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Theorem isausgr 26104
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 4194 . . . . 5 (𝑣 = 𝑉 → 𝒫 𝑣 = 𝒫 𝑉)
32adantr 480 . . . 4 ((𝑣 = 𝑉𝑒 = 𝐸) → 𝒫 𝑣 = 𝒫 𝑉)
43rabeqdv 3225 . . 3 ((𝑣 = 𝑉𝑒 = 𝐸) → {𝑥 ∈ 𝒫 𝑣 ∣ (#‘𝑥) = 2} = {𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2})
51, 4sseq12d 3667 . 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 5018 1 ((𝑉𝑊𝐸𝑋) → (𝑉𝐺𝐸𝐸 ⊆ {𝑥 ∈ 𝒫 𝑉 ∣ (#‘𝑥) = 2}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  {crab 2945  wss 3607  𝒫 cpw 4191   class class class wbr 4685  {copab 4745  cfv 5926  2c2 11108  #chash 13157
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-opab 4746
This theorem is referenced by:  ausgrusgrb  26105  usgrausgri  26106  ausgrumgri  26107  ausgrusgri  26108
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