Metamath Proof Explorer

Theorem fneq2

Description: Equality theorem for function predicate with domain. (Contributed by NM, 1-Aug-1994)

Ref Expression
Assertion fneq2 
|- ( A = B -> ( F Fn A <-> F Fn B ) )

Proof

Step Hyp Ref Expression
1 eqeq2 
 |-  ( A = B -> ( dom F = A <-> dom F = B ) )
2 1 anbi2d 
 |-  ( A = B -> ( ( Fun F /\ dom F = A ) <-> ( Fun F /\ dom F = B ) ) )
3 df-fn 
 |-  ( F Fn A <-> ( Fun F /\ dom F = A ) )
4 df-fn 
 |-  ( F Fn B <-> ( Fun F /\ dom F = B ) )
5 2 3 4 3bitr4g 
 |-  ( A = B -> ( F Fn A <-> F Fn B ) )