Theorem bj-idreseqb

Description: Characterization for two classes to be related under the restricted identity relation. (Contributed by BJ, 24-Dec-2023)

	Ref	Expression
Assertion	bj-idreseqb	⊢ ( 𝐴 ( I ↾ 𝐶 ) 𝐵 ↔ ( 𝐴 ∈ 𝐶 ∧ 𝐴 = 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		relres	⊢ Rel ( I ↾ 𝐶 )
2	1	brrelex12i	⊢ ( 𝐴 ( I ↾ 𝐶 ) 𝐵 → ( 𝐴 ∈ V ∧ 𝐵 ∈ V ) )
3		simpl	⊢ ( ( 𝐴 ∈ 𝐶 ∧ 𝐴 = 𝐵 ) → 𝐴 ∈ 𝐶 )
4	3	elexd	⊢ ( ( 𝐴 ∈ 𝐶 ∧ 𝐴 = 𝐵 ) → 𝐴 ∈ V )
5		eleq1	⊢ ( 𝐴 = 𝐵 → ( 𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐶 ) )
6	5	biimpac	⊢ ( ( 𝐴 ∈ 𝐶 ∧ 𝐴 = 𝐵 ) → 𝐵 ∈ 𝐶 )
7	6	elexd	⊢ ( ( 𝐴 ∈ 𝐶 ∧ 𝐴 = 𝐵 ) → 𝐵 ∈ V )
8	4 7	jca	⊢ ( ( 𝐴 ∈ 𝐶 ∧ 𝐴 = 𝐵 ) → ( 𝐴 ∈ V ∧ 𝐵 ∈ V ) )
9		brres	⊢ ( 𝐵 ∈ V → ( 𝐴 ( I ↾ 𝐶 ) 𝐵 ↔ ( 𝐴 ∈ 𝐶 ∧ 𝐴 I 𝐵 ) ) )
10	9	adantl	⊢ ( ( 𝐴 ∈ V ∧ 𝐵 ∈ V ) → ( 𝐴 ( I ↾ 𝐶 ) 𝐵 ↔ ( 𝐴 ∈ 𝐶 ∧ 𝐴 I 𝐵 ) ) )
11		eqeq12	⊢ ( ( 𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ) → ( 𝑥 = 𝑦 ↔ 𝐴 = 𝐵 ) )
12		df-id	⊢ I = { ⟨ 𝑥 , 𝑦 ⟩ ∣ 𝑥 = 𝑦 }
13	11 12	brabga	⊢ ( ( 𝐴 ∈ V ∧ 𝐵 ∈ V ) → ( 𝐴 I 𝐵 ↔ 𝐴 = 𝐵 ) )
14	13	anbi2d	⊢ ( ( 𝐴 ∈ V ∧ 𝐵 ∈ V ) → ( ( 𝐴 ∈ 𝐶 ∧ 𝐴 I 𝐵 ) ↔ ( 𝐴 ∈ 𝐶 ∧ 𝐴 = 𝐵 ) ) )
15	10 14	bitrd	⊢ ( ( 𝐴 ∈ V ∧ 𝐵 ∈ V ) → ( 𝐴 ( I ↾ 𝐶 ) 𝐵 ↔ ( 𝐴 ∈ 𝐶 ∧ 𝐴 = 𝐵 ) ) )
16	2 8 15	pm5.21nii	⊢ ( 𝐴 ( I ↾ 𝐶 ) 𝐵 ↔ ( 𝐴 ∈ 𝐶 ∧ 𝐴 = 𝐵 ) )