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Theorem dicelvalN 35181
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 35179 . . 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 2524 . 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 3081 . . . . . 6  |-  f  e. 
_V
11 vex 3081 . . . . . 6  |-  s  e. 
_V
1210, 11op1std 6700 . . . . 5  |-  ( Y  =  <. f ,  s
>.  ->  ( 1st `  Y
)  =  f )
1310, 11op2ndd 6701 . . . . . 6  |-  ( Y  =  <. f ,  s
>.  ->  ( 2nd `  Y
)  =  s )
1413fveq1d 5804 . . . . 5  |-  ( Y  =  <. f ,  s
>.  ->  ( ( 2nd `  Y ) `  ( iota_ g  e.  T  ( g `  P )  =  Q ) )  =  ( s `  ( iota_ g  e.  T  ( g `  P
)  =  Q ) ) )
1512, 14eqeq12d 2476 . . . 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 2523 . . . 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 5063 . 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 1370    e. wcel 1758   _Vcvv 3078   <.cop 3994   class class class wbr 4403   {copab 4460    X. cxp 4949   ` cfv 5529   iota_crio 6163   1stc1st 6688   2ndc2nd 6689   lecple 14367   occoc 14368   Atomscatm 33266   LHypclh 33986   LTrncltrn 34103   TEndoctendo 34754   DIsoCcdic 35175
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 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485
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 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-1st 6690  df-2nd 6691  df-dic 35176
This theorem is referenced by:  dicelval2N  35185
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