Metamath Proof Explorer

Theorem eqfnfv3

Description: Derive equality of functions from equality of their values. (Contributed by Jeff Madsen, 2-Sep-2009)

Ref Expression
Assertion eqfnfv3 
|- ( ( F Fn A /\ G Fn B ) -> ( F = G <-> ( B C_ A /\ A. x e. A ( x e. B /\ ( F ` x ) = ( G ` x ) ) ) ) )

Proof

Step Hyp Ref Expression
1 eqfnfv2 
 |-  ( ( F Fn A /\ G Fn B ) -> ( F = G <-> ( A = B /\ A. x e. A ( F ` x ) = ( G ` x ) ) ) )
2 eqss 
 |-  ( A = B <-> ( A C_ B /\ B C_ A ) )
3 2 biancomi 
 |-  ( A = B <-> ( B C_ A /\ A C_ B ) )
4 3 anbi1i 
 |-  ( ( A = B /\ A. x e. A ( F ` x ) = ( G ` x ) ) <-> ( ( B C_ A /\ A C_ B ) /\ A. x e. A ( F ` x ) = ( G ` x ) ) )
5 anass 
 |-  ( ( ( B C_ A /\ A C_ B ) /\ A. x e. A ( F ` x ) = ( G ` x ) ) <-> ( B C_ A /\ ( A C_ B /\ A. x e. A ( F ` x ) = ( G ` x ) ) ) )
6 dfss3 
 |-  ( A C_ B <-> A. x e. A x e. B )
7 6 anbi1i 
 |-  ( ( A C_ B /\ A. x e. A ( F ` x ) = ( G ` x ) ) <-> ( A. x e. A x e. B /\ A. x e. A ( F ` x ) = ( G ` x ) ) )
8 r19.26 
 |-  ( A. x e. A ( x e. B /\ ( F ` x ) = ( G ` x ) ) <-> ( A. x e. A x e. B /\ A. x e. A ( F ` x ) = ( G ` x ) ) )
9 7 8 bitr4i 
 |-  ( ( A C_ B /\ A. x e. A ( F ` x ) = ( G ` x ) ) <-> A. x e. A ( x e. B /\ ( F ` x ) = ( G ` x ) ) )
10 9 anbi2i 
 |-  ( ( B C_ A /\ ( A C_ B /\ A. x e. A ( F ` x ) = ( G ` x ) ) ) <-> ( B C_ A /\ A. x e. A ( x e. B /\ ( F ` x ) = ( G ` x ) ) ) )
11 4 5 10 3bitri 
 |-  ( ( A = B /\ A. x e. A ( F ` x ) = ( G ` x ) ) <-> ( B C_ A /\ A. x e. A ( x e. B /\ ( F ` x ) = ( G ` x ) ) ) )
12 1 11 bitrdi 
 |-  ( ( F Fn A /\ G Fn B ) -> ( F = G <-> ( B C_ A /\ A. x e. A ( x e. B /\ ( F ` x ) = ( G ` x ) ) ) ) )