Theorem rnmptssf

Description: The range of a function given by the maps-to notation as a subset. (Contributed by Glauco Siliprandi, 23-Oct-2021)

	Ref	Expression
Hypotheses	rnmptssf.1	|- F/_ x C
	rnmptssf.2	|- F = ( x e. A |-> B )
Assertion	rnmptssf	|- ( A. x e. A B e. C -> ran F C_ C )

Proof

Step	Hyp	Ref	Expression
1		rnmptssf.1	|- F/_ x C
2		rnmptssf.2	|- F = ( x e. A |-> B )
3	1 2	fmptf	|- ( A. x e. A B e. C <-> F : A --> C )
4		frn	|- ( F : A --> C -> ran F C_ C )
5	3 4	sylbi	|- ( A. x e. A B e. C -> ran F C_ C )