MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  uhgrspansubgr Structured version   Visualization version   GIF version

Theorem uhgrspansubgr 26595
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 3848 . . 3 (Vtx‘𝑆) ⊆ (Vtx‘𝑆)
2 uhgrspan.q . . 3 (𝜑 → (Vtx‘𝑆) = 𝑉)
31, 2syl5sseq 3878 . 2 (𝜑 → (Vtx‘𝑆) ⊆ 𝑉)
4 uhgrspan.r . . 3 (𝜑 → (iEdg‘𝑆) = (𝐸𝐴))
5 resss 5662 . . 3 (𝐸𝐴) ⊆ 𝐸
64, 5syl6eqss 3880 . 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 26594 . 2 (𝜑 → (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆))
128uhgrfun 26371 . . . 4 (𝐺 ∈ UHGraph → Fun 𝐸)
1310, 12syl 17 . . 3 (𝜑 → Fun 𝐸)
14 eqid 2825 . . . 4 (Vtx‘𝑆) = (Vtx‘𝑆)
15 eqid 2825 . . . 4 (iEdg‘𝑆) = (iEdg‘𝑆)
16 eqid 2825 . . . 4 (Edg‘𝑆) = (Edg‘𝑆)
1714, 7, 15, 8, 16issubgr2 26576 . . 3 ((𝐺 ∈ UHGraph ∧ Fun 𝐸𝑆𝑊) → (𝑆 SubGraph 𝐺 ↔ ((Vtx‘𝑆) ⊆ 𝑉 ∧ (iEdg‘𝑆) ⊆ 𝐸 ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆))))
1810, 13, 9, 17syl3anc 1494 . 2 (𝜑 → (𝑆 SubGraph 𝐺 ↔ ((Vtx‘𝑆) ⊆ 𝑉 ∧ (iEdg‘𝑆) ⊆ 𝐸 ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆))))
193, 6, 11, 18mpbir3and 1446 1 (𝜑𝑆 SubGraph 𝐺)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  w3a 1111   = wceq 1656  wcel 2164  wss 3798  𝒫 cpw 4380   class class class wbr 4875  cres 5348  Fun wfun 6121  cfv 6127  Vtxcvtx 26301  iEdgciedg 26302  Edgcedg 26352  UHGraphcuhgr 26361   SubGraph csubgr 26571
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-br 4876  df-opab 4938  df-mpt 4955  df-id 5252  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-fv 6135  df-edg 26353  df-uhgr 26363  df-subgr 26572
This theorem is referenced by:  uhgrspan  26596  upgrspan  26597  umgrspan  26598  usgrspan  26599
  Copyright terms: Public domain W3C validator