Theorem pw2f1o2val2

Description: Membership in a mapped set under 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	pw2f1o2val2	|- ( ( X e. ( 2o ^m A ) /\ Y e. A ) -> ( Y e. ( F ` X ) <-> ( X ` Y ) = 1o ) )

Proof

Step	Hyp	Ref	Expression
1		pw2f1o2.f	|- F = ( x e. ( 2o ^m A ) |-> ( `' x " { 1o } ) )
2	1	pw2f1o2val	|- ( X e. ( 2o ^m A ) -> ( F ` X ) = ( `' X " { 1o } ) )
3	2	eleq2d	|- ( X e. ( 2o ^m A ) -> ( Y e. ( F ` X ) <-> Y e. ( `' X " { 1o } ) ) )
4	3	adantr	|- ( ( X e. ( 2o ^m A ) /\ Y e. A ) -> ( Y e. ( F ` X ) <-> Y e. ( `' X " { 1o } ) ) )
5		elmapi	|- ( X e. ( 2o ^m A ) -> X : A --> 2o )
6		ffn	|- ( X : A --> 2o -> X Fn A )
7		fniniseg	|- ( X Fn A -> ( Y e. ( `' X " { 1o } ) <-> ( Y e. A /\ ( X ` Y ) = 1o ) ) )
8	5 6 7	3syl	|- ( X e. ( 2o ^m A ) -> ( Y e. ( `' X " { 1o } ) <-> ( Y e. A /\ ( X ` Y ) = 1o ) ) )
9	8	baibd	|- ( ( X e. ( 2o ^m A ) /\ Y e. A ) -> ( Y e. ( `' X " { 1o } ) <-> ( X ` Y ) = 1o ) )
10	4 9	bitrd	|- ( ( X e. ( 2o ^m A ) /\ Y e. A ) -> ( Y e. ( F ` X ) <-> ( X ` Y ) = 1o ) )