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Theorem uhgrspan 26165
Description: A spanning subgraph 𝑆 of a hypergraph 𝐺 is a hypergraph. (Contributed by AV, 11-Oct-2020.) (Proof shortened by AV, 18-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
uhgrspan (𝜑𝑆 ∈ UHGraph )

Proof of Theorem uhgrspan
StepHypRef Expression
1 uhgrspan.g . 2 (𝜑𝐺 ∈ UHGraph )
2 uhgrspan.v . . 3 𝑉 = (Vtx‘𝐺)
3 uhgrspan.e . . 3 𝐸 = (iEdg‘𝐺)
4 uhgrspan.s . . 3 (𝜑𝑆𝑊)
5 uhgrspan.q . . 3 (𝜑 → (Vtx‘𝑆) = 𝑉)
6 uhgrspan.r . . 3 (𝜑 → (iEdg‘𝑆) = (𝐸𝐴))
72, 3, 4, 5, 6, 1uhgrspansubgr 26164 . 2 (𝜑𝑆 SubGraph 𝐺)
8 subuhgr 26159 . 2 ((𝐺 ∈ UHGraph ∧ 𝑆 SubGraph 𝐺) → 𝑆 ∈ UHGraph )
91, 7, 8syl2anc 692 1 (𝜑𝑆 ∈ UHGraph )
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1481  wcel 1988   class class class wbr 4644  cres 5106  cfv 5876  Vtxcvtx 25855  iEdgciedg 25856   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-ne 2792  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:  uhgrspanop  26169
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