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Theorem dibopelval2 37269
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 37268 . . 3  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  (
I `  X )  =  ( ( J `
 X )  X. 
{  .0.  } ) )
98eleq2d 2524 . 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 5018 . . 3  |-  ( <. F ,  S >.  e.  ( ( J `  X )  X.  {  .0.  } )  <->  ( F  e.  ( J `  X
)  /\  S  e.  {  .0.  } ) )
11 fvex 5858 . . . . . . . 8  |-  ( (
LTrn `  K ) `  W )  e.  _V
124, 11eqeltri 2538 . . . . . . 7  |-  T  e. 
_V
1312mptex 6118 . . . . . 6  |-  ( f  e.  T  |->  (  _I  |`  B ) )  e. 
_V
145, 13eqeltri 2538 . . . . 5  |-  .0.  e.  _V
1514elsnc2 4047 . . . 4  |-  ( S  e.  {  .0.  }  <->  S  =  .0.  )
1615anbi2i 692 . . 3  |-  ( ( F  e.  ( J `
 X )  /\  S  e.  {  .0.  } )  <->  ( F  e.  ( J `  X
)  /\  S  =  .0.  ) )
1710, 16bitri 249 . 2  |-  ( <. F ,  S >.  e.  ( ( J `  X )  X.  {  .0.  } )  <->  ( F  e.  ( J `  X
)  /\  S  =  .0.  ) )
189, 17syl6bb 261 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1398    e. wcel 1823   _Vcvv 3106   {csn 4016   <.cop 4022   class class class wbr 4439    |-> cmpt 4497    _I cid 4779    X. cxp 4986    |` cres 4990   ` cfv 5570   Basecbs 14716   lecple 14791   LHypclh 36105   LTrncltrn 36222   DIsoAcdia 37152   DIsoBcdib 37262
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-disoa 37153  df-dib 37263
This theorem is referenced by:  dibopelval3  37272  dibglbN  37290  diblsmopel  37295  dib2dim  37367  dih2dimbALTN  37369  dihord6apre  37380
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