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Theorem diaelval 31831
Description: Member of the partial isomorphism A for a lattice  K. (Contributed by NM, 3-Dec-2013.)
Hypotheses
Ref Expression
diaval.b  |-  B  =  ( Base `  K
)
diaval.l  |-  .<_  =  ( le `  K )
diaval.h  |-  H  =  ( LHyp `  K
)
diaval.t  |-  T  =  ( ( LTrn `  K
) `  W )
diaval.r  |-  R  =  ( ( trL `  K
) `  W )
diaval.i  |-  I  =  ( ( DIsoA `  K
) `  W )
Assertion
Ref Expression
diaelval  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  ( F  e.  ( I `  X )  <->  ( F  e.  T  /\  ( R `  F )  .<_  X ) ) )

Proof of Theorem diaelval
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 diaval.b . . . 4  |-  B  =  ( Base `  K
)
2 diaval.l . . . 4  |-  .<_  =  ( le `  K )
3 diaval.h . . . 4  |-  H  =  ( LHyp `  K
)
4 diaval.t . . . 4  |-  T  =  ( ( LTrn `  K
) `  W )
5 diaval.r . . . 4  |-  R  =  ( ( trL `  K
) `  W )
6 diaval.i . . . 4  |-  I  =  ( ( DIsoA `  K
) `  W )
71, 2, 3, 4, 5, 6diaval 31830 . . 3  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  (
I `  X )  =  { f  e.  T  |  ( R `  f )  .<_  X }
)
87eleq2d 2503 . 2  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  ( F  e.  ( I `  X )  <->  F  e.  { f  e.  T  | 
( R `  f
)  .<_  X } ) )
9 fveq2 5728 . . . 4  |-  ( f  =  F  ->  ( R `  f )  =  ( R `  F ) )
109breq1d 4222 . . 3  |-  ( f  =  F  ->  (
( R `  f
)  .<_  X  <->  ( R `  F )  .<_  X ) )
1110elrab 3092 . 2  |-  ( F  e.  { f  e.  T  |  ( R `
 f )  .<_  X }  <->  ( F  e.  T  /\  ( R `
 F )  .<_  X ) )
128, 11syl6bb 253 1  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  ( F  e.  ( I `  X )  <->  ( F  e.  T  /\  ( R `  F )  .<_  X ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   {crab 2709   class class class wbr 4212   ` cfv 5454   Basecbs 13469   lecple 13536   LHypclh 30781   LTrncltrn 30898   trLctrl 30955   DIsoAcdia 31826
This theorem is referenced by:  dian0  31837  diatrl  31842  dialss  31844  diaglbN  31853  dibelval3  31945  dibopelval3  31946  diblss  31968
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-disoa 31827
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