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Theorem upgrspan 26112
 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 25926 . . . 4 (𝐺 ∈ UPGraph → 𝐺 ∈ UHGraph )
81, 7syl 17 . . 3 (𝜑𝐺 ∈ UHGraph )
92, 3, 4, 5, 6, 8uhgrspansubgr 26110 . 2 (𝜑𝑆 SubGraph 𝐺)
10 subupgr 26106 . 2 ((𝐺 ∈ UPGraph ∧ 𝑆 SubGraph 𝐺) → 𝑆 ∈ UPGraph )
111, 9, 10syl2anc 692 1 (𝜑𝑆 ∈ UPGraph )
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1480   ∈ wcel 1987   class class class wbr 4623   ↾ cres 5086  ‘cfv 5857  Vtxcvtx 25808  iEdgciedg 25809   UHGraph cuhgr 25881   UPGraph cupgr 25905   SubGraph csubgr 26086 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pr 4877  ax-un 6914 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-fv 5865  df-edg 25874  df-uhgr 25883  df-upgr 25907  df-subgr 26087 This theorem is referenced by:  upgrspanop  26116
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