MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  upgrss Structured version   Visualization version   GIF version

Theorem upgrss 29105
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 4080 . . . 4 {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} ⊆ (𝒫 𝑉 ∖ {∅})
2 difss 4136 . . . 4 (𝒫 𝑉 ∖ {∅}) ⊆ 𝒫 𝑉
31, 2sstri 3993 . . 3 {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} ⊆ 𝒫 𝑉
4 isupgr.v . . . . 5 𝑉 = (Vtx‘𝐺)
5 isupgr.e . . . . 5 𝐸 = (iEdg‘𝐺)
64, 5upgrf 29103 . . . 4 (𝐺 ∈ UPGraph → 𝐸:dom 𝐸⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
76ffvelcdmda 7104 . . 3 ((𝐺 ∈ UPGraph ∧ 𝐹 ∈ dom 𝐸) → (𝐸𝐹) ∈ {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
83, 7sselid 3981 . 2 ((𝐺 ∈ UPGraph ∧ 𝐹 ∈ dom 𝐸) → (𝐸𝐹) ∈ 𝒫 𝑉)
98elpwid 4609 1 ((𝐺 ∈ UPGraph ∧ 𝐹 ∈ dom 𝐸) → (𝐸𝐹) ⊆ 𝑉)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  {crab 3436  cdif 3948  wss 3951  c0 4333  𝒫 cpw 4600  {csn 4626   class class class wbr 5143  dom cdm 5685  cfv 6561  cle 11296  2c2 12321  chash 14369  Vtxcvtx 29013  iEdgciedg 29014  UPGraphcupgr 29097
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-upgr 29099
This theorem is referenced by:  upgrex  29109  isuspgrim0  47872
  Copyright terms: Public domain W3C validator