Theorem mptresid

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

	Ref	Expression
Assertion	mptresid	$$⊢{\mathrm{I}\upharpoonright }_A=\left(x\in A⟼x\right)$$

Proof

Step	Hyp	Ref	Expression
1		opabresid	$$⊢{\mathrm{I}\upharpoonright }_A=\left\{⟨x,y⟩|\left(x\in A\wedge y=x\right)\right\}$$
2		df-mpt	$$⊢\left(x\in A⟼x\right)=\left\{⟨x,y⟩|\left(x\in A\wedge y=x\right)\right\}$$
3	1 2	eqtr4i	$$⊢{\mathrm{I}\upharpoonright }_A=\left(x\in A⟼x\right)$$