Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dibopelval2 Structured version   Unicode version

Theorem dibopelval2 31880
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 31879 . . 3  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  (
I `  X )  =  ( ( J `
 X )  X. 
{  .0.  } ) )
98eleq2d 2502 . 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 4900 . . 3  |-  ( <. F ,  S >.  e.  ( ( J `  X )  X.  {  .0.  } )  <->  ( F  e.  ( J `  X
)  /\  S  e.  {  .0.  } ) )
11 fvex 5734 . . . . . . . 8  |-  ( (
LTrn `  K ) `  W )  e.  _V
124, 11eqeltri 2505 . . . . . . 7  |-  T  e. 
_V
1312mptex 5958 . . . . . 6  |-  ( f  e.  T  |->  (  _I  |`  B ) )  e. 
_V
145, 13eqeltri 2505 . . . . 5  |-  .0.  e.  _V
1514elsnc2 3835 . . . 4  |-  ( S  e.  {  .0.  }  <->  S  =  .0.  )
1615anbi2i 676 . . 3  |-  ( ( F  e.  ( J `
 X )  /\  S  e.  {  .0.  } )  <->  ( F  e.  ( J `  X
)  /\  S  =  .0.  ) )
1710, 16bitri 241 . 2  |-  ( <. F ,  S >.  e.  ( ( J `  X )  X.  {  .0.  } )  <->  ( F  e.  ( J `  X
)  /\  S  =  .0.  ) )
189, 17syl6bb 253 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 177    /\ wa 359    = wceq 1652    e. wcel 1725   _Vcvv 2948   {csn 3806   <.cop 3809   class class class wbr 4204    e. cmpt 4258    _I cid 4485    X. cxp 4868    |` cres 4872   ` cfv 5446   Basecbs 13461   lecple 13528   LHypclh 30718   LTrncltrn 30835   DIsoAcdia 31763   DIsoBcdib 31873
This theorem is referenced by:  dibopelval3  31883  dibglbN  31901  diblsmopel  31906  dib2dim  31978  dih2dimbALTN  31980  dihord6apre  31991
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-disoa 31764  df-dib 31874
  Copyright terms: Public domain W3C validator