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Theorem dicopelval 31989
Description: Membership in value of the partial isomorphism C for a lattice  K. (Contributed by NM, 15-Feb-2014.)
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 )
dicelval.f  |-  F  e. 
_V
dicelval.s  |-  S  e. 
_V
Assertion
Ref Expression
dicopelval  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( <. F ,  S >.  e.  ( I `  Q )  <->  ( F  =  ( S `  ( iota_ g  e.  T
( g `  P
)  =  Q ) )  /\  S  e.  E ) ) )
Distinct variable groups:    g, K    T, g    g, W    Q, g
Allowed substitution hints:    A( g)    P( g)    S( g)    E( g)    F( g)    H( g)    I(
g)    .<_ ( g)    V( g)

Proof of Theorem dicopelval
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 31988 . . 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 2363 . 2  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( <. F ,  S >.  e.  ( I `  Q )  <->  <. F ,  S >.  e.  { <. f ,  s >.  |  ( f  =  ( s `
 ( iota_ g  e.  T ( g `  P )  =  Q ) )  /\  s  e.  E ) } ) )
10 dicelval.f . . 3  |-  F  e. 
_V
11 dicelval.s . . 3  |-  S  e. 
_V
12 eqeq1 2302 . . . 4  |-  ( f  =  F  ->  (
f  =  ( s `
 ( iota_ g  e.  T ( g `  P )  =  Q ) )  <->  F  =  ( s `  ( iota_ g  e.  T ( g `  P )  =  Q ) ) ) )
1312anbi1d 685 . . 3  |-  ( f  =  F  ->  (
( f  =  ( s `  ( iota_ g  e.  T ( g `
 P )  =  Q ) )  /\  s  e.  E )  <->  ( F  =  ( s `
 ( iota_ g  e.  T ( g `  P )  =  Q ) )  /\  s  e.  E ) ) )
14 fveq1 5540 . . . . 5  |-  ( s  =  S  ->  (
s `  ( iota_ g  e.  T ( g `  P )  =  Q ) )  =  ( S `  ( iota_ g  e.  T ( g `
 P )  =  Q ) ) )
1514eqeq2d 2307 . . . 4  |-  ( s  =  S  ->  ( F  =  ( s `  ( iota_ g  e.  T
( g `  P
)  =  Q ) )  <->  F  =  ( S `  ( iota_ g  e.  T ( g `  P )  =  Q ) ) ) )
16 eleq1 2356 . . . 4  |-  ( s  =  S  ->  (
s  e.  E  <->  S  e.  E ) )
1715, 16anbi12d 691 . . 3  |-  ( s  =  S  ->  (
( F  =  ( s `  ( iota_ g  e.  T ( g `
 P )  =  Q ) )  /\  s  e.  E )  <->  ( F  =  ( S `
 ( iota_ g  e.  T ( g `  P )  =  Q ) )  /\  S  e.  E ) ) )
1810, 11, 13, 17opelopab 4302 . 2  |-  ( <. F ,  S >.  e. 
{ <. f ,  s
>.  |  ( f  =  ( s `  ( iota_ g  e.  T
( g `  P
)  =  Q ) )  /\  s  e.  E ) }  <->  ( F  =  ( S `  ( iota_ g  e.  T
( g `  P
)  =  Q ) )  /\  S  e.  E ) )
199, 18syl6bb 252 1  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( <. F ,  S >.  e.  ( I `  Q )  <->  ( F  =  ( S `  ( iota_ g  e.  T
( g `  P
)  =  Q ) )  /\  S  e.  E ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   _Vcvv 2801   <.cop 3656   class class class wbr 4039   {copab 4092   ` cfv 5271   iota_crio 6313   lecple 13231   occoc 13232   Atomscatm 30075   LHypclh 30795   LTrncltrn 30912   TEndoctendo 31563   DIsoCcdic 31984
This theorem is referenced by:  dicopelval2  31993  dicvaddcl  32002  dicvscacl  32003  dicn0  32004
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-riota 6320  df-dic 31985
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