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Theorem dibelval3 34792
Description: Member of the partial isomorphism B. (Contributed by NM, 26-Feb-2014.)
Hypotheses
Ref Expression
dibval3.b  |-  B  =  ( Base `  K
)
dibval3.l  |-  .<_  =  ( le `  K )
dibval3.h  |-  H  =  ( LHyp `  K
)
dibval3.t  |-  T  =  ( ( LTrn `  K
) `  W )
dibval3.r  |-  R  =  ( ( trL `  K
) `  W )
dibval3.o  |-  .0.  =  ( g  e.  T  |->  (  _I  |`  B ) )
dibval3.i  |-  I  =  ( ( DIsoB `  K
) `  W )
Assertion
Ref Expression
dibelval3  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  ( Y  e.  ( I `  X )  <->  E. f  e.  T  ( Y  =  <. f ,  .0.  >.  /\  ( R `  f
)  .<_  X ) ) )
Distinct variable groups:    f, K    g, K    T, f    f, W   
g, W    f, X    .<_ , f    B, f    f, H    .0. , f    T, g    f, V   
f, Y
Allowed substitution hints:    B( g)    R( f, g)    H( g)    I(
f, g)    .<_ ( g)    V( g)    X( g)    Y( g)    .0. ( g)

Proof of Theorem dibelval3
Dummy variable  s is distinct from all other variables.
StepHypRef Expression
1 dibval3.b . . . 4  |-  B  =  ( Base `  K
)
2 dibval3.l . . . 4  |-  .<_  =  ( le `  K )
3 dibval3.h . . . 4  |-  H  =  ( LHyp `  K
)
4 dibval3.t . . . 4  |-  T  =  ( ( LTrn `  K
) `  W )
5 dibval3.o . . . 4  |-  .0.  =  ( g  e.  T  |->  (  _I  |`  B ) )
6 eqid 2443 . . . 4  |-  ( (
DIsoA `  K ) `  W )  =  ( ( DIsoA `  K ) `  W )
7 dibval3.i . . . 4  |-  I  =  ( ( DIsoB `  K
) `  W )
81, 2, 3, 4, 5, 6, 7dibval2 34789 . . 3  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  (
I `  X )  =  ( ( ( ( DIsoA `  K ) `  W ) `  X
)  X.  {  .0.  } ) )
98eleq2d 2510 . 2  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  ( Y  e.  ( I `  X )  <->  Y  e.  ( ( ( (
DIsoA `  K ) `  W ) `  X
)  X.  {  .0.  } ) ) )
10 dibval3.r . . . . . . 7  |-  R  =  ( ( trL `  K
) `  W )
111, 2, 3, 4, 10, 6diaelval 34678 . . . . . 6  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  (
f  e.  ( ( ( DIsoA `  K ) `  W ) `  X
)  <->  ( f  e.  T  /\  ( R `
 f )  .<_  X ) ) )
1211anbi1d 704 . . . . 5  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  (
( f  e.  ( ( ( DIsoA `  K
) `  W ) `  X )  /\  Y  =  <. f ,  .0.  >.
)  <->  ( ( f  e.  T  /\  ( R `  f )  .<_  X )  /\  Y  =  <. f ,  .0.  >.
) ) )
13 an13 797 . . . . . . . 8  |-  ( ( Y  =  <. f ,  s >.  /\  (
f  e.  ( ( ( DIsoA `  K ) `  W ) `  X
)  /\  s  e.  {  .0.  } ) )  <-> 
( s  e.  {  .0.  }  /\  ( f  e.  ( ( (
DIsoA `  K ) `  W ) `  X
)  /\  Y  =  <. f ,  s >.
) ) )
14 elsn 3891 . . . . . . . . 9  |-  ( s  e.  {  .0.  }  <->  s  =  .0.  )
1514anbi1i 695 . . . . . . . 8  |-  ( ( s  e.  {  .0.  }  /\  ( f  e.  ( ( ( DIsoA `  K ) `  W
) `  X )  /\  Y  =  <. f ,  s >. )
)  <->  ( s  =  .0.  /\  ( f  e.  ( ( (
DIsoA `  K ) `  W ) `  X
)  /\  Y  =  <. f ,  s >.
) ) )
1613, 15bitri 249 . . . . . . 7  |-  ( ( Y  =  <. f ,  s >.  /\  (
f  e.  ( ( ( DIsoA `  K ) `  W ) `  X
)  /\  s  e.  {  .0.  } ) )  <-> 
( s  =  .0. 
/\  ( f  e.  ( ( ( DIsoA `  K ) `  W
) `  X )  /\  Y  =  <. f ,  s >. )
) )
1716exbii 1634 . . . . . 6  |-  ( E. s ( Y  = 
<. f ,  s >.  /\  ( f  e.  ( ( ( DIsoA `  K
) `  W ) `  X )  /\  s  e.  {  .0.  } ) )  <->  E. s ( s  =  .0.  /\  (
f  e.  ( ( ( DIsoA `  K ) `  W ) `  X
)  /\  Y  =  <. f ,  s >.
) ) )
18 fvex 5701 . . . . . . . . . 10  |-  ( (
LTrn `  K ) `  W )  e.  _V
194, 18eqeltri 2513 . . . . . . . . 9  |-  T  e. 
