Theorem imassrn

Description: The image of a class is a subset of its range. Theorem 3.16(xi) of Monk1 p. 39. (Contributed by NM, 31-Mar-1995)

	Ref	Expression
Assertion	imassrn	$$⊢A\left[B\right]\subseteq \mathrm{ran}A$$

Proof

Step	Hyp	Ref	Expression
1		exsimpr	$$⊢\exists x\left(x\in B\wedge ⟨x,y⟩\in A\right)\to \exists x⟨x,y⟩\in A$$
2	1	ss2abi	$$⊢\left\{y|\exists x\left(x\in B\wedge ⟨x,y⟩\in A\right)\right\}\subseteq \left\{y|\exists x⟨x,y⟩\in A\right\}$$
3		dfima3	$$⊢A\left[B\right]=\left\{y|\exists x\left(x\in B\wedge ⟨x,y⟩\in A\right)\right\}$$
4		dfrn3	$$⊢\mathrm{ran}A=\left\{y|\exists x⟨x,y⟩\in A\right\}$$
5	2 3 4	3sstr4i	$$⊢A\left[B\right]\subseteq \mathrm{ran}A$$