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Mirrors > Home > MPE Home > Th. List > Mathboxes > 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 4859 | . . 3 ⊢ 〈𝑉, (𝐸 ↾ 𝐴)〉 ∈ V | |
4 | 3 | a1i 11 | . 2 ⊢ (𝐺 ∈ UHGraph → 〈𝑉, (𝐸 ↾ 𝐴)〉 ∈ V) |
5 | fvex 6113 | . . . . 5 ⊢ (Vtx‘𝐺) ∈ V | |
6 | 1, 5 | eqeltri 2684 | . . . 4 ⊢ 𝑉 ∈ V |
7 | fvex 6113 | . . . . . 6 ⊢ (iEdg‘𝐺) ∈ V | |
8 | 2, 7 | eqeltri 2684 | . . . . 5 ⊢ 𝐸 ∈ V |
9 | 8 | resex 5363 | . . . 4 ⊢ (𝐸 ↾ 𝐴) ∈ V |
10 | opvtxfv 25681 | . . . 4 ⊢ ((𝑉 ∈ V ∧ (𝐸 ↾ 𝐴) ∈ V) → (Vtx‘〈𝑉, (𝐸 ↾ 𝐴)〉) = 𝑉) | |
11 | 6, 9, 10 | mp2an 704 | . . 3 ⊢ (Vtx‘〈𝑉, (𝐸 ↾ 𝐴)〉) = 𝑉 |
12 | 11 | a1i 11 | . 2 ⊢ (𝐺 ∈ UHGraph → (Vtx‘〈𝑉, (𝐸 ↾ 𝐴)〉) = 𝑉) |
13 | opiedgfv 25684 | . . . 4 ⊢ ((𝑉 ∈ V ∧ (𝐸 ↾ 𝐴) ∈ V) → (iEdg‘〈𝑉, (𝐸 ↾ 𝐴)〉) = (𝐸 ↾ 𝐴)) | |
14 | 6, 9, 13 | mp2an 704 | . . 3 ⊢ (iEdg‘〈𝑉, (𝐸 ↾ 𝐴)〉) = (𝐸 ↾ 𝐴) |
15 | 14 | a1i 11 | . 2 ⊢ (𝐺 ∈ UHGraph → (iEdg‘〈𝑉, (𝐸 ↾ 𝐴)〉) = (𝐸 ↾ 𝐴)) |
16 | id 22 | . 2 ⊢ (𝐺 ∈ UHGraph → 𝐺 ∈ UHGraph ) | |
17 | 1, 2, 4, 12, 15, 16 | uhgrspan 40516 | 1 ⊢ (𝐺 ∈ UHGraph → 〈𝑉, (𝐸 ↾ 𝐴)〉 ∈ UHGraph ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 Vcvv 3173 〈cop 4131 ↾ cres 5040 ‘cfv 5804 Vtxcvtx 25673 iEdgciedg 25674 UHGraph cuhgr 25722 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-fv 5812 df-1st 7059 df-2nd 7060 df-vtx 25675 df-iedg 25676 df-uhgr 25724 df-edga 25793 df-subgr 40492 |
This theorem is referenced by: (None) |
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