Theorem pw2f1o2val

Description: Function value of the pw2f1o2 bijection. (Contributed by Stefan O'Rear, 18-Jan-2015) (Revised by Stefan O'Rear, 6-May-2015)

	Ref	Expression
Hypothesis	pw2f1o2.f	|- F = ( x e. ( 2o ^m A ) |-> ( `' x " { 1o } ) )
Assertion	pw2f1o2val	|- ( X e. ( 2o ^m A ) -> ( F ` X ) = ( `' X " { 1o } ) )

Proof

Step	Hyp	Ref	Expression
1		pw2f1o2.f	|- F = ( x e. ( 2o ^m A ) |-> ( `' x " { 1o } ) )
2		cnvexg	|- ( X e. ( 2o ^m A ) -> `' X e. _V )
3		imaexg	|- ( `' X e. _V -> ( `' X " { 1o } ) e. _V )
4	2 3	syl	|- ( X e. ( 2o ^m A ) -> ( `' X " { 1o } ) e. _V )
5		cnveq	|- ( x = X -> `' x = `' X )
6	5	imaeq1d	|- ( x = X -> ( `' x " { 1o } ) = ( `' X " { 1o } ) )
7	6 1	fvmptg	|- ( ( X e. ( 2o ^m A ) /\ ( `' X " { 1o } ) e. _V ) -> ( F ` X ) = ( `' X " { 1o } ) )
8	4 7	mpdan	|- ( X e. ( 2o ^m A ) -> ( F ` X ) = ( `' X " { 1o } ) )