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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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