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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 5425 | . . 3 ⊢ ⟨𝑉, (𝐸 ↾ 𝐴)⟩ ∈ V | |
4 | 3 | a1i 11 | . 2 ⊢ (𝐺 ∈ UHGraph → ⟨𝑉, (𝐸 ↾ 𝐴)⟩ ∈ V) |
5 | 1 | fvexi 6860 | . . . 4 ⊢ 𝑉 ∈ V |
6 | 2 | fvexi 6860 | . . . . 5 ⊢ 𝐸 ∈ V |
7 | 6 | resex 5989 | . . . 4 ⊢ (𝐸 ↾ 𝐴) ∈ V |
8 | 5, 7 | opvtxfvi 28009 | . . 3 ⊢ (Vtx‘⟨𝑉, (𝐸 ↾ 𝐴)⟩) = 𝑉 |
9 | 8 | a1i 11 | . 2 ⊢ (𝐺 ∈ UHGraph → (Vtx‘⟨𝑉, (𝐸 ↾ 𝐴)⟩) = 𝑉) |
10 | 5, 7 | opiedgfvi 28010 | . . 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 28289 | 1 ⊢ (𝐺 ∈ UHGraph → ⟨𝑉, (𝐸 ↾ 𝐴)⟩ ∈ UHGraph) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 Vcvv 3447 ⟨cop 4596 ↾ cres 5639 ‘cfv 6500 Vtxcvtx 27996 iEdgciedg 27997 UHGraphcuhgr 28056 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pr 5388 ax-un 7676 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3449 df-sbc 3744 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-br 5110 df-opab 5172 df-mpt 5193 df-id 5535 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-fv 6508 df-1st 7925 df-2nd 7926 df-vtx 27998 df-iedg 27999 df-edg 28048 df-uhgr 28058 df-subgr 28265 |
This theorem is referenced by: (None) |
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