Theorem fnrndomg

Description: The range of a function is dominated by its domain. This theorem requires the axiom of choice ax-ac2 ; see fnrndomnum for a version that does not. (Contributed by NM, 1-Sep-2004) (Proof shortened by Vincent Gonzalez, 17-Aug-2026)

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

Proof

Step	Hyp	Ref	Expression
1		numth3	$$⊢A\in B\to A\in \mathrm{dom}\mathrm{card}$$
2		fnrndomnum	$$⊢A\in \mathrm{dom}\mathrm{card}\to \left(FFnA\to \mathrm{ran}F\preccurlyeq A\right)$$
3	1 2	syl	$$⊢A\in B\to \left(FFnA\to \mathrm{ran}F\preccurlyeq A\right)$$