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Mirrors > Home > MPE Home > Th. List > uhgrspansubgr | Structured version Visualization version GIF version |
Description: A spanning subgraph 𝑆 of a hypergraph 𝐺 is actually a subgraph of 𝐺. A subgraph 𝑆 of a graph 𝐺 which has the same vertices as 𝐺 and is obtained by removing some edges of 𝐺 is called a spanning subgraph (see section I.1 in [Bollobas] p. 2 and section 1.1 in [Diestel] p. 4). Formally, the edges are "removed" by restricting the edge function of the original graph by an arbitrary class (which actually needs not to be a subset of the domain of the edge function). (Contributed by AV, 18-Nov-2020.) (Proof shortened by AV, 21-Nov-2020.) |
Ref | Expression |
---|---|
uhgrspan.v | ⊢ 𝑉 = (Vtx‘𝐺) |
uhgrspan.e | ⊢ 𝐸 = (iEdg‘𝐺) |
uhgrspan.s | ⊢ (𝜑 → 𝑆 ∈ 𝑊) |
uhgrspan.q | ⊢ (𝜑 → (Vtx‘𝑆) = 𝑉) |
uhgrspan.r | ⊢ (𝜑 → (iEdg‘𝑆) = (𝐸 ↾ 𝐴)) |
uhgrspan.g | ⊢ (𝜑 → 𝐺 ∈ UHGraph) |
Ref | Expression |
---|---|
uhgrspansubgr | ⊢ (𝜑 → 𝑆 SubGraph 𝐺) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssid 3937 | . . 3 ⊢ (Vtx‘𝑆) ⊆ (Vtx‘𝑆) | |
2 | uhgrspan.q | . . 3 ⊢ (𝜑 → (Vtx‘𝑆) = 𝑉) | |
3 | 1, 2 | sseqtrid 3967 | . 2 ⊢ (𝜑 → (Vtx‘𝑆) ⊆ 𝑉) |
4 | uhgrspan.r | . . 3 ⊢ (𝜑 → (iEdg‘𝑆) = (𝐸 ↾ 𝐴)) | |
5 | resss 5843 | . . 3 ⊢ (𝐸 ↾ 𝐴) ⊆ 𝐸 | |
6 | 4, 5 | eqsstrdi 3969 | . 2 ⊢ (𝜑 → (iEdg‘𝑆) ⊆ 𝐸) |
7 | uhgrspan.v | . . 3 ⊢ 𝑉 = (Vtx‘𝐺) | |
8 | uhgrspan.e | . . 3 ⊢ 𝐸 = (iEdg‘𝐺) | |
9 | uhgrspan.s | . . 3 ⊢ (𝜑 → 𝑆 ∈ 𝑊) | |
10 | uhgrspan.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ UHGraph) | |
11 | 7, 8, 9, 2, 4, 10 | uhgrspansubgrlem 27080 | . 2 ⊢ (𝜑 → (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) |
12 | 8 | uhgrfun 26859 | . . . 4 ⊢ (𝐺 ∈ UHGraph → Fun 𝐸) |
13 | 10, 12 | syl 17 | . . 3 ⊢ (𝜑 → Fun 𝐸) |
14 | eqid 2798 | . . . 4 ⊢ (Vtx‘𝑆) = (Vtx‘𝑆) | |
15 | eqid 2798 | . . . 4 ⊢ (iEdg‘𝑆) = (iEdg‘𝑆) | |
16 | eqid 2798 | . . . 4 ⊢ (Edg‘𝑆) = (Edg‘𝑆) | |
17 | 14, 7, 15, 8, 16 | issubgr2 27062 | . . 3 ⊢ ((𝐺 ∈ UHGraph ∧ Fun 𝐸 ∧ 𝑆 ∈ 𝑊) → (𝑆 SubGraph 𝐺 ↔ ((Vtx‘𝑆) ⊆ 𝑉 ∧ (iEdg‘𝑆) ⊆ 𝐸 ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)))) |
18 | 10, 13, 9, 17 | syl3anc 1368 | . 2 ⊢ (𝜑 → (𝑆 SubGraph 𝐺 ↔ ((Vtx‘𝑆) ⊆ 𝑉 ∧ (iEdg‘𝑆) ⊆ 𝐸 ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)))) |
19 | 3, 6, 11, 18 | mpbir3and 1339 | 1 ⊢ (𝜑 → 𝑆 SubGraph 𝐺) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ⊆ wss 3881 𝒫 cpw 4497 class class class wbr 5030 ↾ cres 5521 Fun wfun 6318 ‘cfv 6324 Vtxcvtx 26789 iEdgciedg 26790 Edgcedg 26840 UHGraphcuhgr 26849 SubGraph csubgr 27057 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-fv 6332 df-edg 26841 df-uhgr 26851 df-subgr 27058 |
This theorem is referenced by: uhgrspan 27082 upgrspan 27083 umgrspan 27084 usgrspan 27085 |
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