Theorem rnmptssdff

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	rnmptssdff.1	|- F/ x ph
	rnmptssdff.2	|- F/_ x A
	rnmptssdff.3	|- F/_ x C
	rnmptssdff.4	|- F = ( x e. A |-> B )
	rnmptssdff.5	|- ( ( ph /\ x e. A ) -> B e. C )
Assertion	rnmptssdff	|- ( ph -> ran F C_ C )

Proof

Step	Hyp	Ref	Expression
1		rnmptssdff.1	|- F/ x ph
2		rnmptssdff.2	|- F/_ x A
3		rnmptssdff.3	|- F/_ x C
4		rnmptssdff.4	|- F = ( x e. A |-> B )
5		rnmptssdff.5	|- ( ( ph /\ x e. A ) -> B e. C )
6	1 5	ralrimia	|- ( ph -> A. x e. A B e. C )
7	2 3 4	rnmptssff	|- ( A. x e. A B e. C -> ran F C_ C )
8	6 7	syl	|- ( ph -> ran F C_ C )