Metamath Proof Explorer


Theorem rnmptssd

Description: The range of an operation given by the maps-to notation as a subset. (Contributed by Glauco Siliprandi, 11-Oct-2020)

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

Proof

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