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Theorem dicelvalN 37006
Description: Membership in value of the partial isomorphism C for a lattice  K. (Contributed by NM, 25-Feb-2014.) (New usage is discouraged.)
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
dicelvalN  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( Y  e.  ( I `  Q )  <-> 
( Y  e.  ( _V  X.  _V )  /\  ( ( 1st `  Y
)  =  ( ( 2nd `  Y ) `
 ( iota_ g  e.  T  ( g `  P )  =  Q ) )  /\  ( 2nd `  Y )  e.  E ) ) ) )
Distinct variable groups:    g, K    T, g    g, W    Q, g
Allowed substitution hints:    A( g)    P( g)    E( g)    H( g)    I( g)    .<_ ( g)    V( g)    Y( g)

Proof of Theorem dicelvalN
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 37004 . . 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 2527 . 2  |-  ( ( ( 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.  E ) } ) )
10 vex 3112 . . . . . 6  |-  f  e. 
_V
11 vex 3112 . . . . . 6  |-  s  e. 
_V
1210, 11op1std 6809 . . . . 5  |-  ( Y  =  <. f ,  s
>.  ->  ( 1st `  Y
)  =  f )
1310, 11op2ndd 6810 . . . . . 6  |-  ( Y  =  <. f ,  s
>.  ->  ( 2nd `  Y
)  =  s )
1413fveq1d 5874 . . . . 5  |-  ( Y  =  <. f ,  s
>.  ->  ( ( 2nd `  Y ) `  ( iota_ g  e.  T  ( g `  P )  =  Q ) )  =  ( s `  ( iota_ g  e.  T  ( g `  P
)  =  Q ) ) )
1512, 14eqeq12d 2479 . . . 4  |-  ( Y  =  <. f ,  s
>.  ->  ( ( 1st `  Y )  =  ( ( 2nd `  Y
) `  ( iota_ g  e.  T  ( g `  P )  =  Q ) )  <->  f  =  ( s `  ( iota_ g  e.  T  ( g `  P )  =  Q ) ) ) )
1613eleq1d 2526 . . . 4  |-  ( Y  =  <. f ,  s
>.  ->  ( ( 2nd `  Y )  e.  E  <->  s  e.  E ) )
1715, 16anbi12d 710 . . 3  |-  ( Y  =  <. f ,  s
>.  ->  ( ( ( 1st `  Y )  =  ( ( 2nd `  Y ) `  ( iota_ g  e.  T  ( g `  P )  =  Q ) )  /\  ( 2nd `  Y
)  e.  E )  <-> 
( f  =  ( s `  ( iota_ g  e.  T  ( g `
 P )  =  Q ) )  /\  s  e.  E )
) )
1817elopaba 5124 . 2  |-  ( Y  e.  { <. f ,  s >.  |  ( f  =  ( s `
 ( iota_ g  e.  T  ( g `  P )  =  Q ) )  /\  s  e.  E ) }  <->  ( Y  e.  ( _V  X.  _V )  /\  ( ( 1st `  Y )  =  ( ( 2nd `  Y
) `  ( iota_ g  e.  T  ( g `  P )  =  Q ) )  /\  ( 2nd `  Y )  e.  E ) ) )
199, 18syl6bb 261 1  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( Y  e.  ( I `  Q )  <-> 
( Y  e.  ( _V  X.  _V )  /\  ( ( 1st `  Y
)  =  ( ( 2nd `  Y ) `
 ( iota_ g  e.  T  ( g `  P )  =  Q ) )  /\  ( 2nd `  Y )  e.  E ) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1395    e. wcel 1819   _Vcvv 3109   <.cop 4038   class class class wbr 4456   {copab 4514    X. cxp 5006   ` cfv 5594   iota_crio 6257   1stc1st 6797   2ndc2nd 6798   lecple 14718   occoc 14719   Atomscatm 35089   LHypclh 35809   LTrncltrn 35926   TEndoctendo 36579   DIsoCcdic 37000
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-1st 6799  df-2nd 6800  df-dic 37001
This theorem is referenced by:  dicelval2N  37010
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