Theorem imaeq1

Description: Equality theorem for image. (Contributed by NM, 14-Aug-1994)

	Ref	Expression
Assertion	imaeq1	$$⊢A=B\to A\left[C\right]=B\left[C\right]$$

Proof

Step	Hyp	Ref	Expression
1		reseq1	$$⊢A=B\to {A\upharpoonright }_C={B\upharpoonright }_C$$
2	1	rneqd	$$⊢A=B\to \mathrm{ran}\left({A\upharpoonright }_C\right)=\mathrm{ran}\left({B\upharpoonright }_C\right)$$
3		df-ima	$$⊢A\left[C\right]=\mathrm{ran}\left({A\upharpoonright }_C\right)$$
4		df-ima	$$⊢B\left[C\right]=\mathrm{ran}\left({B\upharpoonright }_C\right)$$
5	2 3 4	3eqtr4g	$$⊢A=B\to A\left[C\right]=B\left[C\right]$$