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Theorem diaelval 31516
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 31515 . . 3  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  (
I `  X )  =  { f  e.  T  |  ( R `  f )  .<_  X }
)
87eleq2d 2471 . 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 5687 . . . 4  |-  ( f  =  F  ->  ( R `  f )  =  ( R `  F ) )
109breq1d 4182 . . 3  |-  ( f  =  F  ->  (
( R `  f
)  .<_  X  <->  ( R `  F )  .<_  X ) )
1110elrab 3052 . 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 1649    e. wcel 1721   {crab 2670   class class class wbr 4172   ` cfv 5413   Basecbs 13424   lecple 13491   LHypclh 30466   LTrncltrn 30583   trLctrl 30640   DIsoAcdia 31511
This theorem is referenced by:  dian0  31522  diatrl  31527  dialss  31529  diaglbN  31538  dibelval3  31630  dibopelval3  31631  diblss  31653
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pr 4363
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-disoa 31512
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