Theorem cnveq

Description: Equality theorem for converse relation. (Contributed by NM, 13-Aug-1995)

	Ref	Expression
Assertion	cnveq	$$⊢A=B\to A^{-1}=B^{-1}$$

Proof

Step	Hyp	Ref	Expression
1		cnvss	$$⊢A\subseteq B\to A^{-1}\subseteq B^{-1}$$
2		cnvss	$$⊢B\subseteq A\to B^{-1}\subseteq A^{-1}$$
3	1 2	anim12i	$$⊢\left(A\subseteq B\wedge B\subseteq A\right)\to \left(A^{-1}\subseteq B^{-1}\wedge B^{-1}\subseteq A^{-1}\right)$$
4		eqss	$$⊢A=B\leftrightarrow \left(A\subseteq B\wedge B\subseteq A\right)$$
5		eqss	$$⊢A^{-1}=B^{-1}\leftrightarrow \left(A^{-1}\subseteq B^{-1}\wedge B^{-1}\subseteq A^{-1}\right)$$
6	3 4 5	3imtr4i	$$⊢A=B\to A^{-1}=B^{-1}$$