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Theorem upgrspan 40517
Description: A spanning subgraph 𝑆 of a pseudograph 𝐺 is a pseudograph. (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))
upgrspan.g (𝜑𝐺 ∈ UPGraph )
Assertion
Ref Expression
upgrspan (𝜑𝑆 ∈ UPGraph )

Proof of Theorem upgrspan
StepHypRef Expression
1 upgrspan.g . 2 (𝜑𝐺 ∈ UPGraph )
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))
7 upgruhgr 25768 . . . 4 (𝐺 ∈ UPGraph → 𝐺 ∈ UHGraph )
81, 7syl 17 . . 3 (𝜑𝐺 ∈ UHGraph )
92, 3, 4, 5, 6, 8uhgrspansubgr 40515 . 2 (𝜑𝑆 SubGraph 𝐺)
10 subupgr 40511 . 2 ((𝐺 ∈ UPGraph ∧ 𝑆 SubGraph 𝐺) → 𝑆 ∈ UPGraph )
111, 9, 10syl2anc 691 1 (𝜑𝑆 ∈ UPGraph )
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1475  wcel 1977   class class class wbr 4583  cres 5040  cfv 5804  Vtxcvtx 25673  iEdgciedg 25674   UHGraph cuhgr 25722   UPGraph cupgr 25747   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-ne 2782  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-upgr 25749  df-edga 25793  df-subgr 40492
This theorem is referenced by:  upgrspanop  40521
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