Metamath Proof Explorer

Theorem rerebi

Description: A real number equals its real part. Proposition 10-3.4(f) of Gleason p. 133. (Contributed by NM, 27-Oct-1999)

Ref Expression
Hypothesis recl.1 
|- A e. CC
Assertion rerebi 
|- ( A e. RR <-> ( Re ` A ) = A )

Proof

Step Hyp Ref Expression
1 recl.1 
 |-  A e. CC
2 rereb 
 |-  ( A e. CC -> ( A e. RR <-> ( Re ` A ) = A ) )
3 1 2 ax-mp 
 |-  ( A e. RR <-> ( Re ` A ) = A )