvtx 29014 iEdgciedg 29015 UPGraphcupgr 29098 USPGraphcuspgr 29166 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-sbc 3788 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fv 6568 df-1st 8015 df-2nd 8016 df-vtx 29016 df-iedg 29017 df-upgr 29100 df-uspgr 29168 | 
| This theorem is referenced by: (None) | 
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