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Theorem uhgrspansubgr 40512
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 3581 . . 3 (Vtx‘𝑆) ⊆ (Vtx‘𝑆)
2 uhgrspan.q . . 3 (𝜑 → (Vtx‘𝑆) = 𝑉)
31, 2syl5sseq 3610 . 2 (𝜑 → (Vtx‘𝑆) ⊆ 𝑉)
4 uhgrspan.r . . 3 (𝜑 → (iEdg‘𝑆) = (𝐸𝐴))
5 resss 5324 . . 3 (𝐸𝐴) ⊆ 𝐸
64, 5syl6eqss 3612 . 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 40511 . 2 (𝜑 → (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆))
128uhgrfun 40285 . . . 4 (𝐺 ∈ UHGraph → Fun 𝐸)
1310, 12syl 17 . . 3 (𝜑 → Fun 𝐸)
14 eqid 2604 . . . 4 (Vtx‘𝑆) = (Vtx‘𝑆)
15 eqid 2604 . . . 4 (iEdg‘𝑆) = (iEdg‘𝑆)
16 eqid 2604 . . . 4 (Edg‘𝑆) = (Edg‘𝑆)
1714, 7, 15, 8, 16issubgr2 40493 . . 3 ((𝐺 ∈ UHGraph ∧ Fun 𝐸𝑆𝑊) → (𝑆 SubGraph 𝐺 ↔ ((Vtx‘𝑆) ⊆ 𝑉 ∧ (iEdg‘𝑆) ⊆ 𝐸 ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆))))
1810, 13, 9, 17syl3anc 1317 . 2 (𝜑 → (𝑆 SubGraph 𝐺 ↔ ((Vtx‘𝑆) ⊆ 𝑉 ∧ (iEdg‘𝑆) ⊆ 𝐸 ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆))))
193, 6, 11, 18mpbir3and 1237 1 (𝜑𝑆 SubGraph 𝐺)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  w3a 1030   = wceq 1474  wcel 1975  wss 3534  𝒫 cpw 4102   class class class wbr 4572  cres 5025  Fun wfun 5779  cfv 5785  Vtxcvtx 40226  iEdgciedg 40227   UHGraph cuhgr 40275  Edgcedga 40348   SubGraph csubgr 40488
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2227  ax-ext 2584  ax-sep 4698  ax-nul 4707  ax-pr 4823  ax-un 6819
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2456  df-mo 2457  df-clab 2591  df-cleq 2597  df-clel 2600  df-nfc 2734  df-ral 2895  df-rex 2896  df-rab 2899  df-v 3169  df-sbc 3397  df-csb 3494  df-dif 3537  df-un 3539  df-in 3541  df-ss 3548  df-nul 3869  df-if 4031  df-pw 4104  df-sn 4120  df-pr 4122  df-op 4126  df-uni 4362  df-br 4573  df-opab 4633  df-mpt 4634  df-id 4938  df-xp 5029  df-rel 5030  df-cnv 5031  df-co 5032  df-dm 5033  df-rn 5034  df-res 5035  df-iota 5749  df-fun 5787  df-fn 5788  df-f 5789  df-fv 5793  df-uhgr 40277  df-edga 40349  df-subgr 40489
This theorem is referenced by:  uhgrspan  40513  upgrspan  40514  umgrspan  40515  usgrspan  40516
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