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

Theorem uhgrss 26179
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 26177 . . . 4 (𝐺 ∈ UHGraph → 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅}))
43ffvelrnda 6502 . . 3 ((𝐺 ∈ UHGraph ∧ 𝐹 ∈ dom 𝐸) → (𝐸𝐹) ∈ (𝒫 𝑉 ∖ {∅}))
54eldifad 3733 . 2 ((𝐺 ∈ UHGraph ∧ 𝐹 ∈ dom 𝐸) → (𝐸𝐹) ∈ 𝒫 𝑉)
65elpwid 4307 1 ((𝐺 ∈ UHGraph ∧ 𝐹 ∈ dom 𝐸) → (𝐸𝐹) ⊆ 𝑉)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1630  wcel 2144  cdif 3718  wss 3721  c0 4061  𝒫 cpw 4295  {csn 4314  dom cdm 5249  cfv 6031  Vtxcvtx 26094  iEdgciedg 26095  UHGraphcuhgr 26171
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-sbc 3586  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-br 4785  df-opab 4845  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-fv 6039  df-uhgr 26173
This theorem is referenced by:  lpvtx  26183  umgredgprv  26222  uhgrspansubgrlem  26404  uhgrspan1  26417
  Copyright terms: Public domain W3C validator