Theorem imadmres

Description: The image of the domain of a restriction. (Contributed by NM, 8-Apr-2007)

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

Proof

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