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Theorem dibopelval2 31957
Description: Member of the partial isomorphism B. (Contributed by NM, 3-Mar-2014.) (Revised by Mario Carneiro, 6-May-2015.)
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
dibopelval2  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  ( <. F ,  S >.  e.  ( I `  X
)  <->  ( F  e.  ( J `  X
)  /\  S  =  .0.  ) ) )
Distinct variable groups:    f, K    f, W    T, f
Allowed substitution hints:    B( f)    S( f)    F( f)    H( f)    I( f)    J( f)    .<_ ( f)    V( f)    X( f)    .0. ( f)

Proof of Theorem dibopelval2
StepHypRef Expression
1 dibval2.b . . . 4  |-  B  =  ( Base `  K
)
2 dibval2.l . . . 4  |-  .<_  =  ( le `  K )
3 dibval2.h . . . 4  |-  H  =  ( LHyp `  K
)
4 dibval2.t . . . 4  |-  T  =  ( ( LTrn `  K
) `  W )
5 dibval2.o . . . 4  |-  .0.  =  ( f  e.  T  |->  (  _I  |`  B ) )
6 dibval2.j . . . 4  |-  J  =  ( ( DIsoA `  K
) `  W )
7 dibval2.i . . . 4  |-  I  =  ( ( DIsoB `  K
) `  W )
81, 2, 3, 4, 5, 6, 7dibval2 31956 . . 3  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  (
I `  X )  =  ( ( J `
 X )  X. 
{  .0.  } ) )
98eleq2d 2363 . 2  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  ( <. F ,  S >.  e.  ( I `  X
)  <->  <. F ,  S >.  e.  ( ( J `
 X )  X. 
{  .0.  } ) ) )
10 opelxp 4735 . . 3  |-  ( <. F ,  S >.  e.  ( ( J `  X )  X.  {  .0.  } )  <->  ( F  e.  ( J `  X
)  /\  S  e.  {  .0.  } ) )
11 fvex 5555 . . . . . . . 8  |-  ( (
LTrn `  K ) `  W )  e.  _V
124, 11eqeltri 2366 . . . . . . 7  |-  T  e. 
_V
1312mptex 5762 . . . . . 6  |-  ( f  e.  T  |->  (  _I  |`  B ) )  e. 
_V
145, 13eqeltri 2366 . . . . 5  |-  .0.  e.  _V
1514elsnc2 3682 . . . 4  |-  ( S  e.  {  .0.  }  <->  S  =  .0.  )
1615anbi2i 675 . . 3  |-  ( ( F  e.  ( J `
 X )  /\  S  e.  {  .0.  } )  <->  ( F  e.  ( J `  X
)  /\  S  =  .0.  ) )
1710, 16bitri 240 . 2  |-  ( <. F ,  S >.  e.  ( ( J `  X )  X.  {  .0.  } )  <->  ( F  e.  ( J `  X
)  /\  S  =  .0.  ) )
189, 17syl6bb 252 1  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  ( <. F ,  S >.  e.  ( I `  X
)  <->  ( F  e.  ( J `  X
)  /\  S  =  .0.  ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   _Vcvv 2801   {csn 3653   <.cop 3656   class class class wbr 4039    e. cmpt 4093    _I cid 4320    X. cxp 4703    |` cres 4707   ` cfv 5271   Basecbs 13164   lecple 13231   LHypclh 30795   LTrncltrn 30912   DIsoAcdia 31840   DIsoBcdib 31950
This theorem is referenced by:  dibopelval3  31960  dibglbN  31978  diblsmopel  31983  dib2dim  32055  dih2dimbALTN  32057  dihord6apre  32068
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-disoa 31841  df-dib 31951
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