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Theorem uhgrspansubgr 27067
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.)
Hypotheses
Ref Expression
uhgrspan.v 𝑉 = (Vtx‘𝐺)
uhgrspan.e 𝐸 = (iEdg‘𝐺)
uhgrspan.s (𝜑𝑆𝑊)
uhgrspan.q (𝜑 → (Vtx‘𝑆) = 𝑉)
uhgrspan.r (𝜑 → (iEdg‘𝑆) = (𝐸𝐴))
uhgrspan.g (𝜑𝐺 ∈ UHGraph)
Assertion
Ref Expression
uhgrspansubgr (𝜑𝑆 SubGraph 𝐺)

Proof of Theorem uhgrspansubgr
StepHypRef Expression
1 ssid 3989 . . 3 (Vtx‘𝑆) ⊆ (Vtx‘𝑆)
2 uhgrspan.q . . 3 (𝜑 → (Vtx‘𝑆) = 𝑉)
31, 2sseqtrid 4019 . 2 (𝜑 → (Vtx‘𝑆) ⊆ 𝑉)
4 uhgrspan.r . . 3 (𝜑 → (iEdg‘𝑆) = (𝐸𝐴))
5 resss 5873 . . 3 (𝐸𝐴) ⊆ 𝐸
64, 5eqsstrdi 4021 . 2 (𝜑 → (iEdg‘𝑆) ⊆ 𝐸)
7 uhgrspan.v . . 3 𝑉 = (Vtx‘𝐺)
8 uhgrspan.e . . 3 𝐸 = (iEdg‘𝐺)
9 uhgrspan.s . . 3 (𝜑𝑆𝑊)
10 uhgrspan.g . . 3 (𝜑𝐺 ∈ UHGraph)
117, 8, 9, 2, 4, 10uhgrspansubgrlem 27066 . 2 (𝜑 → (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆))
128uhgrfun 26845 . . . 4 (𝐺 ∈ UHGraph → Fun 𝐸)
1310, 12syl 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)
1714, 7, 15, 8, 16issubgr2 27048 . . 3 ((𝐺 ∈ UHGraph ∧ Fun 𝐸𝑆𝑊) → (𝑆 SubGraph 𝐺 ↔ ((Vtx‘𝑆) ⊆ 𝑉 ∧ (iEdg‘𝑆) ⊆ 𝐸 ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆))))
1810, 13, 9, 17syl3anc 1367 . 2 (𝜑 → (𝑆 SubGraph 𝐺 ↔ ((Vtx‘𝑆) ⊆ 𝑉 ∧ (iEdg‘𝑆) ⊆ 𝐸 ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆))))
193, 6, 11, 18mpbir3and 1338 1 (𝜑𝑆 SubGraph 𝐺)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  w3a 1083   = wceq 1533  wcel 2110  wss 3936  𝒫 cpw 4539   class class class wbr 5059  cres 5552  Fun wfun 6344  cfv 6350  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 2156  ax-12 2172  ax-ext 2793  ax-sep 5196  ax-nul 5203  ax-pow 5259  ax-pr 5322  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 3497  df-sbc 3773  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4833  df-br 5060  df-opab 5122  df-mpt 5140  df-id 5455  df-xp 5556  df-rel 5557  df-cnv 5558  df-co 5559  df-dm 5560  df-rn 5561  df-res 5562  df-iota 6309  df-fun 6352  df-fn 6353  df-f 6354  df-fv 6358  df-edg 26827  df-uhgr 26837  df-subgr 27044
This theorem is referenced by:  uhgrspan  27068  upgrspan  27069  umgrspan  27070  usgrspan  27071
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