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| Mirrors > Home > MPE Home > Th. List > uhgrspanop | Structured version Visualization version GIF version | ||
| Description: A spanning subgraph of a hypergraph represented by an ordered pair is a hypergraph. (Contributed by Alexander van der Vekens, 27-Dec-2017.) (Revised by AV, 11-Oct-2020.) |
| Ref | Expression |
|---|---|
| uhgrspanop.v | ⊢ 𝑉 = (Vtx‘𝐺) |
| uhgrspanop.e | ⊢ 𝐸 = (iEdg‘𝐺) |
| Ref | Expression |
|---|---|
| uhgrspanop | ⊢ (𝐺 ∈ UHGraph → 〈𝑉, (𝐸 ↾ 𝐴)〉 ∈ UHGraph) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | uhgrspanop.v | . 2 ⊢ 𝑉 = (Vtx‘𝐺) | |
| 2 | uhgrspanop.e | . 2 ⊢ 𝐸 = (iEdg‘𝐺) | |
| 3 | opex 5469 | . . 3 ⊢ 〈𝑉, (𝐸 ↾ 𝐴)〉 ∈ V | |
| 4 | 3 | a1i 11 | . 2 ⊢ (𝐺 ∈ UHGraph → 〈𝑉, (𝐸 ↾ 𝐴)〉 ∈ V) |
| 5 | 1 | fvexi 6920 | . . . 4 ⊢ 𝑉 ∈ V |
| 6 | 2 | fvexi 6920 | . . . . 5 ⊢ 𝐸 ∈ V |
| 7 | 6 | resex 6047 | . . . 4 ⊢ (𝐸 ↾ 𝐴) ∈ V |
| 8 | 5, 7 | opvtxfvi 29026 | . . 3 ⊢ (Vtx‘〈𝑉, (𝐸 ↾ 𝐴)〉) = 𝑉 |
| 9 | 8 | a1i 11 | . 2 ⊢ (𝐺 ∈ UHGraph → (Vtx‘〈𝑉, (𝐸 ↾ 𝐴)〉) = 𝑉) |
| 10 | 5, 7 | opiedgfvi 29027 | . . 3 ⊢ (iEdg‘〈𝑉, (𝐸 ↾ 𝐴)〉) = (𝐸 ↾ 𝐴) |
| 11 | 10 | a1i 11 | . 2 ⊢ (𝐺 ∈ UHGraph → (iEdg‘〈𝑉, (𝐸 ↾ 𝐴)〉) = (𝐸 ↾ 𝐴)) |
| 12 | id 22 | . 2 ⊢ (𝐺 ∈ UHGraph → 𝐺 ∈ UHGraph) | |
| 13 | 1, 2, 4, 9, 11, 12 | uhgrspan 29309 | 1 ⊢ (𝐺 ∈ UHGraph → 〈𝑉, (𝐸 ↾ 𝐴)〉 ∈ UHGraph) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 Vcvv 3480 〈cop 4632 ↾ cres 5687 ‘cfv 6561 Vtxcvtx 29013 iEdgciedg 29014 UHGraphcuhgr 29073 |
| 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-res 5697 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-edg 29065 df-uhgr 29075 df-subgr 29285 |
| This theorem is referenced by: (None) |
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