Theorem cnvi

Description: The converse of the identity relation. Theorem 3.7(ii) of Monk1 p. 36. (Contributed by NM, 26-Apr-1998) (Proof shortened by Andrew Salmon, 27-Aug-2011)

	Ref	Expression
Assertion	cnvi	$$⊢{\mathrm{I}}^{-1}=\mathrm{I}$$

Proof

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