Metamath Proof Explorer

Theorem fvreseq0

Description: Equality of restricted functions is determined by their values (for functions with different domains). (Contributed by AV, 6-Jan-2019)

Ref Expression
Assertion fvreseq0 
|- ( ( ( F Fn A /\ G Fn C ) /\ ( B C_ A /\ B C_ C ) ) -> ( ( F |` B ) = ( G |` B ) <-> A. x e. B ( F ` x ) = ( G ` x ) ) )

Proof

Step Hyp Ref Expression
1 fnssres 
 |-  ( ( F Fn A /\ B C_ A ) -> ( F |` B ) Fn B )
2 fnssres 
 |-  ( ( G Fn C /\ B C_ C ) -> ( G |` B ) Fn B )
3 eqfnfv 
 |-  ( ( ( F |` B ) Fn B /\ ( G |` B ) Fn B ) -> ( ( F |` B ) = ( G |` B ) <-> A. x e. B ( ( F |` B ) ` x ) = ( ( G |` B ) ` x ) ) )
4 fvres 
 |-  ( x e. B -> ( ( F |` B ) ` x ) = ( F ` x ) )
5 fvres 
 |-  ( x e. B -> ( ( G |` B ) ` x ) = ( G ` x ) )
6 4 5 eqeq12d 
 |-  ( x e. B -> ( ( ( F |` B ) ` x ) = ( ( G |` B ) ` x ) <-> ( F ` x ) = ( G ` x ) ) )
7 6 ralbiia 
 |-  ( A. x e. B ( ( F |` B ) ` x ) = ( ( G |` B ) ` x ) <-> A. x e. B ( F ` x ) = ( G ` x ) )
8 3 7 bitrdi 
 |-  ( ( ( F |` B ) Fn B /\ ( G |` B ) Fn B ) -> ( ( F |` B ) = ( G |` B ) <-> A. x e. B ( F ` x ) = ( G ` x ) ) )
9 1 2 8 syl2an 
 |-  ( ( ( F Fn A /\ B C_ A ) /\ ( G Fn C /\ B C_ C ) ) -> ( ( F |` B ) = ( G |` B ) <-> A. x e. B ( F ` x ) = ( G ` x ) ) )
10 9 an4s 
 |-  ( ( ( F Fn A /\ G Fn C ) /\ ( B C_ A /\ B C_ C ) ) -> ( ( F |` B ) = ( G |` B ) <-> A. x e. B ( F ` x ) = ( G ` x ) ) )