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Metamath Proof Explorer

Theorem fssrescdmd

Description: Restriction of a function to a subclass of its domain as a function with domain and codomain. (Contributed by AV, 13-May-2025)

Ref Expression
Hypotheses fssrescdmd.f 
|- ( ph -> F : A --> B )
fssrescdmd.c 
|- ( ph -> C C_ A )
fssrescdmd.d 
|- ( ph -> ( F " C ) C_ D )
Assertion fssrescdmd 
|- ( ph -> ( F |` C ) : C --> D )

Proof

Step Hyp Ref Expression
1 fssrescdmd.f 
 |-  ( ph -> F : A --> B )
2 fssrescdmd.c 
 |-  ( ph -> C C_ A )
3 fssrescdmd.d 
 |-  ( ph -> ( F " C ) C_ D )
4 1 ffnd 
 |-  ( ph -> F Fn A )
5 4 2 fnssresd 
 |-  ( ph -> ( F |` C ) Fn C )
6 resima 
 |-  ( ( F |` C ) " C ) = ( F " C )
7 6 3 eqsstrid 
 |-  ( ph -> ( ( F |` C ) " C ) C_ D )
8 1 ffund 
 |-  ( ph -> Fun F )
9 8 funresd 
 |-  ( ph -> Fun ( F |` C ) )
10 1 fdmd 
 |-  ( ph -> dom F = A )
11 2 10 sseqtrrd 
 |-  ( ph -> C C_ dom F )
12 ssdmres 
 |-  ( C C_ dom F <-> dom ( F |` C ) = C )
13 12 a1i 
 |-  ( ph -> ( C C_ dom F <-> dom ( F |` C ) = C ) )
14 eqcom 
 |-  ( dom ( F |` C ) = C <-> C = dom ( F |` C ) )
15 13 14 bitrdi 
 |-  ( ph -> ( C C_ dom F <-> C = dom ( F |` C ) ) )
16 11 15 mpbid 
 |-  ( ph -> C = dom ( F |` C ) )
17 16 eqimssd 
 |-  ( ph -> C C_ dom ( F |` C ) )
18 funimass4 
 |-  ( ( Fun ( F |` C ) /\ C C_ dom ( F |` C ) ) -> ( ( ( F |` C ) " C ) C_ D <-> A. x e. C ( ( F |` C ) ` x ) e. D ) )
19 9 17 18 syl2anc 
 |-  ( ph -> ( ( ( F |` C ) " C ) C_ D <-> A. x e. C ( ( F |` C ) ` x ) e. D ) )
20 7 19 mpbid 
 |-  ( ph -> A. x e. C ( ( F |` C ) ` x ) e. D )
21 ffnfv 
 |-  ( ( F |` C ) : C --> D <-> ( ( F |` C ) Fn C /\ A. x e. C ( ( F |` C ) ` x ) e. D ) )
22 5 20 21 sylanbrc 
 |-  ( ph -> ( F |` C ) : C --> D )