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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 5464 | . . 3 ⊢ ⟨𝑉, (𝐸 ↾ 𝐴)⟩ ∈ V | |
4 | 3 | a1i 11 | . 2 ⊢ (𝐺 ∈ UHGraph → ⟨𝑉, (𝐸 ↾ 𝐴)⟩ ∈ V) |
5 | 1 | fvexi 6905 | . . . 4 ⊢ 𝑉 ∈ V |
6 | 2 | fvexi 6905 | . . . . 5 ⊢ 𝐸 ∈ V |
7 | 6 | resex 6029 | . . . 4 ⊢ (𝐸 ↾ 𝐴) ∈ V |
8 | 5, 7 | opvtxfvi 28266 | . . 3 ⊢ (Vtx‘⟨𝑉, (𝐸 ↾ 𝐴)⟩) = 𝑉 |
9 | 8 | a1i 11 | . 2 ⊢ (𝐺 ∈ UHGraph → (Vtx‘⟨𝑉, (𝐸 ↾ 𝐴)⟩) = 𝑉) |
10 | 5, 7 | opiedgfvi 28267 | . . 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 28546 | 1 ⊢ (𝐺 ∈ UHGraph → ⟨𝑉, (𝐸 ↾ 𝐴)⟩ ∈ UHGraph) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 Vcvv 3474 ⟨cop 4634 ↾ cres 5678 ‘cfv 6543 Vtxcvtx 28253 iEdgciedg 28254 UHGraphcuhgr 28313 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pr 5427 ax-un 7724 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-sbc 3778 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-fv 6551 df-1st 7974 df-2nd 7975 df-vtx 28255 df-iedg 28256 df-edg 28305 df-uhgr 28315 df-subgr 28522 |
This theorem is referenced by: (None) |
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