Theorem ideqg

Description: For sets, the identity relation is the same as equality. (Contributed by NM, 30-Apr-2004) (Proof shortened by Andrew Salmon, 27-Aug-2011)

	Ref	Expression
Assertion	ideqg	⊢ ( 𝐵 ∈ 𝑉 → ( 𝐴 I 𝐵 ↔ 𝐴 = 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		id	⊢ ( 𝐵 ∈ 𝑉 → 𝐵 ∈ 𝑉 )
2		reli	⊢ Rel I
3	2	brrelex1i	⊢ ( 𝐴 I 𝐵 → 𝐴 ∈ V )
4	1 3	anim12ci	⊢ ( ( 𝐵 ∈ 𝑉 ∧ 𝐴 I 𝐵 ) → ( 𝐴 ∈ V ∧ 𝐵 ∈ 𝑉 ) )
5		eleq1	⊢ ( 𝐴 = 𝐵 → ( 𝐴 ∈ 𝑉 ↔ 𝐵 ∈ 𝑉 ) )
6	5	biimparc	⊢ ( ( 𝐵 ∈ 𝑉 ∧ 𝐴 = 𝐵 ) → 𝐴 ∈ 𝑉 )
7	6	elexd	⊢ ( ( 𝐵 ∈ 𝑉 ∧ 𝐴 = 𝐵 ) → 𝐴 ∈ V )
8		simpl	⊢ ( ( 𝐵 ∈ 𝑉 ∧ 𝐴 = 𝐵 ) → 𝐵 ∈ 𝑉 )
9	7 8	jca	⊢ ( ( 𝐵 ∈ 𝑉 ∧ 𝐴 = 𝐵 ) → ( 𝐴 ∈ V ∧ 𝐵 ∈ 𝑉 ) )
10		eqeq1	⊢ ( 𝑥 = 𝐴 → ( 𝑥 = 𝑦 ↔ 𝐴 = 𝑦 ) )
11		eqeq2	⊢ ( 𝑦 = 𝐵 → ( 𝐴 = 𝑦 ↔ 𝐴 = 𝐵 ) )
12		df-id	⊢ I = { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝑥 = 𝑦 }
13	10 11 12	brabg	⊢ ( ( 𝐴 ∈ V ∧ 𝐵 ∈ 𝑉 ) → ( 𝐴 I 𝐵 ↔ 𝐴 = 𝐵 ) )
14	4 9 13	pm5.21nd	⊢ ( 𝐵 ∈ 𝑉 → ( 𝐴 I 𝐵 ↔ 𝐴 = 𝐵 ) )