Metamath Proof Explorer


Theorem fovcdmd

Description: An operation's value belongs to its codomain. (Contributed by Mario Carneiro, 29-Dec-2016)

Ref Expression
Hypotheses fovcdmd.1
|- ( ph -> F : ( R X. S ) --> C )
fovcdmd.2
|- ( ph -> A e. R )
fovcdmd.3
|- ( ph -> B e. S )
Assertion fovcdmd
|- ( ph -> ( A F B ) e. C )

Proof

Step Hyp Ref Expression
1 fovcdmd.1
 |-  ( ph -> F : ( R X. S ) --> C )
2 fovcdmd.2
 |-  ( ph -> A e. R )
3 fovcdmd.3
 |-  ( ph -> B e. S )
4 fovcdm
 |-  ( ( F : ( R X. S ) --> C /\ A e. R /\ B e. S ) -> ( A F B ) e. C )
5 1 2 3 4 syl3anc
 |-  ( ph -> ( A F B ) e. C )