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Theorem dicelval1sta 31377
Description: Membership in value of the partial isomorphism C for a lattice  K. (Contributed by NM, 16-Feb-2014.)
Hypotheses
Ref Expression
dicelval1sta.l  |-  .<_  =  ( le `  K )
dicelval1sta.a  |-  A  =  ( Atoms `  K )
dicelval1sta.h  |-  H  =  ( LHyp `  K
)
dicelval1sta.p  |-  P  =  ( ( oc `  K ) `  W
)
dicelval1sta.t  |-  T  =  ( ( LTrn `  K
) `  W )
dicelval1sta.i  |-  I  =  ( ( DIsoC `  K
) `  W )
Assertion
Ref Expression
dicelval1sta  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  Y  e.  ( I `  Q
) )  ->  ( 1st `  Y )  =  ( ( 2nd `  Y
) `  ( iota_ g  e.  T ( g `  P )  =  Q ) ) )
Distinct variable groups:    g, K    Q, g    T, g    g, W
Allowed substitution hints:    A( g)    P( g)    H( g)    I( g)    .<_ ( g)    V( g)    Y( g)

Proof of Theorem dicelval1sta
Dummy variables  f 
s are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dicelval1sta.l . . . . . 6  |-  .<_  =  ( le `  K )
2 dicelval1sta.a . . . . . 6  |-  A  =  ( Atoms `  K )
3 dicelval1sta.h . . . . . 6  |-  H  =  ( LHyp `  K
)
4 dicelval1sta.p . . . . . 6  |-  P  =  ( ( oc `  K ) `  W
)
5 dicelval1sta.t . . . . . 6  |-  T  =  ( ( LTrn `  K
) `  W )
6 eqid 2283 . . . . . 6  |-  ( (
TEndo `  K ) `  W )  =  ( ( TEndo `  K ) `  W )
7 dicelval1sta.i . . . . . 6  |-  I  =  ( ( DIsoC `  K
) `  W )
81, 2, 3, 4, 5, 6, 7dicval 31366 . . . . 5  |-  ( ( ( 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.  ( ( TEndo `  K
) `  W )
) } )
98eleq2d 2350 . . . 4  |-  ( ( ( 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.  ( ( TEndo `  K
) `  W )
) } ) )
109biimp3a 1281 . . 3  |-  ( ( ( 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.  ( ( TEndo `  K
) `  W )
) } )
11 eqeq1 2289 . . . . 5  |-  ( f  =  ( 1st `  Y
)  ->  ( f  =  ( s `  ( iota_ g  e.  T
( g `  P
)  =  Q ) )  <->  ( 1st `  Y
)  =  ( s `
 ( iota_ g  e.  T ( g `  P )  =  Q ) ) ) )
1211anbi1d 685 . . . 4  |-  ( f  =  ( 1st `  Y
)  ->  ( (
f  =  ( s `
 ( iota_ g  e.  T ( g `  P )  =  Q ) )  /\  s  e.  ( ( TEndo `  K
) `  W )
)  <->  ( ( 1st `  Y )  =  ( s `  ( iota_ g  e.  T ( g `
 P )  =  Q ) )  /\  s  e.  ( ( TEndo `  K ) `  W ) ) ) )
13 fveq1 5524 . . . . . 6  |-  ( s  =  ( 2nd `  Y
)  ->  ( s `  ( iota_ g  e.  T
( g `  P
)  =  Q ) )  =  ( ( 2nd `  Y ) `
 ( iota_ g  e.  T ( g `  P )  =  Q ) ) )
1413eqeq2d 2294 . . . . 5  |-  ( s  =  ( 2nd `  Y
)  ->  ( ( 1st `  Y )  =  ( s `  ( iota_ g  e.  T ( g `  P )  =  Q ) )  <-> 
( 1st `  Y
)  =  ( ( 2nd `  Y ) `
 ( iota_ g  e.  T ( g `  P )  =  Q ) ) ) )
15 eleq1 2343 . . . . 5  |-  ( s  =  ( 2nd `  Y
)  ->  ( s  e.  ( ( TEndo `  K
) `  W )  <->  ( 2nd `  Y )  e.  ( ( TEndo `  K ) `  W
) ) )
1614, 15anbi12d 691 . . . 4  |-  ( s  =  ( 2nd `  Y
)  ->  ( (
( 1st `  Y
)  =  ( s `
 ( iota_ g  e.  T ( g `  P )  =  Q ) )  /\  s  e.  ( ( TEndo `  K
) `  W )
)  <->  ( ( 1st `  Y )  =  ( ( 2nd `  Y
) `  ( iota_ g  e.  T ( g `  P )  =  Q ) )  /\  ( 2nd `  Y )  e.  ( ( TEndo `  K
) `  W )
) ) )
1712, 16elopabi 6185 . . 3  |-  ( Y  e.  { <. f ,  s >.  |  ( f  =  ( s `
 ( iota_ g  e.  T ( g `  P )  =  Q ) )  /\  s  e.  ( ( TEndo `  K
) `  W )
) }  ->  (
( 1st `  Y
)  =  ( ( 2nd `  Y ) `
 ( iota_ g  e.  T ( g `  P )  =  Q ) )  /\  ( 2nd `  Y )  e.  ( ( TEndo `  K
) `  W )
) )
1810, 17syl 15 . 2  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  Y  e.  ( I `  Q
) )  ->  (
( 1st `  Y
)  =  ( ( 2nd `  Y ) `
 ( iota_ g  e.  T ( g `  P )  =  Q ) )  /\  ( 2nd `  Y )  e.  ( ( TEndo `  K
) `  W )
) )
1918simpld 445 1  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( Q  e.  A  /\  -.  Q  .<_  W )  /\  Y  e.  ( I `  Q
) )  ->  ( 1st `  Y )  =  ( ( 2nd `  Y
) `  ( iota_ g  e.  T ( g `  P )  =  Q ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   class class class wbr 4023   {copab 4076   ` cfv 5255   1stc1st 6120   2ndc2nd 6121   iota_crio 6297   lecple 13215   occoc 13216   Atomscatm 29453   LHypclh 30173   LTrncltrn 30290   TEndoctendo 30941   DIsoCcdic 31362
This theorem is referenced by:  dicvaddcl  31380  dicvscacl  31381
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-1st 6122  df-2nd 6123  df-riota 6304  df-dic 31363
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