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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 3996 | . . 3 ⊢ (Vtx‘𝑆) ⊆ (Vtx‘𝑆) | |
2 | uhgrspan.q | . . 3 ⊢ (𝜑 → (Vtx‘𝑆) = 𝑉) | |
3 | 1, 2 | sseqtrid 4026 | . 2 ⊢ (𝜑 → (Vtx‘𝑆) ⊆ 𝑉) |
4 | uhgrspan.r | . . 3 ⊢ (𝜑 → (iEdg‘𝑆) = (𝐸 ↾ 𝐴)) | |
5 | resss 5996 | . . 3 ⊢ (𝐸 ↾ 𝐴) ⊆ 𝐸 | |
6 | 4, 5 | eqsstrdi 4028 | . 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 28982 | . 2 ⊢ (𝜑 → (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) |
12 | 8 | uhgrfun 28761 | . . . 4 ⊢ (𝐺 ∈ UHGraph → Fun 𝐸) |
13 | 10, 12 | syl 17 | . . 3 ⊢ (𝜑 → Fun 𝐸) |
14 | eqid 2724 | . . . 4 ⊢ (Vtx‘𝑆) = (Vtx‘𝑆) | |
15 | eqid 2724 | . . . 4 ⊢ (iEdg‘𝑆) = (iEdg‘𝑆) | |
16 | eqid 2724 | . . . 4 ⊢ (Edg‘𝑆) = (Edg‘𝑆) | |
17 | 14, 7, 15, 8, 16 | issubgr2 28964 | . . 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 205 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ⊆ wss 3940 𝒫 cpw 4594 class class class wbr 5138 ↾ cres 5668 Fun wfun 6527 ‘cfv 6533 Vtxcvtx 28691 iEdgciedg 28692 Edgcedg 28742 UHGraphcuhgr 28751 SubGraph csubgr 28959 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5289 ax-nul 5296 ax-pr 5417 ax-un 7718 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-sbc 3770 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-br 5139 df-opab 5201 df-mpt 5222 df-id 5564 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-fv 6541 df-edg 28743 df-uhgr 28753 df-subgr 28960 |
This theorem is referenced by: uhgrspan 28984 upgrspan 28985 umgrspan 28986 usgrspan 28987 |
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