Theorem isausgr 29745

Description: The property of an ordered 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	⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → (𝑉𝐺𝐸 ↔ 𝐸 ⊆ {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

Step	Hyp	Ref	Expression
1		simpr 490	. . 3 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → 𝑒 = 𝐸)
2		pweq 4570	. . . . 5 ⊢ (𝑣 = 𝑉 → 𝒫 𝑣 = 𝒫 𝑉)
3	2	adantr 486	. . . 4 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → 𝒫 𝑣 = 𝒫 𝑉)
4	3	rabeqdv 3427	. . 3 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → {𝑥 ∈ 𝒫 𝑣 ∣ (♯‘𝑥) = 2} = {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2})
5	1, 4	sseq12d 3941	. 2 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 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}}
7	5, 6	brabga 5507	1 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → (𝑉𝐺𝐸 ↔ 𝐸 ⊆ {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2}))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2144  {crab 3412   ⊆ wss 3895  𝒫 cpw 4556   class class class wbr 5102  {copab 5166  ‘cfv 6538  2c2 12397  ♯chash 14474

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-br 5103  df-opab 5167

This theorem is used by:  ausgrusgrb  29746  usgrausgri  29747  ausgrumgri  29748  ausgrusgri  29749