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Theorem isausgr 40394
 Description: The property of an unordered pair to be an alternatively defined simple graph, defined as a pair (V,E) of a set V (vertex set) and a set of unordered pairs of elements of V (edge set). (Contributed by Alexander van der Vekens, 28-Aug-2017.)
Hypothesis
Ref Expression
ausgr.1 𝐺 = {⟨𝑣, 𝑒⟩ ∣ 𝑒 ⊆ {𝑥 ∈ 𝒫 𝑣 ∣ (#‘𝑥) = 2}}
Assertion
Ref Expression
isausgr ((𝑉𝑊𝐸𝑋) → (𝑉𝐺𝐸𝐸 ⊆ {𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2}))
Distinct variable groups:   𝑣,𝑒,𝑥,𝐸   𝑒,𝑉,𝑣,𝑥   𝑥,𝑋
Allowed substitution hints:   𝐺(𝑥,𝑣,𝑒)   𝑊(𝑥,𝑣,𝑒)   𝑋(𝑣,𝑒)

Proof of Theorem isausgr
StepHypRef Expression
1 simpr 476 . . 3 ((𝑣 = 𝑉𝑒 = 𝐸) → 𝑒 = 𝐸)
2 pweq 4111 . . . . 5 (𝑣 = 𝑉 → 𝒫 𝑣 = 𝒫 𝑉)
32adantr 480 . . . 4 ((𝑣 = 𝑉𝑒 = 𝐸) → 𝒫 𝑣 = 𝒫 𝑉)
43rabeqdv 3167 . . 3 ((𝑣 = 𝑉𝑒 = 𝐸) → {𝑥 ∈ 𝒫 𝑣 ∣ (#‘𝑥) = 2} = {𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2})
51, 4sseq12d 3597 . 2 ((𝑣 = 𝑉𝑒 = 𝐸) → (𝑒 ⊆ {𝑥 ∈ 𝒫 𝑣 ∣ (#‘𝑥) = 2} ↔ 𝐸 ⊆ {𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2}))
6 ausgr.1 . 2 𝐺 = {⟨𝑣, 𝑒⟩ ∣ 𝑒 ⊆ {𝑥 ∈ 𝒫 𝑣 ∣ (#‘𝑥) = 2}}
75, 6brabga 4914 1 ((𝑉𝑊𝐸𝑋) → (𝑉𝐺𝐸𝐸 ⊆ {𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2}))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {crab 2900   ⊆ wss 3540  𝒫 cpw 4108   class class class wbr 4583  {copab 4642  ‘cfv 5804  2c2 10947  #chash 12979 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-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 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-rab 2905  df-v 3175  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-br 4584  df-opab 4644 This theorem is referenced by:  ausgrusgrb  40395  usgrausgri  40396  ausgrumgri  40397  ausgrusgri  40398
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