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Theorem dibval2 31334
Description: Value of the partial isomorphism B. (Contributed by NM, 18-Jan-2014.)
Hypotheses
Ref Expression
dibval2.b  |-  B  =  ( Base `  K
)
dibval2.l  |-  .<_  =  ( le `  K )
dibval2.h  |-  H  =  ( LHyp `  K
)
dibval2.t  |-  T  =  ( ( LTrn `  K
) `  W )
dibval2.o  |-  .0.  =  ( f  e.  T  |->  (  _I  |`  B ) )
dibval2.j  |-  J  =  ( ( DIsoA `  K
) `  W )
dibval2.i  |-  I  =  ( ( DIsoB `  K
) `  W )
Assertion
Ref Expression
dibval2  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  (
I `  X )  =  ( ( J `
 X )  X. 
{  .0.  } ) )
Distinct variable groups:    f, K    f, W
Allowed substitution hints:    B( f)    T( f)    H( f)    I( f)    J( f)    .<_ ( f)    V( f)    X( f)    .0. ( f)

Proof of Theorem dibval2
StepHypRef Expression
1 dibval2.b . . . 4  |-  B  =  ( Base `  K
)
2 dibval2.l . . . 4  |-  .<_  =  ( le `  K )
3 dibval2.h . . . 4  |-  H  =  ( LHyp `  K
)
4 dibval2.j . . . 4  |-  J  =  ( ( DIsoA `  K
) `  W )
51, 2, 3, 4diaeldm 31226 . . 3  |-  ( ( K  e.  V  /\  W  e.  H )  ->  ( X  e.  dom  J  <-> 
( X  e.  B  /\  X  .<_  W ) ) )
65biimpar 471 . 2  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  X  e.  dom  J )
7 dibval2.t . . 3  |-  T  =  ( ( LTrn `  K
) `  W )
8 dibval2.o . . 3  |-  .0.  =  ( f  e.  T  |->  (  _I  |`  B ) )
9 dibval2.i . . 3  |-  I  =  ( ( DIsoB `  K
) `  W )
101, 3, 7, 8, 4, 9dibval 31332 . 2  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  X  e.  dom  J )  ->  (
I `  X )  =  ( ( J `
 X )  X. 
{  .0.  } ) )
116, 10syldan 456 1  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  (
I `  X )  =  ( ( J `
 X )  X. 
{  .0.  } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   {csn 3640   class class class wbr 4023    e. cmpt 4077    _I cid 4304    X. cxp 4687   dom cdm 4689    |` cres 4691   ` cfv 5255   Basecbs 13148   lecple 13215   LHypclh 30173   LTrncltrn 30290   DIsoAcdia 31218   DIsoBcdib 31328
This theorem is referenced by:  dibopelval2  31335  dibval3N  31336  dibelval3  31337  dibelval1st  31339  dibelval2nd  31342  dibn0  31343  dibord  31349  dib0  31354  dib1dim  31355  dibss  31359  diblss  31360  dihwN  31479
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-disoa 31219  df-dib 31329
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