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Metamath Proof Explorer

Theorem elpmap

Description: Member of a projective map. (Contributed by NM, 27-Jan-2012)

Ref Expression
Hypotheses pmapfval.b 
|- B = ( Base ` K )
pmapfval.l 
|- .<_ = ( le ` K )
pmapfval.a 
|- A = ( Atoms ` K )
pmapfval.m 
|- M = ( pmap ` K )
Assertion elpmap 
|- ( ( K e. C /\ X e. B ) -> ( P e. ( M ` X ) <-> ( P e. A /\ P .<_ X ) ) )

Proof

Step Hyp Ref Expression
1 pmapfval.b 
 |-  B = ( Base ` K )
2 pmapfval.l 
 |-  .<_ = ( le ` K )
3 pmapfval.a 
 |-  A = ( Atoms ` K )
4 pmapfval.m 
 |-  M = ( pmap ` K )
5 1 2 3 4 pmapval 
 |-  ( ( K e. C /\ X e. B ) -> ( M ` X ) = { x e. A | x .<_ X } )
6 5 eleq2d 
 |-  ( ( K e. C /\ X e. B ) -> ( P e. ( M ` X ) <-> P e. { x e. A | x .<_ X } ) )
7 breq1 
 |-  ( x = P -> ( x .<_ X <-> P .<_ X ) )
8 7 elrab 
 |-  ( P e. { x e. A | x .<_ X } <-> ( P e. A /\ P .<_ X ) )
9 6 8 bitrdi 
 |-  ( ( K e. C /\ X e. B ) -> ( P e. ( M ` X ) <-> ( P e. A /\ P .<_ X ) ) )