Theorem uspgrymrelen

Description: The set G of the "simple pseudographs" with a fixed set of vertices V and the class R of the symmetric relations on the fixed set V are equinumerous. For more details about the class G of all "simple pseudographs" see comments on uspgrbisymrel . (Contributed by AV, 27-Nov-2021)

	Ref	Expression
Hypotheses	uspgrbisymrel.g	⊢ 𝐺 = { ⟨ 𝑣 , 𝑒 ⟩ ∣ ( 𝑣 = 𝑉 ∧ ∃ 𝑞 ∈ USPGraph ( ( Vtx ‘ 𝑞 ) = 𝑣 ∧ ( Edg ‘ 𝑞 ) = 𝑒 ) ) }
	uspgrbisymrel.r	⊢ 𝑅 = { 𝑟 ∈ 𝒫 ( 𝑉 × 𝑉 ) ∣ ∀ 𝑥 ∈ 𝑉 ∀ 𝑦 ∈ 𝑉 ( 𝑥 𝑟 𝑦 ↔ 𝑦 𝑟 𝑥 ) }
Assertion	uspgrymrelen	⊢ ( 𝑉 ∈ 𝑊 → 𝐺 ≈ 𝑅 )

Proof

Step	Hyp	Ref	Expression
1		uspgrbisymrel.g	⊢ 𝐺 = { ⟨ 𝑣 , 𝑒 ⟩ ∣ ( 𝑣 = 𝑉 ∧ ∃ 𝑞 ∈ USPGraph ( ( Vtx ‘ 𝑞 ) = 𝑣 ∧ ( Edg ‘ 𝑞 ) = 𝑒 ) ) }
2		uspgrbisymrel.r	⊢ 𝑅 = { 𝑟 ∈ 𝒫 ( 𝑉 × 𝑉 ) ∣ ∀ 𝑥 ∈ 𝑉 ∀ 𝑦 ∈ 𝑉 ( 𝑥 𝑟 𝑦 ↔ 𝑦 𝑟 𝑥 ) }
3		eqid	⊢ 𝒫 ( Pairs ‘ 𝑉 ) = 𝒫 ( Pairs ‘ 𝑉 )
4	3 1	uspgrspren	⊢ ( 𝑉 ∈ 𝑊 → 𝐺 ≈ 𝒫 ( Pairs ‘ 𝑉 ) )
5	3 2	sprsymrelen	⊢ ( 𝑉 ∈ 𝑊 → 𝒫 ( Pairs ‘ 𝑉 ) ≈ 𝑅 )
6		entr	⊢ ( ( 𝐺 ≈ 𝒫 ( Pairs ‘ 𝑉 ) ∧ 𝒫 ( Pairs ‘ 𝑉 ) ≈ 𝑅 ) → 𝐺 ≈ 𝑅 )
7	4 5 6	syl2anc	⊢ ( 𝑉 ∈ 𝑊 → 𝐺 ≈ 𝑅 )