Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 3988 | . . 3 ⊢ (Vtx‘𝑆) ⊆ (Vtx‘𝑆) | |
2 | uhgrspan.q | . . 3 ⊢ (𝜑 → (Vtx‘𝑆) = 𝑉) | |
3 | 1, 2 | sseqtrid 4018 | . 2 ⊢ (𝜑 → (Vtx‘𝑆) ⊆ 𝑉) |
4 | uhgrspan.r | . . 3 ⊢ (𝜑 → (iEdg‘𝑆) = (𝐸 ↾ 𝐴)) | |
5 | resss 5872 | . . 3 ⊢ (𝐸 ↾ 𝐴) ⊆ 𝐸 | |
6 | 4, 5 | eqsstrdi 4020 | . 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 27066 | . 2 ⊢ (𝜑 → (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) |
12 | 8 | uhgrfun 26845 | . . . 4 ⊢ (𝐺 ∈ UHGraph → Fun 𝐸) |
13 | 10, 12 | syl 17 | . . 3 ⊢ (𝜑 → Fun 𝐸) |
14 | eqid 2821 | . . . 4 ⊢ (Vtx‘𝑆) = (Vtx‘𝑆) | |
15 | eqid 2821 | . . . 4 ⊢ (iEdg‘𝑆) = (iEdg‘𝑆) | |
16 | eqid 2821 | . . . 4 ⊢ (Edg‘𝑆) = (Edg‘𝑆) | |
17 | 14, 7, 15, 8, 16 | issubgr2 27048 | . . 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 1367 | . 2 ⊢ (𝜑 → (𝑆 SubGraph 𝐺 ↔ ((Vtx‘𝑆) ⊆ 𝑉 ∧ (iEdg‘𝑆) ⊆ 𝐸 ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)))) |
19 | 3, 6, 11, 18 | mpbir3and 1338 | 1 ⊢ (𝜑 → 𝑆 SubGraph 𝐺) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ⊆ wss 3935 𝒫 cpw 4538 class class class wbr 5058 ↾ cres 5551 Fun wfun 6343 ‘cfv 6349 Vtxcvtx 26775 iEdgciedg 26776 Edgcedg 26826 UHGraphcuhgr 26835 SubGraph csubgr 27043 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-fv 6357 df-edg 26827 df-uhgr 26837 df-subgr 27044 |
This theorem is referenced by: uhgrspan 27068 upgrspan 27069 umgrspan 27070 usgrspan 27071 |
Copyright terms: Public domain | W3C validator |