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Theorem dicelvalN 31293
Description: Membership in value of the partial isomorphism C for a lattice  K. (Contributed by NM, 25-Feb-2014.) (New usage is discouraged.)
Hypotheses
Ref Expression
dicval.l  |-  .<_  =  ( le `  K )
dicval.a  |-  A  =  ( Atoms `  K )
dicval.h  |-  H  =  ( LHyp `  K
)
dicval.p  |-  P  =  ( ( oc `  K ) `  W
)
dicval.t  |-  T  =  ( ( LTrn `  K
) `  W )
dicval.e  |-  E  =  ( ( TEndo `  K
) `  W )
dicval.i  |-  I  =  ( ( DIsoC `  K
) `  W )
Assertion
Ref Expression
dicelvalN  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( Y  e.  ( I `  Q )  <-> 
( Y  e.  ( _V  X.  _V )  /\  ( ( 1st `  Y
)  =  ( ( 2nd `  Y ) `
 ( iota_ g  e.  T ( g `  P )  =  Q ) )  /\  ( 2nd `  Y )  e.  E ) ) ) )
Distinct variable groups:    g, K    T, g    g, W    Q, g
Allowed substitution hints:    A( g)    P( g)    E( g)    H( g)    I( g)    .<_ ( g)    V( g)    Y( g)

Proof of Theorem dicelvalN
Dummy variables  f 
s are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dicval.l . . . 4  |-  .<_  =  ( le `  K )
2 dicval.a . . . 4  |-  A  =  ( Atoms `  K )
3 dicval.h . . . 4  |-  H  =  ( LHyp `  K
)
4 dicval.p . . . 4  |-  P  =  ( ( oc `  K ) `  W
)
5 dicval.t . . . 4  |-  T  =  ( ( LTrn `  K
) `  W )
6 dicval.e . . . 4  |-  E  =  ( ( TEndo `  K
) `  W )
7 dicval.i . . . 4  |-  I  =  ( ( DIsoC `  K
) `  W )
81, 2, 3, 4, 5, 6, 7dicval 31291 . . 3  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( I `  Q
)  =  { <. f ,  s >.  |  ( f  =  ( s `
 ( iota_ g  e.  T ( g `  P )  =  Q ) )  /\  s  e.  E ) } )
98eleq2d 2454 . 2  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( Y  e.  ( I `  Q )  <-> 
Y  e.  { <. f ,  s >.  |  ( f  =  ( s `
 ( iota_ g  e.  T ( g `  P )  =  Q ) )  /\  s  e.  E ) } ) )
10 vex 2902 . . . . . 6  |-  f  e. 
_V
11 vex 2902 . . . . . 6  |-  s  e. 
_V
1210, 11op1std 6296 . . . . 5  |-  ( Y  =  <. f ,  s
>.  ->  ( 1st `  Y
)  =  f )
1310, 11op2ndd 6297 . . . . . 6  |-  ( Y  =  <. f ,  s
>.  ->  ( 2nd `  Y
)  =  s )
1413fveq1d 5670 . . . . 5  |-  ( Y  =  <. f ,  s
>.  ->  ( ( 2nd `  Y ) `  ( iota_ g  e.  T ( g `  P )  =  Q ) )  =  ( s `  ( iota_ g  e.  T
( g `  P
)  =  Q ) ) )
1512, 14eqeq12d 2401 . . . 4  |-  ( Y  =  <. f ,  s
>.  ->  ( ( 1st `  Y )  =  ( ( 2nd `  Y
) `  ( iota_ g  e.  T ( g `  P )  =  Q ) )  <->  f  =  ( s `  ( iota_ g  e.  T ( g `  P )  =  Q ) ) ) )
1613eleq1d 2453 . . . 4  |-  ( Y  =  <. f ,  s
>.  ->  ( ( 2nd `  Y )  e.  E  <->  s  e.  E ) )
1715, 16anbi12d 692 . . 3  |-  ( Y  =  <. f ,  s
>.  ->  ( ( ( 1st `  Y )  =  ( ( 2nd `  Y ) `  ( iota_ g  e.  T ( g `  P )  =  Q ) )  /\  ( 2nd `  Y
)  e.  E )  <-> 
( f  =  ( s `  ( iota_ g  e.  T ( g `
 P )  =  Q ) )  /\  s  e.  E )
) )
1817elopaba 6348 . 2  |-  ( Y  e.  { <. f ,  s >.  |  ( f  =  ( s `
 ( iota_ g  e.  T ( g `  P )  =  Q ) )  /\  s  e.  E ) }  <->  ( Y  e.  ( _V  X.  _V )  /\  ( ( 1st `  Y )  =  ( ( 2nd `  Y
) `  ( iota_ g  e.  T ( g `  P )  =  Q ) )  /\  ( 2nd `  Y )  e.  E ) ) )
199, 18syl6bb 253 1  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( Y  e.  ( I `  Q )  <-> 
( Y  e.  ( _V  X.  _V )  /\  ( ( 1st `  Y
)  =  ( ( 2nd `  Y ) `
 ( iota_ g  e.  T ( g `  P )  =  Q ) )  /\  ( 2nd `  Y )  e.  E ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717   _Vcvv 2899   <.cop 3760   class class class wbr 4153   {copab 4206    X. cxp 4816   ` cfv 5394   1stc1st 6286   2ndc2nd 6287   iota_crio 6478   lecple 13463   occoc 13464   Atomscatm 29378   LHypclh 30098   LTrncltrn 30215   TEndoctendo 30866   DIsoCcdic 31287
This theorem is referenced by:  dicelval2N  31297
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-reu 2656  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-1st 6288  df-2nd 6289  df-riota 6485  df-dic 31288
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