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Theorem uhgrspansubgr 26164
 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 3616 . . 3 (Vtx‘𝑆) ⊆ (Vtx‘𝑆)
2 uhgrspan.q . . 3 (𝜑 → (Vtx‘𝑆) = 𝑉)
31, 2syl5sseq 3645 . 2 (𝜑 → (Vtx‘𝑆) ⊆ 𝑉)
4 uhgrspan.r . . 3 (𝜑 → (iEdg‘𝑆) = (𝐸𝐴))
5 resss 5410 . . 3 (𝐸𝐴) ⊆ 𝐸
64, 5syl6eqss 3647 . 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 26163 . 2 (𝜑 → (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆))
128uhgrfun 25942 . . . 4 (𝐺 ∈ UHGraph → Fun 𝐸)
1310, 12syl 17 . . 3 (𝜑 → Fun 𝐸)
14 eqid 2620 . . . 4 (Vtx‘𝑆) = (Vtx‘𝑆)
15 eqid 2620 . . . 4 (iEdg‘𝑆) = (iEdg‘𝑆)
16 eqid 2620 . . . 4 (Edg‘𝑆) = (Edg‘𝑆)
1714, 7, 15, 8, 16issubgr2 26145 . . 3 ((𝐺 ∈ UHGraph ∧ Fun 𝐸𝑆𝑊) → (𝑆 SubGraph 𝐺 ↔ ((Vtx‘𝑆) ⊆ 𝑉 ∧ (iEdg‘𝑆) ⊆ 𝐸 ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆))))
1810, 13, 9, 17syl3anc 1324 . 2 (𝜑 → (𝑆 SubGraph 𝐺 ↔ ((Vtx‘𝑆) ⊆ 𝑉 ∧ (iEdg‘𝑆) ⊆ 𝐸 ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆))))
193, 6, 11, 18mpbir3and 1243 1 (𝜑𝑆 SubGraph 𝐺)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ w3a 1036   = wceq 1481   ∈ wcel 1988   ⊆ wss 3567  𝒫 cpw 4149   class class class wbr 4644   ↾ cres 5106  Fun wfun 5870  ‘cfv 5876  Vtxcvtx 25855  iEdgciedg 25856  Edgcedg 25920   UHGraph cuhgr 25932   SubGraph csubgr 26140 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-fv 5884  df-edg 25921  df-uhgr 25934  df-subgr 26141 This theorem is referenced by:  uhgrspan  26165  upgrspan  26166  umgrspan  26167  usgrspan  26168
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