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Theorem dibelval2nd 34797
Description: Membership in value of the partial isomorphism B for a lattice  K. (Contributed by NM, 13-Feb-2014.)
Hypotheses
Ref Expression
dibelval2nd.b  |-  B  =  ( Base `  K
)
dibelval2nd.l  |-  .<_  =  ( le `  K )
dibelval2nd.h  |-  H  =  ( LHyp `  K
)
dibelval2nd.t  |-  T  =  ( ( LTrn `  K
) `  W )
dibelval2nd.o  |-  .0.  =  ( f  e.  T  |->  (  _I  |`  B ) )
dibelval2nd.i  |-  I  =  ( ( DIsoB `  K
) `  W )
Assertion
Ref Expression
dibelval2nd  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( X  e.  B  /\  X  .<_  W )  /\  Y  e.  ( I `  X
) )  ->  ( 2nd `  Y )  =  .0.  )
Distinct variable groups:    f, K    f, W
Allowed substitution hints:    B( f)    T( f)    H( f)    I( f)    .<_ ( f)    V( f)    X( f)    Y( f)    .0. ( f)

Proof of Theorem dibelval2nd
StepHypRef Expression
1 dibelval2nd.b . . . . 5  |-  B  =  ( Base `  K
)
2 dibelval2nd.l . . . . 5  |-  .<_  =  ( le `  K )
3 dibelval2nd.h . . . . 5  |-  H  =  ( LHyp `  K
)
4 dibelval2nd.t . . . . 5  |-  T  =  ( ( LTrn `  K
) `  W )
5 dibelval2nd.o . . . . 5  |-  .0.  =  ( f  e.  T  |->  (  _I  |`  B ) )
6 eqid 2443 . . . . 5  |-  ( (
DIsoA `  K ) `  W )  =  ( ( DIsoA `  K ) `  W )
7 dibelval2nd.i . . . . 5  |-  I  =  ( ( DIsoB `  K
) `  W )
81, 2, 3, 4, 5, 6, 7dibval2 34789 . . . 4  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  (
I `  X )  =  ( ( ( ( DIsoA `  K ) `  W ) `  X
)  X.  {  .0.  } ) )
98eleq2d 2510 . . 3  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  ( Y  e.  ( I `  X )  <->  Y  e.  ( ( ( (
DIsoA `  K ) `  W ) `  X
)  X.  {  .0.  } ) ) )
109biimp3a 1318 . 2  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( X  e.  B  /\  X  .<_  W )  /\  Y  e.  ( I `  X
) )  ->  Y  e.  ( ( ( (
DIsoA `  K ) `  W ) `  X
)  X.  {  .0.  } ) )
11 xp2nd 6607 . 2  |-  ( Y  e.  ( ( ( ( DIsoA `  K ) `  W ) `  X
)  X.  {  .0.  } )  ->  ( 2nd `  Y )  e.  {  .0.  } )
12 elsni 3902 . 2  |-  ( ( 2nd `  Y )  e.  {  .0.  }  ->  ( 2nd `  Y
)  =  .0.  )
1310, 11, 123syl 20 1  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( X  e.  B  /\  X  .<_  W )  /\  Y  e.  ( I `  X
) )  ->  ( 2nd `  Y )  =  .0.  )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   {csn 3877   class class class wbr 4292    e. cmpt 4350    _I cid 4631    X. cxp 4838    |` cres 4842   ` cfv 5418   2ndc2nd 6576   Basecbs 14174   lecple 14245   LHypclh 33628   LTrncltrn 33745   DIsoAcdia 34673   DIsoBcdib 34783
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-reu 2722  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-2nd 6578  df-disoa 34674  df-dib 34784
This theorem is referenced by:  diblss  34815
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