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Theorem dibelval2nd 36350
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 2467 . . . . 5  |-  ( (
DIsoA `  K ) `  W )  =  ( ( DIsoA `  K ) `  W )
7 dibelval2nd.i . . . . 5  |-  I  =  ( ( DIsoB `  K
) `  W )
81, 2, 3, 4, 5, 6, 7dibval2 36342 . . . 4  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  (
I `  X )  =  ( ( ( ( DIsoA `  K ) `  W ) `  X
)  X.  {  .0.  } ) )
98eleq2d 2537 . . 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 1328 . 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 6826 . 2  |-  ( Y  e.  ( ( ( ( DIsoA `  K ) `  W ) `  X
)  X.  {  .0.  } )  ->  ( 2nd `  Y )  e.  {  .0.  } )
12 elsni 4058 . 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 973    = wceq 1379    e. wcel 1767   {csn 4033   class class class wbr 4453    |-> cmpt 4511    _I cid 4796    X. cxp 5003    |` cres 5007   ` cfv 5594   2ndc2nd 6794   Basecbs 14507   lecple 14579   LHypclh 35181   LTrncltrn 35298   DIsoAcdia 36226   DIsoBcdib 36336
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-2nd 6796  df-disoa 36227  df-dib 36337
This theorem is referenced by:  diblss  36368
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