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

Theorem ushgruhgr 26134
 Description: An undirected simple hypergraph is an undirected hypergraph. (Contributed by AV, 19-Jan-2020.) (Revised by AV, 9-Oct-2020.)
Assertion
Ref Expression
ushgruhgr (𝐺 ∈ USHGraph → 𝐺 ∈ UHGraph)

Proof of Theorem ushgruhgr
StepHypRef Expression
1 eqid 2748 . . . 4 (Vtx‘𝐺) = (Vtx‘𝐺)
2 eqid 2748 . . . 4 (iEdg‘𝐺) = (iEdg‘𝐺)
31, 2ushgrf 26128 . . 3 (𝐺 ∈ USHGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→(𝒫 (Vtx‘𝐺) ∖ {∅}))
4 f1f 6250 . . 3 ((iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→(𝒫 (Vtx‘𝐺) ∖ {∅}) → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶(𝒫 (Vtx‘𝐺) ∖ {∅}))
53, 4syl 17 . 2 (𝐺 ∈ USHGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶(𝒫 (Vtx‘𝐺) ∖ {∅}))
61, 2isuhgr 26125 . 2 (𝐺 ∈ USHGraph → (𝐺 ∈ UHGraph ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)⟶(𝒫 (Vtx‘𝐺) ∖ {∅})))
75, 6mpbird 247 1 (𝐺 ∈ USHGraph → 𝐺 ∈ UHGraph)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 2127   ∖ cdif 3700  ∅c0 4046  𝒫 cpw 4290  {csn 4309  dom cdm 5254  ⟶wf 6033  –1-1→wf1 6034  ‘cfv 6037  Vtxcvtx 26044  iEdgciedg 26045  UHGraphcuhgr 26121  USHGraphcushgr 26122 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728  ax-nul 4929 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1623  df-ex 1842  df-nf 1847  df-sb 2035  df-eu 2599  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-ral 3043  df-rex 3044  df-rab 3047  df-v 3330  df-sbc 3565  df-dif 3706  df-un 3708  df-in 3710  df-ss 3717  df-nul 4047  df-if 4219  df-pw 4292  df-sn 4310  df-pr 4312  df-op 4316  df-uni 4577  df-br 4793  df-opab 4853  df-rel 5261  df-cnv 5262  df-co 5263  df-dm 5264  df-rn 5265  df-iota 6000  df-fun 6039  df-fn 6040  df-f 6041  df-f1 6042  df-fv 6045  df-uhgr 26123  df-ushgr 26124 This theorem is referenced by:  ushgrun  26141  ushgrunop  26142  ushgredgedg  26291  ushgredgedgloop  26293
 Copyright terms: Public domain W3C validator