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Theorem dibelval3 37017
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 2457 . . . 4  |-  ( (
DIsoA `  K ) `  W )  =  ( ( DIsoA `  K ) `  W )
7 dibval3.i . . . 4  |-  I  =  ( ( DIsoB `  K
) `  W )
81, 2, 3, 4, 5, 6, 7dibval2 37014 . . 3  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  (
I `  X )  =  ( ( ( ( DIsoA `  K ) `  W ) `  X
)  X.  {  .0.  } ) )
98eleq2d 2527 . 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 36903 . . . . . 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 799 . . . . . . . 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 4046 . . . . . . . . 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 1668 . . . . . 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 5882 . . . . . . . . . 10  |-  ( (
LTrn `  K ) `  W )  e.  _V
194, 18eqeltri 2541 . . . . . . . . 9  |-  T  e. 
_V
2019mptex 6144 . . . . . . . 8  |-  ( g  e.  T  |->  (  _I  |`  B ) )  e. 
_V
215, 20eqeltri 2541 . . . . . . 7  |-  .0.  e.  _V
22 opeq2 4220 . . . . . . . . 9  |-  ( s  =  .0.  ->  <. f ,  s >.  =  <. f ,  .0.  >. )
2322eqeq2d 2471 . . . . . . . 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 3146 . . . . . 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 798 . . . . . 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 1715 . . 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 5025 . . 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 2813 . . 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 1395   E.wex 1613    e. wcel 1819   E.wrex 2808   _Vcvv 3109   {csn 4032   <.cop 4038   class class class wbr 4456    |-> cmpt 4515    _I cid 4799    X. cxp 5006    |` cres 5010   ` cfv 5594   Basecbs 14644   lecple 14719   LHypclh 35851   LTrncltrn 35968   trLctrl 36026   DIsoAcdia 36898   DIsoBcdib 37008
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-disoa 36899  df-dib 37009
This theorem is referenced by:  cdlemn11pre  37080  dihord2pre  37095
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