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

Theorem ushgrf 26840
 Description: The edge function of an undirected simple hypergraph is a one-to-one function into the power set of the set of vertices. (Contributed by AV, 9-Oct-2020.)
Hypotheses
Ref Expression
uhgrf.v 𝑉 = (Vtx‘𝐺)
uhgrf.e 𝐸 = (iEdg‘𝐺)
Assertion
Ref Expression
ushgrf (𝐺 ∈ USHGraph → 𝐸:dom 𝐸1-1→(𝒫 𝑉 ∖ {∅}))

Proof of Theorem ushgrf
StepHypRef Expression
1 uhgrf.v . . 3 𝑉 = (Vtx‘𝐺)
2 uhgrf.e . . 3 𝐸 = (iEdg‘𝐺)
31, 2isushgr 26838 . 2 (𝐺 ∈ USHGraph → (𝐺 ∈ USHGraph ↔ 𝐸:dom 𝐸1-1→(𝒫 𝑉 ∖ {∅})))
43ibi 269 1 (𝐺 ∈ USHGraph → 𝐸:dom 𝐸1-1→(𝒫 𝑉 ∖ {∅}))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1531   ∈ wcel 2108   ∖ cdif 3931  ∅c0 4289  𝒫 cpw 4537  {csn 4559  dom cdm 5548  –1-1→wf1 6345  ‘cfv 6348  Vtxcvtx 26773  iEdgciedg 26774  USHGraphcushgr 26834 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-nul 5201 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-br 5058  df-opab 5120  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fv 6356  df-ushgr 26836 This theorem is referenced by:  ushgruhgr  26846  uspgrupgrushgr  26954  ushgredgedg  27003  ushgredgedgloop  27005  isomushgr  43982  ushrisomgr  43997
 Copyright terms: Public domain W3C validator