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Theorem dicelval1sta 36002
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 2467 . . . . . 6  |-  ( (
TEndo `  K ) `  W )  =  ( ( TEndo `  K ) `  W )
7 dicelval1sta.i . . . . . 6  |-  I  =  ( ( DIsoC `  K
) `  W )
81, 2, 3, 4, 5, 6, 7dicval 35991 . . . . 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 2537 . . . 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 1328 . . 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 2471 . . . . 5  |-  ( f  =  ( 1st `  Y
)  ->  ( f  =  ( s `  ( iota_ g  e.  T  ( g `  P
)  =  Q ) )  <->  ( 1st `  Y
)  =  ( s `
 ( iota_ g  e.  T  ( g `  P )  =  Q ) ) ) )
1211anbi1d 704 . . . 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 5865 . . . . . 6  |-  ( s  =  ( 2nd `  Y
)  ->  ( s `  ( iota_ g  e.  T  ( g `  P
)  =  Q ) )  =  ( ( 2nd `  Y ) `
 ( iota_ g  e.  T  ( g `  P )  =  Q ) ) )
1413eqeq2d 2481 . . . . 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 2539 . . . . 5  |-  ( s  =  ( 2nd `  Y
)  ->  ( s  e.  ( ( TEndo `  K
) `  W )  <->  ( 2nd `  Y )  e.  ( ( TEndo `  K ) `  W
) ) )
1614, 15anbi12d 710 . . . 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 6845 . . 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 16 . 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 459 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   class class class wbr 4447   {copab 4504   ` cfv 5588   iota_crio 6244   1stc1st 6782   2ndc2nd 6783   lecple 14562   occoc 14563   Atomscatm 34078   LHypclh 34798   LTrncltrn 34915   TEndoctendo 35566   DIsoCcdic 35987
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6245  df-1st 6784  df-2nd 6785  df-dic 35988
This theorem is referenced by:  dicvaddcl  36005  dicvscacl  36006
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