Theorem cnvimarndmOLD

Description: Obsolete version of cnvimarndm as of 29-Sep-2026. (Contributed by Jeff Hankins, 15-Jul-2009) (Proof modification is discouraged.) (New usage is discouraged.)

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

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