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Theorem dibelval1st 31884
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 2435 . . . . 5  |-  ( (
LTrn `  K ) `  W )  =  ( ( LTrn `  K
) `  W )
5 eqid 2435 . . . . 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 31879 . . . 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 2502 . . 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 1283 . 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 6368 . 2  |-  ( Y  e.  ( ( J `
 X )  X. 
{ ( f  e.  ( ( LTrn `  K
) `  W )  |->  (  _I  |`  B ) ) } )  -> 
( 1st `  Y
)  e.  ( J `
 X ) )
1210, 11syl 16 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 set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   {csn 3806   class class class wbr 4204    e. cmpt 4258    _I cid 4485    X. cxp 4868    |` cres 4872   ` cfv 5446   1stc1st 6339   Basecbs 13461   lecple 13528   LHypclh 30718   LTrncltrn 30835   DIsoAcdia 31763   DIsoBcdib 31873
This theorem is referenced by:  dibelval1st1  31885  dibelval1st2N  31886  diblss  31905
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-1st 6341  df-disoa 31764  df-dib 31874
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