Theorem rere

Description: A real number equals its real part. One direction of Proposition 10-3.4(f) of Gleason p. 133. (Contributed by Paul Chapman, 7-Sep-2007)

	Ref	Expression
Assertion	rere	|- ( A e. RR -> ( Re ` A ) = A )

Proof

Step	Hyp	Ref	Expression
1		recn	|- ( A e. RR -> A e. CC )
2		rereb	|- ( A e. CC -> ( A e. RR <-> ( Re ` A ) = A ) )
3	1 2	syl	|- ( A e. RR -> ( A e. RR <-> ( Re ` A ) = A ) )
4	3	ibi	|- ( A e. RR -> ( Re ` A ) = A )