Theorem resieq

Description: A restricted identity relation is equivalent to equality in its domain. (Contributed by NM, 30-Apr-2004)

	Ref	Expression
Assertion	resieq	⊢ ( ( 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ) → ( 𝐵 ( I ↾ 𝐴 ) 𝐶 ↔ 𝐵 = 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		breq2	⊢ ( 𝑥 = 𝐶 → ( 𝐵 ( I ↾ 𝐴 ) 𝑥 ↔ 𝐵 ( I ↾ 𝐴 ) 𝐶 ) )
2		eqeq2	⊢ ( 𝑥 = 𝐶 → ( 𝐵 = 𝑥 ↔ 𝐵 = 𝐶 ) )
3	1 2	bibi12d	⊢ ( 𝑥 = 𝐶 → ( ( 𝐵 ( I ↾ 𝐴 ) 𝑥 ↔ 𝐵 = 𝑥 ) ↔ ( 𝐵 ( I ↾ 𝐴 ) 𝐶 ↔ 𝐵 = 𝐶 ) ) )
4	3	imbi2d	⊢ ( 𝑥 = 𝐶 → ( ( 𝐵 ∈ 𝐴 → ( 𝐵 ( I ↾ 𝐴 ) 𝑥 ↔ 𝐵 = 𝑥 ) ) ↔ ( 𝐵 ∈ 𝐴 → ( 𝐵 ( I ↾ 𝐴 ) 𝐶 ↔ 𝐵 = 𝐶 ) ) ) )
5		vex	⊢ 𝑥 ∈ V
6	5	opres	⊢ ( 𝐵 ∈ 𝐴 → ( ⟨ 𝐵 , 𝑥 ⟩ ∈ ( I ↾ 𝐴 ) ↔ ⟨ 𝐵 , 𝑥 ⟩ ∈ I ) )
7		df-br	⊢ ( 𝐵 ( I ↾ 𝐴 ) 𝑥 ↔ ⟨ 𝐵 , 𝑥 ⟩ ∈ ( I ↾ 𝐴 ) )
8	5	ideq	⊢ ( 𝐵 I 𝑥 ↔ 𝐵 = 𝑥 )
9		df-br	⊢ ( 𝐵 I 𝑥 ↔ ⟨ 𝐵 , 𝑥 ⟩ ∈ I )
10	8 9	bitr3i	⊢ ( 𝐵 = 𝑥 ↔ ⟨ 𝐵 , 𝑥 ⟩ ∈ I )
11	6 7 10	3bitr4g	⊢ ( 𝐵 ∈ 𝐴 → ( 𝐵 ( I ↾ 𝐴 ) 𝑥 ↔ 𝐵 = 𝑥 ) )
12	4 11	vtoclg	⊢ ( 𝐶 ∈ 𝐴 → ( 𝐵 ∈ 𝐴 → ( 𝐵 ( I ↾ 𝐴 ) 𝐶 ↔ 𝐵 = 𝐶 ) ) )
13	12	impcom	⊢ ( ( 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ) → ( 𝐵 ( I ↾ 𝐴 ) 𝐶 ↔ 𝐵 = 𝐶 ) )