Mathbox for Alexander van der Vekens < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  uhgrspansubgr Structured version   Visualization version   GIF version

Theorem uhgrspansubgr 40515
 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 3587 . . 3 (Vtx‘𝑆) ⊆ (Vtx‘𝑆)
2 uhgrspan.q . . 3 (𝜑 → (Vtx‘𝑆) = 𝑉)
31, 2syl5sseq 3616 . 2 (𝜑 → (Vtx‘𝑆) ⊆ 𝑉)
4 uhgrspan.r . . 3 (𝜑 → (iEdg‘𝑆) = (𝐸𝐴))
5 resss 5342 . . 3 (𝐸𝐴) ⊆ 𝐸
64, 5syl6eqss 3618 . 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 40514 . 2 (𝜑 → (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆))
128uhgrfun 25732 . . . 4 (𝐺 ∈ UHGraph → Fun 𝐸)
1310, 12syl 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)
1714, 7, 15, 8, 16issubgr2 40496 . . 3 ((𝐺 ∈ UHGraph ∧ Fun 𝐸𝑆𝑊) → (𝑆 SubGraph 𝐺 ↔ ((Vtx‘𝑆) ⊆ 𝑉 ∧ (iEdg‘𝑆) ⊆ 𝐸 ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆))))
1810, 13, 9, 17syl3anc 1318 . 2 (𝜑 → (𝑆 SubGraph 𝐺 ↔ ((Vtx‘𝑆) ⊆ 𝑉 ∧ (iEdg‘𝑆) ⊆ 𝐸 ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆))))
193, 6, 11, 18mpbir3and 1238 1 (𝜑𝑆 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
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