Metamath Proof Explorer


Theorem rnmptssdf

Description: The range of an operation 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 )