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| 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 4005 | . . 3 ⊢ (Vtx‘𝑆) ⊆ (Vtx‘𝑆) | |
| 2 | uhgrspan.q | . . 3 ⊢ (𝜑 → (Vtx‘𝑆) = 𝑉) | |
| 3 | 1, 2 | sseqtrid 4025 | . 2 ⊢ (𝜑 → (Vtx‘𝑆) ⊆ 𝑉) | 
| 4 | uhgrspan.r | . . 3 ⊢ (𝜑 → (iEdg‘𝑆) = (𝐸 ↾ 𝐴)) | |
| 5 | resss 6018 | . . 3 ⊢ (𝐸 ↾ 𝐴) ⊆ 𝐸 | |
| 6 | 4, 5 | eqsstrdi 4027 | . 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 29308 | . 2 ⊢ (𝜑 → (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)) | 
| 12 | 8 | uhgrfun 29084 | . . . 4 ⊢ (𝐺 ∈ UHGraph → Fun 𝐸) | 
| 13 | 10, 12 | syl 17 | . . 3 ⊢ (𝜑 → Fun 𝐸) | 
| 14 | eqid 2736 | . . . 4 ⊢ (Vtx‘𝑆) = (Vtx‘𝑆) | |
| 15 | eqid 2736 | . . . 4 ⊢ (iEdg‘𝑆) = (iEdg‘𝑆) | |
| 16 | eqid 2736 | . . . 4 ⊢ (Edg‘𝑆) = (Edg‘𝑆) | |
| 17 | 14, 7, 15, 8, 16 | issubgr2 29290 | . . 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 1372 | . 2 ⊢ (𝜑 → (𝑆 SubGraph 𝐺 ↔ ((Vtx‘𝑆) ⊆ 𝑉 ∧ (iEdg‘𝑆) ⊆ 𝐸 ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆)))) | 
| 19 | 3, 6, 11, 18 | mpbir3and 1342 | 1 ⊢ (𝜑 → 𝑆 SubGraph 𝐺) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ w3a 1086 = wceq 1539 ∈ wcel 2107 ⊆ wss 3950 𝒫 cpw 4599 class class class wbr 5142 ↾ cres 5686 Fun wfun 6554 ‘cfv 6560 Vtxcvtx 29014 iEdgciedg 29015 Edgcedg 29065 UHGraphcuhgr 29074 SubGraph csubgr 29285 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-sbc 3788 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-fv 6568 df-edg 29066 df-uhgr 29076 df-subgr 29286 | 
| This theorem is referenced by: uhgrspan 29310 upgrspan 29311 umgrspan 29312 usgrspan 29313 | 
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