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Theorem dicelval3 31992
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 31991 . . 3  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( Q  e.  A  /\  -.  Q  .<_  W ) )  -> 
( I `  Q
)  =  { <. f ,  s >.  |  ( f  =  ( s `
 G )  /\  s  e.  E ) } )
109eleq2d 2363 . 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 772 . . . . . . 7  |-  ( ( Y  =  <. f ,  s >.  /\  (
f  =  ( s `
 G )  /\  s  e.  E )
)  <->  ( f  =  ( s `  G
)  /\  ( Y  =  <. f ,  s
>.  /\  s  e.  E
) ) )
1312exbii 1572 . . . . . 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 5555 . . . . . . 7  |-  ( s `
 G )  e. 
_V
15 opeq1 3812 . . . . . . . . 9  |-  ( f  =  ( s `  G )  ->  <. f ,  s >.  =  <. ( s `  G ) ,  s >. )
1615eqeq2d 2307 . . . . . . . 8  |-  ( f  =  ( s `  G )  ->  ( Y  =  <. f ,  s >.  <->  Y  =  <. ( s `  G ) ,  s >. )
)
1716anbi1d 685 . . . . . . 7  |-  ( f  =  ( s `  G )  ->  (
( Y  =  <. f ,  s >.  /\  s  e.  E )  <->  ( Y  =  <. ( s `  G ) ,  s
>.  /\  s  e.  E
) ) )
1814, 17ceqsexv 2836 . . . . . 6  |-  ( E. f ( f  =  ( s `  G
)  /\  ( Y  =  <. f ,  s
>.  /\  s  e.  E
) )  <->  ( Y  =  <. ( s `  G ) ,  s
>.  /\  s  e.  E
) )
19 ancom 437 . . . . . 6  |-  ( ( Y  =  <. (
s `  G ) ,  s >.  /\  s  e.  E )  <->  ( s  e.  E  /\  Y  = 
<. ( s `  G
) ,  s >.
) )
2013, 18, 193bitri 262 . . . . 5  |-  ( E. f ( Y  = 
<. f ,  s >.  /\  ( f  =  ( s `  G )  /\  s  e.  E
) )  <->  ( s  e.  E  /\  Y  = 
<. ( s `  G
) ,  s >.
) )
2120exbii 1572 . . . 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 240 . . 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 4288 . . 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 2562 . . 3  |-  ( E. s  e.  E  Y  =  <. ( s `  G ) ,  s
>. 
<->  E. s ( s  e.  E  /\  Y  =  <. ( s `  G ) ,  s
>. ) )
2522, 23, 243bitr4i 268 . 2  |-  ( Y  e.  { <. f ,  s >.  |  ( f  =  ( s `
 G )  /\  s  e.  E ) } 
<->  E. s  e.  E  Y  =  <. ( s `
 G ) ,  s >. )
2610, 25syl6bb 252 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 176    /\ wa 358   E.wex 1531    = wceq 1632    e. wcel 1696   E.wrex 2557   <.cop 3656   class class class wbr 4039   {copab 4092   ` cfv 5271   iota_crio 6313   lecple 13231   occoc 13232   Atomscatm 30075   LHypclh 30795   LTrncltrn 30912   TEndoctendo 31563   DIsoCcdic 31984
This theorem is referenced by:  cdlemn11pre  32022  dihord2pre  32037
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-riota 6320  df-dic 31985
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