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Theorem dicelval3 31979
Description: Member of the partial isomorphism C. (Contributed by NM, 26-Feb-2014.)
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 )
dicval2.g  |-  G  =  ( iota_ g  e.  T
( g `  P
)  =  Q )
Assertion
Ref Expression
dicelval3  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( Y  e.  ( I `  Q )  <->  E. s  e.  E  Y  =  <. ( s `
 G ) ,  s >. ) )
Distinct variable groups:    g, s, K    T, g    g, W, s    E, s    Q, g, s    Y, s
Allowed substitution hints:    A( g, s)    P( g, s)    T( s)    E( g)    G( g, s)    H( g, s)    I( g, s)    .<_ ( g, s)    V( g, s)    Y( g)

Proof of Theorem dicelval3
Dummy variable  f is 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 )
8 dicval2.g . . . 4  |-  G  =  ( iota_ g  e.  T
( g `  P
)  =  Q )
91, 2, 3, 4, 5, 6, 7, 8dicval2 31978 . . 3  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( I `  Q
)  =  { <. f ,  s >.  |  ( f  =  ( s `
 G )  /\  s  e.  E ) } )
109eleq2d 2504 . 2  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( Y  e.  ( I `  Q )  <-> 
Y  e.  { <. f ,  s >.  |  ( f  =  ( s `
 G )  /\  s  e.  E ) } ) )
11 excom 1757 . . . 4  |-  ( E. f E. s ( Y  =  <. f ,  s >.  /\  (
f  =  ( s `
 G )  /\  s  e.  E )
)  <->  E. s E. f
( Y  =  <. f ,  s >.  /\  (
f  =  ( s `
 G )  /\  s  e.  E )
) )
12 an12 774 . . . . . . 7  |-  ( ( Y  =  <. f ,  s >.  /\  (
f  =  ( s `
 G )  /\  s  e.  E )
)  <->  ( f  =  ( s `  G
)  /\  ( Y  =  <. f ,  s
>.  /\  s  e.  E
) ) )
1312exbii 1593 . . . . . 6  |-  ( E. f ( Y  = 
<. f ,  s >.  /\  ( f  =  ( s `  G )  /\  s  e.  E
) )  <->  E. f
( f  =  ( s `  G )  /\  ( Y  = 
<. f ,  s >.  /\  s  e.  E
) ) )
14 fvex 5743 . . . . . . 7  |-  ( s `
 G )  e. 
_V
15 opeq1 3985 . . . . . . . . 9  |-  ( f  =  ( s `  G )  ->  <. f ,  s >.  =  <. ( s `  G ) ,  s >. )
1615eqeq2d 2448 . . . . . . . 8  |-  ( f  =  ( s `  G )  ->  ( Y  =  <. f ,  s >.  <->  Y  =  <. ( s `  G ) ,  s >. )
)
1716anbi1d 687 . . . . . . 7  |-  ( f  =  ( s `  G )  ->  (
( Y  =  <. f ,  s >.  /\  s  e.  E )  <->  ( Y  =  <. ( s `  G ) ,  s
>.  /\  s  e.  E
) ) )
1814, 17ceqsexv 2992 . . . . . 6  |-  ( E. f ( f  =  ( s `  G
)  /\  ( Y  =  <. f ,  s
>.  /\  s  e.  E
) )  <->  ( Y  =  <. ( s `  G ) ,  s
>.  /\  s  e.  E
) )
19 ancom 439 . . . . . 6  |-  ( ( Y  =  <. (
s `  G ) ,  s >.  /\  s  e.  E )  <->  ( s  e.  E  /\  Y  = 
<. ( s `  G
) ,  s >.
) )
2013, 18, 193bitri 264 . . . . 5  |-  ( E. f ( Y  = 
<. f ,  s >.  /\  ( f  =  ( s `  G )  /\  s  e.  E
) )  <->  ( s  e.  E  /\  Y  = 
<. ( s `  G
) ,  s >.
) )
2120exbii 1593 . . . 4  |-  ( E. s E. f ( Y  =  <. f ,  s >.  /\  (
f  =  ( s `
 G )  /\  s  e.  E )
)  <->  E. s ( s  e.  E  /\  Y  =  <. ( s `  G ) ,  s
>. ) )
2211, 21bitri 242 . . 3  |-  ( E. f E. s ( Y  =  <. f ,  s >.  /\  (
f  =  ( s `
 G )  /\  s  e.  E )
)  <->  E. s ( s  e.  E  /\  Y  =  <. ( s `  G ) ,  s
>. ) )
23 elopab 4463 . . 3  |-  ( Y  e.  { <. f ,  s >.  |  ( f  =  ( s `
 G )  /\  s  e.  E ) } 
<->  E. f E. s
( Y  =  <. f ,  s >.  /\  (
f  =  ( s `
 G )  /\  s  e.  E )
) )
24 df-rex 2712 . . 3  |-  ( E. s  e.  E  Y  =  <. ( s `  G ) ,  s
>. 
<->  E. s ( s  e.  E  /\  Y  =  <. ( s `  G ) ,  s
>. ) )
2522, 23, 243bitr4i 270 . 2  |-  ( Y  e.  { <. f ,  s >.  |  ( f  =  ( s `
 G )  /\  s  e.  E ) } 
<->  E. s  e.  E  Y  =  <. ( s `
 G ) ,  s >. )
2610, 25syl6bb 254 1  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( Y  e.  ( I `  Q )  <->  E. s  e.  E  Y  =  <. ( s `
 G ) ,  s >. ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    /\ wa 360   E.wex 1551    = wceq 1653    e. wcel 1726   E.wrex 2707   <.cop 3818   class class class wbr 4213   {copab 4266   ` cfv 5455   iota_crio 6543   lecple 13537   occoc 13538   Atomscatm 30062   LHypclh 30782   LTrncltrn 30899   TEndoctendo 31550   DIsoCcdic 31971
This theorem is referenced by:  cdlemn11pre  32009  dihord2pre  32024
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-rep 4321  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-reu 2713  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-riota 6550  df-dic 31972
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