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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 5358 | . . 3 ⊢ 〈𝑉, (𝐸 ↾ 𝐴)〉 ∈ V | |
4 | 3 | a1i 11 | . 2 ⊢ (𝐺 ∈ UHGraph → 〈𝑉, (𝐸 ↾ 𝐴)〉 ∈ V) |
5 | 1 | fvexi 6686 | . . . 4 ⊢ 𝑉 ∈ V |
6 | 2 | fvexi 6686 | . . . . 5 ⊢ 𝐸 ∈ V |
7 | 6 | resex 5901 | . . . 4 ⊢ (𝐸 ↾ 𝐴) ∈ V |
8 | 5, 7 | opvtxfvi 26796 | . . 3 ⊢ (Vtx‘〈𝑉, (𝐸 ↾ 𝐴)〉) = 𝑉 |
9 | 8 | a1i 11 | . 2 ⊢ (𝐺 ∈ UHGraph → (Vtx‘〈𝑉, (𝐸 ↾ 𝐴)〉) = 𝑉) |
10 | 5, 7 | opiedgfvi 26797 | . . 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 27076 | 1 ⊢ (𝐺 ∈ UHGraph → 〈𝑉, (𝐸 ↾ 𝐴)〉 ∈ UHGraph) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 Vcvv 3496 〈cop 4575 ↾ cres 5559 ‘cfv 6357 Vtxcvtx 26783 iEdgciedg 26784 UHGraphcuhgr 26843 |
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 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-fv 6365 df-1st 7691 df-2nd 7692 df-vtx 26785 df-iedg 26786 df-edg 26835 df-uhgr 26845 df-subgr 27052 |
This theorem is referenced by: (None) |
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