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Theorem dicval 34661
Description: The partial isomorphism C for a lattice  K. (Contributed by NM, 15-Dec-2013.) (Revised by Mario Carneiro, 22-Sep-2015.)
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
dicval  |-  ( ( ( 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 ) } )
Distinct variable groups:    f, g,
s, K    T, g    f, W, g, s    f, E, s    P, f    Q, f, g, s    T, f
Allowed substitution hints:    A( f, g, s)    P( g, s)    T( s)    E( g)    H( f, g, s)    I( f, g, s)    .<_ ( f, g, s)    V( f, g, s)

Proof of Theorem dicval
Dummy variables  r 
q are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dicval.l . . . . 5  |-  .<_  =  ( le `  K )
2 dicval.a . . . . 5  |-  A  =  ( Atoms `  K )
3 dicval.h . . . . 5  |-  H  =  ( LHyp `  K
)
4 dicval.p . . . . 5  |-  P  =  ( ( oc `  K ) `  W
)
5 dicval.t . . . . 5  |-  T  =  ( ( LTrn `  K
) `  W )
6 dicval.e . . . . 5  |-  E  =  ( ( TEndo `  K
) `  W )
7 dicval.i . . . . 5  |-  I  =  ( ( DIsoC `  K
) `  W )
81, 2, 3, 4, 5, 6, 7dicfval 34660 . . . 4  |-  ( ( K  e.  V  /\  W  e.  H )  ->  I  =  ( q  e.  { r  e.  A  |  -.  r  .<_  W }  |->  { <. f ,  s >.  |  ( f  =  ( s `
 ( iota_ g  e.  T  ( g `  P )  =  q ) )  /\  s  e.  E ) } ) )
98adantr 465 . . 3  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  I  =  ( q  e.  { r  e.  A  |  -.  r  .<_  W }  |->  { <. f ,  s
>.  |  ( f  =  ( s `  ( iota_ g  e.  T  ( g `  P
)  =  q ) )  /\  s  e.  E ) } ) )
109fveq1d 5688 . 2  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( I `  Q
)  =  ( ( q  e.  { r  e.  A  |  -.  r  .<_  W }  |->  {
<. f ,  s >.  |  ( f  =  ( s `  ( iota_ g  e.  T  ( g `  P )  =  q ) )  /\  s  e.  E
) } ) `  Q ) )
11 simpr 461 . . . 4  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( Q  e.  A  /\  -.  Q  .<_  W ) )
12 breq1 4290 . . . . . 6  |-  ( r  =  Q  ->  (
r  .<_  W  <->  Q  .<_  W ) )
1312notbid 294 . . . . 5  |-  ( r  =  Q  ->  ( -.  r  .<_  W  <->  -.  Q  .<_  W ) )
1413elrab 3112 . . . 4  |-  ( Q  e.  { r  e.  A  |  -.  r  .<_  W }  <->  ( Q  e.  A  /\  -.  Q  .<_  W ) )
1511, 14sylibr 212 . . 3  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  ->  Q  e.  { r  e.  A  |  -.  r  .<_  W } )
16 eqeq2 2447 . . . . . . . . 9  |-  ( q  =  Q  ->  (
( g `  P
)  =  q  <->  ( g `  P )  =  Q ) )
1716riotabidv 6049 . . . . . . . 8  |-  ( q  =  Q  ->  ( iota_ g  e.  T  ( g `  P )  =  q )  =  ( iota_ g  e.  T  ( g `  P
)  =  Q ) )
1817fveq2d 5690 . . . . . . 7  |-  ( q  =  Q  ->  (
s `  ( iota_ g  e.  T  ( g `  P )  =  q ) )  =  ( s `  ( iota_ g  e.  T  ( g `
 P )  =  Q ) ) )
1918eqeq2d 2449 . . . . . 6  |-  ( q  =  Q  ->  (
f  =  ( s `
 ( iota_ g  e.  T  ( g `  P )  =  q ) )  <->  f  =  ( s `  ( iota_ g  e.  T  ( g `  P )  =  Q ) ) ) )
2019anbi1d 704 . . . . 5  |-  ( q  =  Q  ->  (
( f  =  ( s `  ( iota_ g  e.  T  ( g `
 P )  =  q ) )  /\  s  e.  E )  <->  ( f  =  ( s `
 ( iota_ g  e.  T  ( g `  P )  =  Q ) )  /\  s  e.  E ) ) )
2120opabbidv 4350 . . . 4  |-  ( q  =  Q  ->  { <. f ,  s >.  |  ( f  =  ( s `
 ( iota_ g  e.  T  ( g `  P )  =  q ) )  /\  s  e.  E ) }  =  { <. f ,  s
>.  |  ( f  =  ( s `  ( iota_ g  e.  T  ( g `  P
)  =  Q ) )  /\  s  e.  E ) } )
22 eqid 2438 . . . 4  |-  ( q  e.  { r  e.  A  |  -.  r  .<_  W }  |->  { <. f ,  s >.  |  ( f  =  ( s `
 ( iota_ g  e.  T  ( g `  P )  =  q ) )  /\  s  e.  E ) } )  =  ( q  e. 
{ r  e.  A  |  -.  r  .<_  W }  |->  { <. f ,  s
>.  |  ( f  =  ( s `  ( iota_ g  e.  T  ( g `  P
)  =  q ) )  /\  s  e.  E ) } )
23 fvex 5696 . . . . . . . . . . 11  |-  ( (
TEndo `  K ) `  W )  e.  _V
246, 23eqeltri 2508 . . . . . . . . . 10  |-  E  e. 
_V
2524uniex 6371 . . . . . . . . 9  |-  U. E  e.  _V
2625rnex 6507 . . . . . . . 8  |-  ran  U. E  e.  _V
2726uniex 6371 . . . . . . 7  |-  U. ran  U. E  e.  _V
2827pwex 4470 . . . . . 6  |-  ~P U. ran  U. E  e.  _V
2928, 24xpex 6503 . . . . 5  |-  ( ~P
U. ran  U. E  X.  E )  e.  _V
30 simpl 457 . . . . . . . . 9  |-  ( ( f  =  ( s `
 ( iota_ g  e.  T  ( g `  P )  =  Q ) )  /\  s  e.  E )  ->  f  =  ( s `  ( iota_ g  e.  T  ( g `  P
)  =  Q ) ) )
31 fvssunirn 5708 . . . . . . . . . . 11  |-  ( s `
 ( iota_ g  e.  T  ( g `  P )  =  Q ) )  C_  U. ran  s
32 elssuni 4116 . . . . . . . . . . . . 13  |-  ( s  e.  E  ->  s  C_ 
U. E )
3332adantl 466 . . . . . . . . . . . 12  |-  ( ( f  =  ( s `
 ( iota_ g  e.  T  ( g `  P )  =  Q ) )  /\  s  e.  E )  ->  s  C_ 
U. E )
34 rnss 5063 . . . . . . . . . . . 12  |-  ( s 
C_  U. E  ->  ran  s  C_  ran  U. E
)
35 uniss 4107 . . . . . . . . . . . 12  |-  ( ran  s  C_  ran  U. E  ->  U. ran  s  C_  U.
ran  U. E )
3633, 34, 353syl 20 . . . . . . . . . . 11  |-  ( ( f  =  ( s `
 ( iota_ g  e.  T  ( g `  P )  =  Q ) )  /\  s  e.  E )  ->  U. ran  s  C_  U. ran  U. E )
3731, 36syl5ss 3362 . . . . . . . . . 10  |-  ( ( f  =  ( s `
 ( iota_ g  e.  T  ( g `  P )  =  Q ) )  /\  s  e.  E )  ->  (
s `  ( iota_ g  e.  T  ( g `  P )  =  Q ) )  C_  U. ran  U. E )
3827elpw2 4451 . . . . . . . . . 10  |-  ( ( s `  ( iota_ g  e.  T  ( g `
 P )  =  Q ) )  e. 
~P U. ran  U. E  <->  ( s `  ( iota_ g  e.  T  ( g `
 P )  =  Q ) )  C_  U.
ran  U. E )
3937, 38sylibr 212 . . . . . . . . 9  |-  ( ( f  =  ( s `
 ( iota_ g  e.  T  ( g `  P )  =  Q ) )  /\  s  e.  E )  ->  (
s `  ( iota_ g  e.  T  ( g `  P )  =  Q ) )  e.  ~P U.
ran  U. E )
4030, 39eqeltrd 2512 . . . . . . . 8  |-  ( ( f  =  ( s `
 ( iota_ g  e.  T  ( g `  P )  =  Q ) )  /\  s  e.  E )  ->  f  e.  ~P U. ran  U. E )
41 simpr 461 . . . . . . . 8  |-  ( ( f  =  ( s `
 ( iota_ g  e.  T  ( g `  P )  =  Q ) )  /\  s  e.  E )  ->  s  e.  E )
4240, 41jca 532 . . . . . . 7  |-  ( ( f  =  ( s `
 ( iota_ g  e.  T  ( g `  P )  =  Q ) )  /\  s  e.  E )  ->  (
f  e.  ~P U. ran  U. E  /\  s  e.  E ) )
4342ssopab2i 4611 . . . . . 6  |-  { <. f ,  s >.  |  ( f  =  ( s `
 ( iota_ g  e.  T  ( g `  P )  =  Q ) )  /\  s  e.  E ) }  C_  {
<. f ,  s >.  |  ( f  e. 
~P U. ran  U. E  /\  s  e.  E
) }
44 df-xp 4841 . . . . . 6  |-  ( ~P
U. ran  U. E  X.  E )  =  { <. f ,  s >.  |  ( f  e. 
~P U. ran  U. E  /\  s  e.  E
) }
4543, 44sseqtr4i 3384 . . . . 5  |-  { <. f ,  s >.  |  ( f  =  ( s `
 ( iota_ g  e.  T  ( g `  P )  =  Q ) )  /\  s  e.  E ) }  C_  ( ~P U. ran  U. E  X.  E )
4629, 45ssexi 4432 . . . 4  |-  { <. f ,  s >.  |  ( f  =  ( s `
 ( iota_ g  e.  T  ( g `  P )  =  Q ) )  /\  s  e.  E ) }  e.  _V
4721, 22, 46fvmpt 5769 . . 3  |-  ( Q  e.  { r  e.  A  |  -.  r  .<_  W }  ->  (
( q  e.  {
r  e.  A  |  -.  r  .<_  W }  |->  { <. f ,  s
>.  |  ( f  =  ( s `  ( iota_ g  e.  T  ( g `  P
)  =  q ) )  /\  s  e.  E ) } ) `
 Q )  =  { <. f ,  s
>.  |  ( f  =  ( s `  ( iota_ g  e.  T  ( g `  P
)  =  Q ) )  /\  s  e.  E ) } )
4815, 47syl 16 . 2  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( ( q  e. 
{ r  e.  A  |  -.  r  .<_  W }  |->  { <. f ,  s
>.  |  ( f  =  ( s `  ( iota_ g  e.  T  ( g `  P
)  =  q ) )  /\  s  e.  E ) } ) `
 Q )  =  { <. f ,  s
>.  |  ( f  =  ( s `  ( iota_ g  e.  T  ( g `  P
)  =  Q ) )  /\  s  e.  E ) } )
4910, 48eqtrd 2470 1  |-  ( ( ( 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 ) } )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   {crab 2714   _Vcvv 2967    C_ wss 3323   ~Pcpw 3855   U.cuni 4086   class class class wbr 4287   {copab 4344    e. cmpt 4345    X. cxp 4833   ran crn 4836   ` cfv 5413   iota_crio 6046   lecple 14237   occoc 14238   Atomscatm 32748   LHypclh 33468   LTrncltrn 33585   TEndoctendo 34236   DIsoCcdic 34657
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-reu 2717  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-id 4631  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-dic 34658
This theorem is referenced by:  dicopelval  34662  dicelvalN  34663  dicval2  34664  dicfnN  34668  dicvalrelN  34670  dicssdvh  34671  dicelval1sta  34672  dihpN  34821
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