Theorem cnvimarndm

Description: The preimage of the range of a class is the domain of the class. (Contributed by Jeff Hankins, 15-Jul-2009)

	Ref	Expression
Assertion	cnvimarndm	$$⊢A^{-1}\left[\mathrm{ran}A\right]=\mathrm{dom}A$$

Proof

Step	Hyp	Ref	Expression
1		imadmrn	$$⊢A^{-1}\left[\mathrm{dom}{A}^{-1}\right]=\mathrm{ran}{A}^{-1}$$
2		df-rn	$$⊢\mathrm{ran}A=\mathrm{dom}{A}^{-1}$$
3	2	imaeq2i	$$⊢A^{-1}\left[\mathrm{ran}A\right]=A^{-1}\left[\mathrm{dom}{A}^{-1}\right]$$
4		dfdm4	$$⊢\mathrm{dom}A=\mathrm{ran}{A}^{-1}$$
5	1 3 4	3eqtr4i	$$⊢A^{-1}\left[\mathrm{ran}A\right]=\mathrm{dom}A$$