Theorem rneq

Description: Equality theorem for range. (Contributed by NM, 29-Dec-1996)

	Ref	Expression
Assertion	rneq	$$⊢A=B\to \mathrm{ran}A=\mathrm{ran}B$$

Proof

Step	Hyp	Ref	Expression
1		cnveq	$$⊢A=B\to A^{-1}=B^{-1}$$
2	1	dmeqd	$$⊢A=B\to \mathrm{dom}{A}^{-1}=\mathrm{dom}{B}^{-1}$$
3		df-rn	$$⊢\mathrm{ran}A=\mathrm{dom}{A}^{-1}$$
4		df-rn	$$⊢\mathrm{ran}B=\mathrm{dom}{B}^{-1}$$
5	2 3 4	3eqtr4g	$$⊢A=B\to \mathrm{ran}A=\mathrm{ran}B$$