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Mirrors > Home > MPE Home > Th. List > uspgrun | Structured version Visualization version GIF version |
Description: The union 𝑈 of two simple pseudographs 𝐺 and 𝐻 with the same vertex set 𝑉 is a pseudograph with the vertex 𝑉 and the union (𝐸 ∪ 𝐹) of the (indexed) edges. (Contributed by AV, 16-Oct-2020.) |
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) = ∅) |
uspgrun.u | ⊢ (𝜑 → 𝑈 ∈ 𝑊) |
uspgrun.v | ⊢ (𝜑 → (Vtx‘𝑈) = 𝑉) |
uspgrun.un | ⊢ (𝜑 → (iEdg‘𝑈) = (𝐸 ∪ 𝐹)) |
Ref | Expression |
---|---|
uspgrun | ⊢ (𝜑 → 𝑈 ∈ UPGraph) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uspgrun.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ USPGraph) | |
2 | uspgrupgr 26961 | . . 3 ⊢ (𝐺 ∈ USPGraph → 𝐺 ∈ UPGraph) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝜑 → 𝐺 ∈ UPGraph) |
4 | uspgrun.h | . . 3 ⊢ (𝜑 → 𝐻 ∈ USPGraph) | |
5 | uspgrupgr 26961 | . . 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 | uspgrun.u | . 2 ⊢ (𝜑 → 𝑈 ∈ 𝑊) | |
13 | uspgrun.v | . 2 ⊢ (𝜑 → (Vtx‘𝑈) = 𝑉) | |
14 | uspgrun.un | . 2 ⊢ (𝜑 → (iEdg‘𝑈) = (𝐸 ∪ 𝐹)) | |
15 | 3, 6, 7, 8, 9, 10, 11, 12, 13, 14 | upgrun 26903 | 1 ⊢ (𝜑 → 𝑈 ∈ UPGraph) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 ∪ cun 3934 ∩ cin 3935 ∅c0 4291 dom cdm 5555 ‘cfv 6355 Vtxcvtx 26781 iEdgciedg 26782 UPGraphcupgr 26865 USPGraphcuspgr 26933 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-id 5460 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fv 6363 df-upgr 26867 df-uspgr 26935 |
This theorem is referenced by: (None) |
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