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Theorem uhgrss 25855
Description: An edge is a subset of vertices. (Contributed by Alexander van der Vekens, 26-Dec-2017.) (Revised by AV, 18-Jan-2020.)
Hypotheses
Ref Expression
uhgrf.v 𝑉 = (Vtx‘𝐺)
uhgrf.e 𝐸 = (iEdg‘𝐺)
Assertion
Ref Expression
uhgrss ((𝐺 ∈ UHGraph ∧ 𝐹 ∈ dom 𝐸) → (𝐸𝐹) ⊆ 𝑉)

Proof of Theorem uhgrss
StepHypRef Expression
1 uhgrf.v . . . . 5 𝑉 = (Vtx‘𝐺)
2 uhgrf.e . . . . 5 𝐸 = (iEdg‘𝐺)
31, 2uhgrf 25853 . . . 4 (𝐺 ∈ UHGraph → 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅}))
43ffvelrnda 6315 . . 3 ((𝐺 ∈ UHGraph ∧ 𝐹 ∈ dom 𝐸) → (𝐸𝐹) ∈ (𝒫 𝑉 ∖ {∅}))
54eldifad 3567 . 2 ((𝐺 ∈ UHGraph ∧ 𝐹 ∈ dom 𝐸) → (𝐸𝐹) ∈ 𝒫 𝑉)
65elpwid 4141 1 ((𝐺 ∈ UHGraph ∧ 𝐹 ∈ dom 𝐸) → (𝐸𝐹) ⊆ 𝑉)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  cdif 3552  wss 3555  c0 3891  𝒫 cpw 4130  {csn 4148  dom cdm 5074  cfv 5847  Vtxcvtx 25774  iEdgciedg 25775   UHGraph cuhgr 25847
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-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867
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-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-fv 5855  df-uhgr 25849
This theorem is referenced by:  lpvtx  25859  umgredgprv  25897  uhgrspansubgrlem  26075  uhgrspan1  26088
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