Theorem ereq2

Description: Equality theorem for equivalence predicate. (Contributed by Mario Carneiro, 12-Aug-2015)

	Ref	Expression
Assertion	ereq2	⊢ ( 𝐴 = 𝐵 → ( 𝑅 Er 𝐴 ↔ 𝑅 Er 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		eqeq2	⊢ ( 𝐴 = 𝐵 → ( dom 𝑅 = 𝐴 ↔ dom 𝑅 = 𝐵 ) )
2	1	3anbi2d	⊢ ( 𝐴 = 𝐵 → ( ( Rel 𝑅 ∧ dom 𝑅 = 𝐴 ∧ ( ◡ 𝑅 ∪ ( 𝑅 ∘ 𝑅 ) ) ⊆ 𝑅 ) ↔ ( Rel 𝑅 ∧ dom 𝑅 = 𝐵 ∧ ( ◡ 𝑅 ∪ ( 𝑅 ∘ 𝑅 ) ) ⊆ 𝑅 ) ) )
3		df-er	⊢ ( 𝑅 Er 𝐴 ↔ ( Rel 𝑅 ∧ dom 𝑅 = 𝐴 ∧ ( ◡ 𝑅 ∪ ( 𝑅 ∘ 𝑅 ) ) ⊆ 𝑅 ) )
4		df-er	⊢ ( 𝑅 Er 𝐵 ↔ ( Rel 𝑅 ∧ dom 𝑅 = 𝐵 ∧ ( ◡ 𝑅 ∪ ( 𝑅 ∘ 𝑅 ) ) ⊆ 𝑅 ) )
5	2 3 4	3bitr4g	⊢ ( 𝐴 = 𝐵 → ( 𝑅 Er 𝐴 ↔ 𝑅 Er 𝐵 ) )