Mathbox for Alexander van der Vekens |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > 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 3587 | . . 3 ⊢ (Vtx‘𝑆) ⊆ (Vtx‘𝑆) | |
2 | uhgrspan.q | . . 3 ⊢ (𝜑 → (Vtx‘𝑆) = 𝑉) | |
3 | 1, 2 | syl5sseq 3616 | . 2 ⊢ (𝜑 → (Vtx‘𝑆) ⊆ 𝑉) |
4 | uhgrspan.r | . . 3 ⊢ (𝜑 → (iEdg‘𝑆) = (𝐸 ↾ 𝐴)) | |
5 | resss 5342 | . . 3 ⊢ (𝐸 ↾ 𝐴) ⊆ 𝐸 | |
6 | 4, 5 | syl6eqss 3618 | . 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 40514 | . 2 ⊢ (𝜑 → (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) |
12 | 8 | uhgrfun 25732 | . . . 4 ⊢ (𝐺 ∈ UHGraph → Fun 𝐸) |
13 | 10, 12 | syl 17 | . . 3 ⊢ (𝜑 → Fun 𝐸) |
14 | eqid 2610 | . . . 4 ⊢ (Vtx‘𝑆) = (Vtx‘𝑆) | |
15 | eqid 2610 | . . . 4 ⊢ (iEdg‘𝑆) = (iEdg‘𝑆) | |
16 | eqid 2610 | . . . 4 ⊢ (Edg‘𝑆) = (Edg‘𝑆) | |
17 | 14, 7, 15, 8, 16 | issubgr2 40496 | . . 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 1318 | . 2 ⊢ (𝜑 → (𝑆 SubGraph 𝐺 ↔ ((Vtx‘𝑆) ⊆ 𝑉 ∧ (iEdg‘𝑆) ⊆ 𝐸 ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)))) |
19 | 3, 6, 11, 18 | mpbir3and 1238 | 1 ⊢ (𝜑 → 𝑆 SubGraph 𝐺) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ⊆ wss 3540 𝒫 cpw 4108 class class class wbr 4583 ↾ cres 5040 Fun wfun 5798 ‘cfv 5804 Vtxcvtx 25673 iEdgciedg 25674 UHGraph cuhgr 25722 Edgcedga 25792 SubGraph csubgr 40491 |
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-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-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-uhgr 25724 df-edga 25793 df-subgr 40492 |
This theorem is referenced by: uhgrspan 40516 upgrspan 40517 umgrspan 40518 usgrspan 40519 |
Copyright terms: Public domain | W3C validator |