Theorem eqrel

Description: Extensionality principle for relations. Theorem 3.2(ii) of Monk1 p. 33. (Contributed by NM, 2-Aug-1994)

	Ref	Expression
Assertion	eqrel	⊢ ( ( Rel 𝐴 ∧ Rel 𝐵 ) → ( 𝐴 = 𝐵 ↔ ∀ 𝑥 ∀ 𝑦 ( ⟨ 𝑥 , 𝑦 ⟩ ∈ 𝐴 ↔ ⟨ 𝑥 , 𝑦 ⟩ ∈ 𝐵 ) ) )

Proof

Step	Hyp	Ref	Expression
1		ssrel	⊢ ( Rel 𝐴 → ( 𝐴 ⊆ 𝐵 ↔ ∀ 𝑥 ∀ 𝑦 ( ⟨ 𝑥 , 𝑦 ⟩ ∈ 𝐴 → ⟨ 𝑥 , 𝑦 ⟩ ∈ 𝐵 ) ) )
2		ssrel	⊢ ( Rel 𝐵 → ( 𝐵 ⊆ 𝐴 ↔ ∀ 𝑥 ∀ 𝑦 ( ⟨ 𝑥 , 𝑦 ⟩ ∈ 𝐵 → ⟨ 𝑥 , 𝑦 ⟩ ∈ 𝐴 ) ) )
3	1 2	bi2anan9	⊢ ( ( Rel 𝐴 ∧ Rel 𝐵 ) → ( ( 𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐴 ) ↔ ( ∀ 𝑥 ∀ 𝑦 ( ⟨ 𝑥 , 𝑦 ⟩ ∈ 𝐴 → ⟨ 𝑥 , 𝑦 ⟩ ∈ 𝐵 ) ∧ ∀ 𝑥 ∀ 𝑦 ( ⟨ 𝑥 , 𝑦 ⟩ ∈ 𝐵 → ⟨ 𝑥 , 𝑦 ⟩ ∈ 𝐴 ) ) ) )
4		eqss	⊢ ( 𝐴 = 𝐵 ↔ ( 𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐴 ) )
5		2albiim	⊢ ( ∀ 𝑥 ∀ 𝑦 ( ⟨ 𝑥 , 𝑦 ⟩ ∈ 𝐴 ↔ ⟨ 𝑥 , 𝑦 ⟩ ∈ 𝐵 ) ↔ ( ∀ 𝑥 ∀ 𝑦 ( ⟨ 𝑥 , 𝑦 ⟩ ∈ 𝐴 → ⟨ 𝑥 , 𝑦 ⟩ ∈ 𝐵 ) ∧ ∀ 𝑥 ∀ 𝑦 ( ⟨ 𝑥 , 𝑦 ⟩ ∈ 𝐵 → ⟨ 𝑥 , 𝑦 ⟩ ∈ 𝐴 ) ) )
6	3 4 5	3bitr4g	⊢ ( ( Rel 𝐴 ∧ Rel 𝐵 ) → ( 𝐴 = 𝐵 ↔ ∀ 𝑥 ∀ 𝑦 ( ⟨ 𝑥 , 𝑦 ⟩ ∈ 𝐴 ↔ ⟨ 𝑥 , 𝑦 ⟩ ∈ 𝐵 ) ) )