Theorem rnmptssdf

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

Proof

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