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Theorem upgrss 28337
Description: An edge is a subset of vertices. (Contributed by Mario Carneiro, 11-Mar-2015.) (Revised by AV, 29-Nov-2020.)
Hypotheses
Ref Expression
isupgr.v 𝑉 = (Vtx‘𝐺)
isupgr.e 𝐸 = (iEdg‘𝐺)
Assertion
Ref Expression
upgrss ((𝐺 ∈ UPGraph ∧ 𝐹 ∈ dom 𝐸) → (𝐸𝐹) ⊆ 𝑉)

Proof of Theorem upgrss
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 ssrab2 4076 . . . 4 {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} ⊆ (𝒫 𝑉 ∖ {∅})
2 difss 4130 . . . 4 (𝒫 𝑉 ∖ {∅}) ⊆ 𝒫 𝑉
31, 2sstri 3990 . . 3 {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} ⊆ 𝒫 𝑉
4 isupgr.v . . . . 5 𝑉 = (Vtx‘𝐺)
5 isupgr.e . . . . 5 𝐸 = (iEdg‘𝐺)
64, 5upgrf 28335 . . . 4 (𝐺 ∈ UPGraph → 𝐸:dom 𝐸⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
76ffvelcdmda 7083 . . 3 ((𝐺 ∈ UPGraph ∧ 𝐹 ∈ dom 𝐸) → (𝐸𝐹) ∈ {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
83, 7sselid 3979 . 2 ((𝐺 ∈ UPGraph ∧ 𝐹 ∈ dom 𝐸) → (𝐸𝐹) ∈ 𝒫 𝑉)
98elpwid 4610 1 ((𝐺 ∈ UPGraph ∧ 𝐹 ∈ dom 𝐸) → (𝐸𝐹) ⊆ 𝑉)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  {crab 3432  cdif 3944  wss 3947  c0 4321  𝒫 cpw 4601  {csn 4627   class class class wbr 5147  dom cdm 5675  cfv 6540  cle 11245  2c2 12263  chash 14286  Vtxcvtx 28245  iEdgciedg 28246  UPGraphcupgr 28329
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-fv 6548  df-upgr 28331
This theorem is referenced by:  upgrex  28341
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