Theorem mptresid

Description: The restricted identity relation expressed in maps-to notation. (Contributed by FL, 25-Apr-2012)

	Ref	Expression
Assertion	mptresid	⊢ ( I ↾ 𝐴 ) = ( 𝑥 ∈ 𝐴 ↦ 𝑥 )

Proof

Step	Hyp	Ref	Expression
1		opabresid	⊢ ( I ↾ 𝐴 ) = { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑥 ) }
2		df-mpt	⊢ ( 𝑥 ∈ 𝐴 ↦ 𝑥 ) = { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( 𝑥 ∈ 𝐴 ∧ 𝑦 = 𝑥 ) }
3	1 2	eqtr4i	⊢ ( I ↾ 𝐴 ) = ( 𝑥 ∈ 𝐴 ↦ 𝑥 )