Theorem rnmptssff

Description: The range of a function given by the maps-to notation as a subset. (Contributed by Glauco Siliprandi, 24-Jan-2025)

	Ref	Expression
Hypotheses	rnmptssff.1	|- F/_ x A
	rnmptssff.2	|- F/_ x C
	rnmptssff.3	|- F = ( x e. A |-> B )
Assertion	rnmptssff	|- ( A. x e. A B e. C -> ran F C_ C )

Proof

Step	Hyp	Ref	Expression
1		rnmptssff.1	|- F/_ x A
2		rnmptssff.2	|- F/_ x C
3		rnmptssff.3	|- F = ( x e. A |-> B )
4	1 2 3	fmptff	|- ( A. x e. A B e. C <-> F : A --> C )
5		frn	|- ( F : A --> C -> ran F C_ C )
6	4 5	sylbi	|- ( A. x e. A B e. C -> ran F C_ C )