_V
2019mptex 5948 . . . . . . . 8  |-  ( g  e.  T  |->  (  _I  |`  B ) )  e. 
_V
215, 20eqeltri 2513 . . . . . . 7  |-  .0.  e.  _V
22 opeq2 4060 . . . . . . . . 9  |-  ( s  =  .0.  ->  <. f ,  s >.  =  <. f ,  .0.  >. )
2322eqeq2d 2454 . . . . . . . 8  |-  ( s  =  .0.  ->  ( Y  =  <. f ,  s >.  <->  Y  =  <. f ,  .0.  >. )
)
2423anbi2d 703 . . . . . . 7  |-  ( s  =  .0.  ->  (
( f  e.  ( ( ( DIsoA `  K
) `  W ) `  X )  /\  Y  =  <. f ,  s
>. )  <->  ( f  e.  ( ( ( DIsoA `  K ) `  W
) `  X )  /\  Y  =  <. f ,  .0.  >. )
) )
2521, 24ceqsexv 3009 . . . . . 6  |-  ( E. s ( s  =  .0.  /\  ( f  e.  ( ( (
DIsoA `  K ) `  W ) `  X
)  /\  Y  =  <. f ,  s >.
) )  <->  ( f  e.  ( ( ( DIsoA `  K ) `  W
) `  X )  /\  Y  =  <. f ,  .0.  >. )
)
2617, 25bitri 249 . . . . 5  |-  ( E. s ( Y  = 
<. f ,  s >.  /\  ( f  e.  ( ( ( DIsoA `  K
) `  W ) `  X )  /\  s  e.  {  .0.  } ) )  <->  ( f  e.  ( ( ( DIsoA `  K ) `  W
) `  X )  /\  Y  =  <. f ,  .0.  >. )
)
27 anass 649 . . . . . 6  |-  ( ( ( f  e.  T  /\  Y  =  <. f ,  .0.  >. )  /\  ( R `  f
)  .<_  X )  <->  ( f  e.  T  /\  ( Y  =  <. f ,  .0.  >.  /\  ( R `  f )  .<_  X ) ) )
28 an32 796 . . . . . 6  |-  ( ( ( f  e.  T  /\  Y  =  <. f ,  .0.  >. )  /\  ( R `  f
)  .<_  X )  <->  ( (
f  e.  T  /\  ( R `  f ) 
.<_  X )  /\  Y  =  <. f ,  .0.  >.
) )
2927, 28bitr3i 251 . . . . 5  |-  ( ( f  e.  T  /\  ( Y  =  <. f ,  .0.  >.  /\  ( R `  f )  .<_  X ) )  <->  ( (
f  e.  T  /\  ( R `  f ) 
.<_  X )  /\  Y  =  <. f ,  .0.  >.
) )
3012, 26, 293bitr4g 288 . . . 4  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  ( E. s ( Y  = 
<. f ,  s >.  /\  ( f  e.  ( ( ( DIsoA `  K
) `  W ) `  X )  /\  s  e.  {  .0.  } ) )  <->  ( f  e.  T  /\  ( Y  =  <. f ,  .0.  >.  /\  ( R `  f
)  .<_  X ) ) ) )
3130exbidv 1680 . . 3  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  ( E. f E. s ( Y  =  <. f ,  s >.  /\  (
f  e.  ( ( ( DIsoA `  K ) `  W ) `  X
)  /\  s  e.  {  .0.  } ) )  <->  E. f ( f  e.  T  /\  ( Y  =  <. f ,  .0.  >.  /\  ( R `  f
)  .<_  X ) ) ) )
32 elxp 4857 . . 3  |-  ( Y  e.  ( ( ( ( DIsoA `  K ) `  W ) `  X
)  X.  {  .0.  } )  <->  E. f E. s
( Y  =  <. f ,  s >.  /\  (
f  e.  ( ( ( DIsoA `  K ) `  W ) `  X
)  /\  s  e.  {  .0.  } ) ) )
33 df-rex 2721 . . 3  |-  ( E. f  e.  T  ( Y  =  <. f ,  .0.  >.  /\  ( R `  f )  .<_  X )  <->  E. f ( f  e.  T  /\  ( Y  =  <. f ,  .0.  >.  /\  ( R `  f
)  .<_  X ) ) )
3431, 32, 333bitr4g 288 . 2  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  ( Y  e.  ( (
( ( DIsoA `  K
) `  W ) `  X )  X.  {  .0.  } )  <->  E. f  e.  T  ( Y  =  <. f ,  .0.  >.  /\  ( R `  f
)  .<_  X ) ) )
359, 34bitrd 253 1  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  ( Y  e.  ( I `  X )  <->  E. f  e.  T  ( Y  =  <. f ,  .0.  >.  /\  ( R `  f
)  .<_  X ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369   E.wex 1586    e. wcel 1756   E.wrex 2716   _Vcvv 2972   {csn 3877   <.cop 3883   class class class wbr 4292    e. cmpt 4350    _I cid 4631    X. cxp 4838    |` cres 4842   ` cfv 5418   Basecbs 14174   lecple 14245   LHypclh 33628   LTrncltrn 33745   trLctrl 33802   DIsoAcdia 34673   DIsoBcdib 34783
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-reu 2722  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-disoa 34674  df-dib 34784
This theorem is referenced by:  cdlemn11pre  34855  dihord2pre  34870
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