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Theorem dicelval3 36270
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 36269 . . 3  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( I `  Q
)  =  { <. f ,  s >.  |  ( f  =  ( s `
 G )  /\  s  e.  E ) } )
109eleq2d 2537 . 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 1798 . . . 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 795 . . . . . . 7  |-  ( ( Y  =  <. f ,  s >.  /\  (
f  =  ( s `
 G )  /\  s  e.  E )
)  <->  ( f  =  ( s `  G
)  /\  ( Y  =  <. f ,  s
>.  /\  s  e.  E
) ) )
1312exbii 1644 . . . . . 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 5881 . . . . . . 7  |-  ( s `
 G )  e. 
_V
15 opeq1 4218 . . . . . . . . 9  |-  ( f  =  ( s `  G )  ->  <. f ,  s >.  =  <. ( s `  G ) ,  s >. )
1615eqeq2d 2481 . . . . . . . 8  |-  ( f  =  ( s `  G )  ->  ( Y  =  <. f ,  s >.  <->  Y  =  <. ( s `  G ) ,  s >. )
)
1716anbi1d 704 . . . . . . 7  |-  ( f  =  ( s `  G )  ->  (
( Y  =  <. f ,  s >.  /\  s  e.  E )  <->  ( Y  =  <. ( s `  G ) ,  s
>.  /\  s  e.  E
) ) )
1814, 17ceqsexv 3155 . . . . . 6  |-  ( E. f ( f  =  ( s `  G
)  /\  ( Y  =  <. f ,  s
>.  /\  s  e.  E
) )  <->  ( Y  =  <. ( s `  G ) ,  s
>.  /\  s  e.  E
) )
19 ancom 450 . . . . . 6  |-  ( ( Y  =  <. (
s `  G ) ,  s >.  /\  s  e.  E )  <->  ( s  e.  E  /\  Y  = 
<. ( s `  G
) ,  s >.
) )
2013, 18, 193bitri 271 . . . . 5  |-  ( E. f ( Y  = 
<. f ,  s >.  /\  ( f  =  ( s `  G )  /\  s  e.  E
) )  <->  ( s  e.  E  /\  Y  = 
<. ( s `  G
) ,  s >.
) )
2120exbii 1644 . . . 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 249 . . 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 4760 . . 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 2823 . . 3  |-  ( E. s  e.  E  Y  =  <. ( s `  G ) ,  s
>. 
<->  E. s ( s  e.  E  /\  Y  =  <. ( s `  G ) ,  s
>. ) )
2522, 23, 243bitr4i 277 . 2  |-  ( Y  e.  { <. f ,  s >.  |  ( f  =  ( s `
 G )  /\  s  e.  E ) } 
<->  E. s  e.  E  Y  =  <. ( s `
 G ) ,  s >. )
2610, 25syl6bb 261 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 setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379   E.wex 1596    e. wcel 1767   E.wrex 2818   <.cop 4038   class class class wbr 4452   {copab 4509   ` cfv 5593   iota_crio 6254   lecple 14574   occoc 14575   Atomscatm 34353   LHypclh 35073   LTrncltrn 35190   TEndoctendo 35841   DIsoCcdic 36262
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4563  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691  ax-un 6586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4251  df-iun 4332  df-br 4453  df-opab 4511  df-mpt 4512  df-id 4800  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-f1 5598  df-fo 5599  df-f1o 5600  df-fv 5601  df-riota 6255  df-dic 36263
This theorem is referenced by:  cdlemn11pre  36300  dihord2pre  36315
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