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Theorem dibelval2nd 34638
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 2422 . . . . 5  |-  ( (
DIsoA `  K ) `  W )  =  ( ( DIsoA `  K ) `  W )
7 dibelval2nd.i . . . . 5  |-  I  =  ( ( DIsoB `  K
) `  W )
81, 2, 3, 4, 5, 6, 7dibval2 34630 . . . 4  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  (
I `  X )  =  ( ( ( ( DIsoA `  K ) `  W ) `  X
)  X.  {  .0.  } ) )
98eleq2d 2492 . . 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 1364 . 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 6834 . 2  |-  ( Y  e.  ( ( ( ( DIsoA `  K ) `  W ) `  X
)  X.  {  .0.  } )  ->  ( 2nd `  Y )  e.  {  .0.  } )
12 elsni 4021 . 2  |-  ( ( 2nd `  Y )  e.  {  .0.  }  ->  ( 2nd `  Y
)  =  .0.  )
1310, 11, 123syl 18 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 370    /\ w3a 982    = wceq 1437    e. wcel 1868   {csn 3996   class class class wbr 4420    |-> cmpt 4479    _I cid 4759    X. cxp 4847    |` cres 4851   ` cfv 5597   2ndc2nd 6802   Basecbs 15108   lecple 15184   LHypclh 33467   LTrncltrn 33584   DIsoAcdia 34514   DIsoBcdib 34624
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-rep 4533  ax-sep 4543  ax-nul 4551  ax-pow 4598  ax-pr 4656  ax-un 6593
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-ral 2780  df-rex 2781  df-reu 2782  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-op 4003  df-uni 4217  df-iun 4298  df-br 4421  df-opab 4480  df-mpt 4481  df-id 4764  df-xp 4855  df-rel 4856  df-cnv 4857  df-co 4858  df-dm 4859  df-rn 4860  df-res 4861  df-ima 4862  df-iota 5561  df-fun 5599  df-fn 5600  df-f 5601  df-f1 5602  df-fo 5603  df-f1o 5604  df-fv 5605  df-2nd 6804  df-disoa 34515  df-dib 34625
This theorem is referenced by:  diblss  34656
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