Theorem dfid4

Description: The identity function expressed using maps-to notation. (Contributed by Scott Fenton, 15-Dec-2017)

	Ref	Expression
Assertion	dfid4	$$⊢\mathrm{I}=\left(x\in \mathrm{V}⟼x\right)$$

Proof

Step	Hyp	Ref	Expression
1		equcom	$$⊢x=y\leftrightarrow y=x$$
2		vex	$$⊢x\in \mathrm{V}$$
3	2	biantrur	$$⊢y=x\leftrightarrow \left(x\in \mathrm{V}\wedge y=x\right)$$
4	1 3	bitri	$$⊢x=y\leftrightarrow \left(x\in \mathrm{V}\wedge y=x\right)$$
5	4	opabbii	$$⊢\left\{⟨x,y⟩|x=y\right\}=\left\{⟨x,y⟩|\left(x\in \mathrm{V}\wedge y=x\right)\right\}$$
6		df-id	$$⊢\mathrm{I}=\left\{⟨x,y⟩|x=y\right\}$$
7		df-mpt	$$⊢\left(x\in \mathrm{V}⟼x\right)=\left\{⟨x,y⟩|\left(x\in \mathrm{V}\wedge y=x\right)\right\}$$
8	5 6 7	3eqtr4i	$$⊢\mathrm{I}=\left(x\in \mathrm{V}⟼x\right)$$