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Theorem upgrspan 29126
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 28935 . . . 4 (𝐺 ∈ UPGraph → 𝐺 ∈ UHGraph)
81, 7syl 17 . . 3 (𝜑𝐺 ∈ UHGraph)
92, 3, 4, 5, 6, 8uhgrspansubgr 29124 . 2 (𝜑𝑆 SubGraph 𝐺)
10 subupgr 29120 . 2 ((𝐺 ∈ UPGraph ∧ 𝑆 SubGraph 𝐺) → 𝑆 ∈ UPGraph)
111, 9, 10syl2anc 582 1 (𝜑𝑆 ∈ UPGraph)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098   class class class wbr 5152  cres 5684  cfv 6553  Vtxcvtx 28829  iEdgciedg 28830  UHGraphcuhgr 28889  UPGraphcupgr 28913   SubGraph csubgr 29100
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-fv 6561  df-edg 28881  df-uhgr 28891  df-upgr 28915  df-subgr 29101
This theorem is referenced by:  upgrspanop  29130
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