Metamath Proof Explorer

Theorem funres

Description: A restriction of a function is a function. Compare Exercise 18 of TakeutiZaring p. 25. (Contributed by NM, 16-Aug-1994)

Ref Expression
Assertion funres 
|- ( Fun F -> Fun ( F |` A ) )

Proof

Step Hyp Ref Expression
1 resss 
 |-  ( F |` A ) C_ F
2 funss 
 |-  ( ( F |` A ) C_ F -> ( Fun F -> Fun ( F |` A ) ) )
3 1 2 ax-mp 
 |-  ( Fun F -> Fun ( F |` A ) )