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Theorem dibelval1st 31339
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 2283 . . . . 5  |-  ( (
LTrn `  K ) `  W )  =  ( ( LTrn `  K
) `  W )
5 eqid 2283 . . . . 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 31334 . . . 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 2350 . . 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 1281 . 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 6149 . 2  |-  ( Y  e.  ( ( J `
 X )  X. 
{ ( f  e.  ( ( LTrn `  K
) `  W )  |->  (  _I  |`  B ) ) } )  -> 
( 1st `  Y
)  e.  ( J `
 X ) )
1210, 11syl 15 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 358    /\ w3a 934    = wceq 1623    e. wcel 1684   {csn 3640   class class class wbr 4023    e. cmpt 4077    _I cid 4304    X. cxp 4687    |` cres 4691   ` cfv 5255   1stc1st 6120   Basecbs 13148   lecple 13215   LHypclh 30173   LTrncltrn 30290   DIsoAcdia 31218   DIsoBcdib 31328
This theorem is referenced by:  dibelval1st1  31340  dibelval1st2N  31341  diblss  31360
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-1st 6122  df-disoa 31219  df-dib 31329
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