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Theorem dibelval1st 34680
Description: Membership in value of the partial isomorphism B for a lattice  K. (Contributed by NM, 13-Feb-2014.)
Hypotheses
Ref Expression
dibelval1.b  |-  B  =  ( Base `  K
)
dibelval1.l  |-  .<_  =  ( le `  K )
dibelval1.h  |-  H  =  ( LHyp `  K
)
dibelval1.j  |-  J  =  ( ( DIsoA `  K
) `  W )
dibelval1.i  |-  I  =  ( ( DIsoB `  K
) `  W )
Assertion
Ref Expression
dibelval1st  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( X  e.  B  /\  X  .<_  W )  /\  Y  e.  ( I `  X
) )  ->  ( 1st `  Y )  e.  ( J `  X
) )

Proof of Theorem dibelval1st
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 dibelval1.b . . . . 5  |-  B  =  ( Base `  K
)
2 dibelval1.l . . . . 5  |-  .<_  =  ( le `  K )
3 dibelval1.h . . . . 5  |-  H  =  ( LHyp `  K
)
4 eqid 2423 . . . . 5  |-  ( (
LTrn `  K ) `  W )  =  ( ( LTrn `  K
) `  W )
5 eqid 2423 . . . . 5  |-  ( f  e.  ( ( LTrn `  K ) `  W
)  |->  (  _I  |`  B ) )  =  ( f  e.  ( ( LTrn `  K ) `  W
)  |->  (  _I  |`  B ) )
6 dibelval1.j . . . . 5  |-  J  =  ( ( DIsoA `  K
) `  W )
7 dibelval1.i . . . . 5  |-  I  =  ( ( DIsoB `  K
) `  W )
81, 2, 3, 4, 5, 6, 7dibval2 34675 . . . 4  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  (
I `  X )  =  ( ( J `
 X )  X. 
{ ( f  e.  ( ( LTrn `  K
) `  W )  |->  (  _I  |`  B ) ) } ) )
98eleq2d 2493 . . 3  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  ( Y  e.  ( I `  X )  <->  Y  e.  ( ( J `  X )  X.  {
( f  e.  ( ( LTrn `  K
) `  W )  |->  (  _I  |`  B ) ) } ) ) )
109biimp3a 1365 . 2  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( X  e.  B  /\  X  .<_  W )  /\  Y  e.  ( I `  X
) )  ->  Y  e.  ( ( J `  X )  X.  {
( f  e.  ( ( LTrn `  K
) `  W )  |->  (  _I  |`  B ) ) } ) )
11 xp1st 6835 . 2  |-  ( Y  e.  ( ( J `
 X )  X. 
{ ( f  e.  ( ( LTrn `  K
) `  W )  |->  (  _I  |`  B ) ) } )  -> 
( 1st `  Y
)  e.  ( J `
 X ) )
1210, 11syl 17 1  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( X  e.  B  /\  X  .<_  W )  /\  Y  e.  ( I `  X
) )  ->  ( 1st `  Y )  e.  ( J `  X
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 371    /\ w3a 983    = wceq 1438    e. wcel 1869   {csn 3997   class class class wbr 4421    |-> cmpt 4480    _I cid 4761    X. cxp 4849    |` cres 4853   ` cfv 5599   1stc1st 6803   Basecbs 15114   lecple 15190   LHypclh 33512   LTrncltrn 33629   DIsoAcdia 34559   DIsoBcdib 34669
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-8 1871  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-rep 4534  ax-sep 4544  ax-nul 4553  ax-pow 4600  ax-pr 4658  ax-un 6595
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-ral 2781  df-rex 2782  df-reu 2783  df-rab 2785  df-v 3084  df-sbc 3301  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3763  df-if 3911  df-pw 3982  df-sn 3998  df-pr 4000  df-op 4004  df-uni 4218  df-iun 4299  df-br 4422  df-opab 4481  df-mpt 4482  df-id 4766  df-xp 4857  df-rel 4858  df-cnv 4859  df-co 4860  df-dm 4861  df-rn 4862  df-res 4863  df-ima 4864  df-iota 5563  df-fun 5601  df-fn 5602  df-f 5603  df-f1 5604  df-fo 5605  df-f1o 5606  df-fv 5607  df-1st 6805  df-disoa 34560  df-dib 34670
This theorem is referenced by:  dibelval1st1  34681  dibelval1st2N  34682  diblss  34701
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