Theorem sprsymrelen

Description: The class P of subsets of the set of pairs over a fixed set V and the class R of symmetric relations on the fixed set V are equinumerous. (Contributed by AV, 27-Nov-2021)

	Ref	Expression
Hypotheses	sprsymrelf.p	⊢ 𝑃 = 𝒫 ( Pairs ‘ 𝑉 )
	sprsymrelf.r	⊢ 𝑅 = { 𝑟 ∈ 𝒫 ( 𝑉 × 𝑉 ) ∣ ∀ 𝑥 ∈ 𝑉 ∀ 𝑦 ∈ 𝑉 ( 𝑥 𝑟 𝑦 ↔ 𝑦 𝑟 𝑥 ) }
Assertion	sprsymrelen	⊢ ( 𝑉 ∈ 𝑊 → 𝑃 ≈ 𝑅 )

Proof

Step	Hyp	Ref	Expression
1		sprsymrelf.p	⊢ 𝑃 = 𝒫 ( Pairs ‘ 𝑉 )
2		sprsymrelf.r	⊢ 𝑅 = { 𝑟 ∈ 𝒫 ( 𝑉 × 𝑉 ) ∣ ∀ 𝑥 ∈ 𝑉 ∀ 𝑦 ∈ 𝑉 ( 𝑥 𝑟 𝑦 ↔ 𝑦 𝑟 𝑥 ) }
3	1 2	sprbisymrel	⊢ ( 𝑉 ∈ 𝑊 → ∃ 𝑓 𝑓 : 𝑃 –1-1-onto→ 𝑅 )
4		bren	⊢ ( 𝑃 ≈ 𝑅 ↔ ∃ 𝑓 𝑓 : 𝑃 –1-1-onto→ 𝑅 )
5	3 4	sylibr	⊢ ( 𝑉 ∈ 𝑊 → 𝑃 ≈ 𝑅 )