Theorem imacnvcnv

Description: The image of the double converse of a class. (Contributed by NM, 8-Apr-2007)

	Ref	Expression
Assertion	imacnvcnv	$$⊢{A^{-1}}^{-1}\left[B\right]=A\left[B\right]$$

Proof

Step	Hyp	Ref	Expression
1		rescnvcnv	$$⊢{{A^{-1}}^{-1}\upharpoonright }_B={A\upharpoonright }_B$$
2	1	rneqi	$$⊢\mathrm{ran}\left({{A^{-1}}^{-1}\upharpoonright }_B\right)=\mathrm{ran}\left({A\upharpoonright }_B\right)$$
3		df-ima	$$⊢{A^{-1}}^{-1}\left[B\right]=\mathrm{ran}\left({{A^{-1}}^{-1}\upharpoonright }_B\right)$$
4		df-ima	$$⊢A\left[B\right]=\mathrm{ran}\left({A\upharpoonright }_B\right)$$
5	2 3 4	3eqtr4i	$$⊢{A^{-1}}^{-1}\left[B\right]=A\left[B\right]$$