Theorem isausgr 29127

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	⊢ ((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 484	. . 3 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → 𝑒 = 𝐸)
2		pweq 4567	. . . . 5 ⊢ (𝑣 = 𝑉 → 𝒫 𝑣 = 𝒫 𝑉)
3	2	adantr 480	. . . 4 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → 𝒫 𝑣 = 𝒫 𝑉)
4	3	rabeqdv 3412	. . 3 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → {𝑥 ∈ 𝒫 𝑣 ∣ (♯‘𝑥) = 2} = {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2})
5	1, 4	sseq12d 3971	. 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 5481	1 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → (𝑉𝐺𝐸 ↔ 𝐸 ⊆ {𝑥 ∈ 𝒫 𝑉 ∣ (♯‘𝑥) = 2}))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  {crab 3396   ⊆ wss 3905  𝒫 cpw 4553   class class class wbr 5095  {copab 5157  ‘cfv 6486  2c2 12201  ♯chash 14255

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-rab 3397  df-v 3440  df-dif 3908  df-un 3910  df-ss 3922  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-br 5096  df-opab 5158

This theorem is referenced by:  ausgrusgrb  29128  usgrausgri  29129  ausgrumgri  29130  ausgrusgri  29131