Metamath Proof Explorer

Theorem fnssresb

Description: Restriction of a function with a subclass of its domain. (Contributed by NM, 10-Oct-2007)

Ref Expression
Assertion fnssresb 
|- ( F Fn A -> ( ( F |` B ) Fn B <-> B C_ A ) )

Proof

Step Hyp Ref Expression
1 df-fn 
 |-  ( ( F |` B ) Fn B <-> ( Fun ( F |` B ) /\ dom ( F |` B ) = B ) )
2 fnfun 
 |-  ( F Fn A -> Fun F )
3 2 funresd 
 |-  ( F Fn A -> Fun ( F |` B ) )
4 3 biantrurd 
 |-  ( F Fn A -> ( dom ( F |` B ) = B <-> ( Fun ( F |` B ) /\ dom ( F |` B ) = B ) ) )
5 ssdmres 
 |-  ( B C_ dom F <-> dom ( F |` B ) = B )
6 fndm 
 |-  ( F Fn A -> dom F = A )
7 6 sseq2d 
 |-  ( F Fn A -> ( B C_ dom F <-> B C_ A ) )
8 5 7 bitr3id 
 |-  ( F Fn A -> ( dom ( F |` B ) = B <-> B C_ A ) )
9 4 8 bitr3d 
 |-  ( F Fn A -> ( ( Fun ( F |` B ) /\ dom ( F |` B ) = B ) <-> B C_ A ) )
10 1 9 bitrid 
 |-  ( F Fn A -> ( ( F |` B ) Fn B <-> B C_ A ) )