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Theorem dibelval2nd 32012
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 2438 . . . . 5  |-  ( (
DIsoA `  K ) `  W )  =  ( ( DIsoA `  K ) `  W )
7 dibelval2nd.i . . . . 5  |-  I  =  ( ( DIsoB `  K
) `  W )
81, 2, 3, 4, 5, 6, 7dibval2 32004 . . . 4  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  (
I `  X )  =  ( ( ( ( DIsoA `  K ) `  W ) `  X
)  X.  {  .0.  } ) )
98eleq2d 2505 . . 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 1284 . 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 6379 . 2  |-  ( Y  e.  ( ( ( ( DIsoA `  K ) `  W ) `  X
)  X.  {  .0.  } )  ->  ( 2nd `  Y )  e.  {  .0.  } )
12 elsni 3840 . 2  |-  ( ( 2nd `  Y )  e.  {  .0.  }  ->  ( 2nd `  Y
)  =  .0.  )
1310, 11, 123syl 19 1  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( X  e.  B  /\  X  .<_  W )  /\  Y  e.  ( I `  X
) )  ->  ( 2nd `  Y )  =  .0.  )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   {csn 3816   class class class wbr 4214    e. cmpt 4268    _I cid 4495    X. cxp 4878    |` cres 4882   ` cfv 5456   2ndc2nd 6350   Basecbs 13471   lecple 13538   LHypclh 30843   LTrncltrn 30960   DIsoAcdia 31888   DIsoBcdib 31998
This theorem is referenced by:  diblss  32030
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-2nd 6352  df-disoa 31889  df-dib 31999
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