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Mirrors > Home > MPE Home > Th. List > uspgrunop | Structured version Visualization version GIF version |
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 28944 | . . 3 ⊢ (𝐺 ∈ USPGraph → 𝐺 ∈ UPGraph) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝜑 → 𝐺 ∈ UPGraph) |
4 | uspgrun.h | . . 3 ⊢ (𝜑 → 𝐻 ∈ USPGraph) | |
5 | uspgrupgr 28944 | . . 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 28887 | 1 ⊢ (𝜑 → ⟨𝑉, (𝐸 ∪ 𝐹)⟩ ∈ UPGraph) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ∪ cun 3941 ∩ cin 3942 ∅c0 4317 ⟨cop 4629 dom cdm 5669 ‘cfv 6537 Vtxcvtx 28764 iEdgciedg 28765 UPGraphcupgr 28848 USPGraphcuspgr 28916 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 ax-un 7722 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-sbc 3773 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fv 6545 df-1st 7974 df-2nd 7975 df-vtx 28766 df-iedg 28767 df-upgr 28850 df-uspgr 28918 |
This theorem is referenced by: (None) |
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