Metamath Proof Explorer


Theorem ntrclsfv2

Description: If (pseudo-)interior and (pseudo-)closure functions are related by the duality operator then there is a functional relation between them (Contributed by RP, 28-May-2021)

Ref Expression
Hypotheses ntrcls.o
|- O = ( i e. _V |-> ( k e. ( ~P i ^m ~P i ) |-> ( j e. ~P i |-> ( i \ ( k ` ( i \ j ) ) ) ) ) )
ntrcls.d
|- D = ( O ` B )
ntrcls.r
|- ( ph -> I D K )
Assertion ntrclsfv2
|- ( ph -> ( D ` K ) = I )

Proof

Step Hyp Ref Expression
1 ntrcls.o
 |-  O = ( i e. _V |-> ( k e. ( ~P i ^m ~P i ) |-> ( j e. ~P i |-> ( i \ ( k ` ( i \ j ) ) ) ) ) )
2 ntrcls.d
 |-  D = ( O ` B )
3 ntrcls.r
 |-  ( ph -> I D K )
4 1 2 3 ntrclsnvobr
 |-  ( ph -> K D I )
5 1 2 4 ntrclsfv1
 |-  ( ph -> ( D ` K ) = I )