Theorem fnrndomg

Description: The range of a function is dominated by its domain. (Contributed by NM, 1-Sep-2004)

	Ref	Expression
Assertion	fnrndomg	$$⊢A\in B\to \left(FFnA\to \mathrm{ran}F\preccurlyeq A\right)$$

Proof

Step	Hyp	Ref	Expression
1		dffn4	$$⊢FFnA\leftrightarrow F:A\underset{onto}{⟶}\mathrm{ran}F$$
2		fodomg	$$⊢A\in B\to \left(F:A\underset{onto}{⟶}\mathrm{ran}F\to \mathrm{ran}F\preccurlyeq A\right)$$
3	1 2	biimtrid	$$⊢A\in B\to \left(FFnA\to \mathrm{ran}F\preccurlyeq A\right)$$