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

Theorem uhgrac 25834
 Description: The property of being an undirected hypergraph represented by a class. This representation is useful if the set of vertices and the edge function is/needs not to be known. (Contributed by AV, 1-Jan-2020.)
Assertion
Ref Expression
uhgrac (𝐺 ∈ UHGrph → (2nd𝐺):dom (2nd𝐺)⟶(𝒫 (1st𝐺) ∖ {∅}))

Proof of Theorem uhgrac
Dummy variables 𝑒 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 pweq 4111 . . . . 5 (𝑣 = (1st𝐺) → 𝒫 𝑣 = 𝒫 (1st𝐺))
21difeq1d 3689 . . . 4 (𝑣 = (1st𝐺) → (𝒫 𝑣 ∖ {∅}) = (𝒫 (1st𝐺) ∖ {∅}))
32feq3d 5945 . . 3 (𝑣 = (1st𝐺) → (𝑒:dom 𝑒⟶(𝒫 𝑣 ∖ {∅}) ↔ 𝑒:dom 𝑒⟶(𝒫 (1st𝐺) ∖ {∅})))
4 id 22 . . . 4 (𝑒 = (2nd𝐺) → 𝑒 = (2nd𝐺))
5 dmeq 5246 . . . 4 (𝑒 = (2nd𝐺) → dom 𝑒 = dom (2nd𝐺))
64, 5feq12d 5946 . . 3 (𝑒 = (2nd𝐺) → (𝑒:dom 𝑒⟶(𝒫 (1st𝐺) ∖ {∅}) ↔ (2nd𝐺):dom (2nd𝐺)⟶(𝒫 (1st𝐺) ∖ {∅})))
73, 6elopabi 7120 . 2 (𝐺 ∈ {⟨𝑣, 𝑒⟩ ∣ 𝑒:dom 𝑒⟶(𝒫 𝑣 ∖ {∅})} → (2nd𝐺):dom (2nd𝐺)⟶(𝒫 (1st𝐺) ∖ {∅}))
8 df-uhgra 25821 . 2 UHGrph = {⟨𝑣, 𝑒⟩ ∣ 𝑒:dom 𝑒⟶(𝒫 𝑣 ∖ {∅})}
97, 8eleq2s 2706 1 (𝐺 ∈ UHGrph → (2nd𝐺):dom (2nd𝐺)⟶(𝒫 (1st𝐺) ∖ {∅}))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977   ∖ cdif 3537  ∅c0 3874  𝒫 cpw 4108  {csn 4125  {copab 4642  dom cdm 5038  ⟶wf 5800  ‘cfv 5804  1st c1st 7057  2nd c2nd 7058   UHGrph cuhg 25819 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-1st 7059  df-2nd 7060  df-uhgra 25821 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator