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Theorem dicval 35130
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 35129 . . . 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 5794 . 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 4396 . . . . . 6  |-  ( r  =  Q  ->  (
r  .<_  W  <->  Q  .<_  W ) )
1312notbid 294 . . . . 5  |-  ( r  =  Q  ->  ( -.  r  .<_  W  <->  -.  Q  .<_  W ) )
1413elrab 3217 . . . 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 2466 . . . . . . . . 9  |-  ( q  =  Q  ->  (
( g `  P
)  =  q  <->  ( g `  P )  =  Q ) )
1716riotabidv 6156 . . . . . . . 8  |-  ( q  =  Q  ->  ( iota_ g  e.  T  ( g `  P )  =  q )  =  ( iota_ g  e.  T  ( g `  P
)  =  Q ) )
1817fveq2d 5796 . . . . . . 7  |-  ( q  =  Q  ->  (
s `  ( iota_ g  e.  T  ( g `  P )  =  q ) )  =  ( s `  ( iota_ g  e.  T  ( g `
 P )  =  Q ) ) )
1918eqeq2d 2465 . . . . . 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 4456 . . . 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 2451 . . . 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 5802 . . . . . . . . . . 11  |-  ( (
TEndo `  K ) `  W )  e.  _V
246, 23eqeltri 2535 . . . . . . . . . 10  |-  E  e. 
_V
2524uniex 6479 . . . . . . . . 9  |-  U. E  e.  _V
2625rnex 6615 . . . . . . . 8  |-  ran  U. E  e.  _V
2726uniex 6479 . . . . . . 7  |-  U. ran  U. E  e.  _V
2827pwex 4576 . . . . . 6  |-  ~P U. ran  U. E  e.  _V
2928, 24xpex 6611 . . . . 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 5815 . . . . . . . . . . 11  |-  ( s `
 ( iota_ g  e.  T  ( g `  P )  =  Q ) )  C_  U. ran  s
32 elssuni 4222 . . . . . . . . . . . . 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 5169 . . . . . . . . . . . 12  |-  ( s 
C_  U. E  ->  ran  s  C_  ran  U. E
)
35 uniss 4213 . . . . . . . . . . . 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 3468 . . . . . . . . . 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 4557 . . . . . . . . . 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 2539 . . . . . . . 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 4717 . . . . . 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 4947 . . . . . 6  |-  ( ~P
U. ran  U. E  X.  E )  =  { <. f ,  s >.  |  ( f  e. 
~P U. ran  U. E  /\  s  e.  E
) }
4543, 44sseqtr4i 3490 . . . . 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 4538 . . . 4  |-  { <. f ,  s >.  |  ( f  =  ( s `
 ( iota_ g  e.  T  ( g `  P )  =  Q ) )  /\  s  e.  E ) }  e.  _V
4721, 22, 46fvmpt 5876 . . 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 2492 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 1370    e. wcel 1758   {crab 2799   _Vcvv 3071    C_ wss 3429   ~Pcpw 3961   U.cuni 4192   class class class wbr 4393   {copab 4450    |-> cmpt 4451    X. cxp 4939   ran crn 4942   ` cfv 5519   iota_crio 6153   lecple 14356   occoc 14357   Atomscatm 33217   LHypclh 33937   LTrncltrn 34054   TEndoctendo 34705   DIsoCcdic 35126
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-iun 4274  df-br 4394  df-opab 4452  df-mpt 4453  df-id 4737  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-riota 6154  df-dic 35127
This theorem is referenced by:  dicopelval  35131  dicelvalN  35132  dicval2  35133  dicfnN  35137  dicvalrelN  35139  dicssdvh  35140  dicelval1sta  35141  dihpN  35290
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