|   | Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | |
| 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 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 | 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 29136 | 1 ⊢ (𝜑 → 𝑈 ∈ UPGraph) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2107 ∪ cun 3948 ∩ cin 3949 ∅c0 4332 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-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| 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-clab 2714 df-cleq 2728 df-clel 2815 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-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-upgr 29100 df-uspgr 29168 | 
| This theorem is referenced by: (None) | 
| Copyright terms: Public domain | W3C validator |