Theorem resima

Description: A restriction to an image. (Contributed by NM, 29-Sep-2004)

	Ref	Expression
Assertion	resima	$$⊢\left({A\upharpoonright }_B\right)\left[B\right]=A\left[B\right]$$

Proof

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