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Theorem dicopelval 34816
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 34815 . . 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 2534 . 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 2475 . . . 4  |-  ( f  =  F  ->  (
f  =  ( s `
 ( iota_ g  e.  T  ( g `  P )  =  Q ) )  <->  F  =  ( s `  ( iota_ g  e.  T  ( g `  P )  =  Q ) ) ) )
1312anbi1d 719 . . 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 5878 . . . . 5  |-  ( s  =  S  ->  (
s `  ( iota_ g  e.  T  ( g `  P )  =  Q ) )  =  ( S `  ( iota_ g  e.  T  ( g `
 P )  =  Q ) ) )
1514eqeq2d 2481 . . . 4  |-  ( s  =  S  ->  ( F  =  ( s `  ( iota_ g  e.  T  ( g `  P
)  =  Q ) )  <->  F  =  ( S `  ( iota_ g  e.  T  ( g `  P )  =  Q ) ) ) )
16 eleq1 2537 . . . 4  |-  ( s  =  S  ->  (
s  e.  E  <->  S  e.  E ) )
1715, 16anbi12d 725 . . 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 4723 . 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 269 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 189    /\ wa 376    = wceq 1452    e. wcel 1904   _Vcvv 3031   <.cop 3965   class class class wbr 4395   {copab 4453   ` cfv 5589   iota_crio 6269   lecple 15275   occoc 15276   Atomscatm 32900   LHypclh 33620   LTrncltrn 33737   TEndoctendo 34390   DIsoCcdic 34811
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-dic 34812
This theorem is referenced by:  dicopelval2  34820  dicvaddcl  34829  dicvscacl  34830  dicn0  34831
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