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Theorem dicfnN 31995
Description: Functionality and domain of the partial isomorphism C. (Contributed by NM, 8-Mar-2014.) (New usage is discouraged.)
Hypotheses
Ref Expression
dicfn.l  |-  .<_  =  ( le `  K )
dicfn.a  |-  A  =  ( Atoms `  K )
dicfn.h  |-  H  =  ( LHyp `  K
)
dicfn.i  |-  I  =  ( ( DIsoC `  K
) `  W )
Assertion
Ref Expression
dicfnN  |-  ( ( K  e.  V  /\  W  e.  H )  ->  I  Fn  { p  e.  A  |  -.  p  .<_  W } )
Distinct variable groups:    .<_ , p    A, p    K, p    W, p
Allowed substitution hints:    H( p)    I( p)    V( p)

Proof of Theorem dicfnN
Dummy variables  q 
f  s  u are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 breq1 4042 . . . . . . 7  |-  ( p  =  q  ->  (
p  .<_  W  <->  q  .<_  W ) )
21notbid 285 . . . . . 6  |-  ( p  =  q  ->  ( -.  p  .<_  W  <->  -.  q  .<_  W ) )
32elrab 2936 . . . . 5  |-  ( q  e.  { p  e.  A  |  -.  p  .<_  W }  <->  ( q  e.  A  /\  -.  q  .<_  W ) )
4 dicfn.l . . . . . . 7  |-  .<_  =  ( le `  K )
5 dicfn.a . . . . . . 7  |-  A  =  ( Atoms `  K )
6 dicfn.h . . . . . . 7  |-  H  =  ( LHyp `  K
)
7 eqid 2296 . . . . . . 7  |-  ( ( oc `  K ) `
 W )  =  ( ( oc `  K ) `  W
)
8 eqid 2296 . . . . . . 7  |-  ( (
LTrn `  K ) `  W )  =  ( ( LTrn `  K
) `  W )
9 eqid 2296 . . . . . . 7  |-  ( (
TEndo `  K ) `  W )  =  ( ( TEndo `  K ) `  W )
10 dicfn.i . . . . . . 7  |-  I  =  ( ( DIsoC `  K
) `  W )
114, 5, 6, 7, 8, 9, 10dicval 31988 . . . . . 6  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( q  e.  A  /\  -.  q  .<_  W ) )  -> 
( I `  q
)  =  { <. f ,  s >.  |  ( f  =  ( s `
 ( iota_ u  e.  ( ( LTrn `  K
) `  W )
( u `  (
( oc `  K
) `  W )
)  =  q ) )  /\  s  e.  ( ( TEndo `  K
) `  W )
) } )
12 fvex 5555 . . . . . 6  |-  ( I `
 q )  e. 
_V
1311, 12syl6eqelr 2385 . . . . 5  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( q  e.  A  /\  -.  q  .<_  W ) )  ->  { <. f ,  s
>.  |  ( f  =  ( s `  ( iota_ u  e.  ( ( LTrn `  K
) `  W )
( u `  (
( oc `  K
) `  W )
)  =  q ) )  /\  s  e.  ( ( TEndo `  K
) `  W )
) }  e.  _V )
143, 13sylan2b 461 . . . 4  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  q  e.  { p  e.  A  |  -.  p  .<_  W }
)  ->  { <. f ,  s >.  |  ( f  =  ( s `
 ( iota_ u  e.  ( ( LTrn `  K
) `  W )
( u `  (
( oc `  K
) `  W )
)  =  q ) )  /\  s  e.  ( ( TEndo `  K
) `  W )
) }  e.  _V )
1514ralrimiva 2639 . . 3  |-  ( ( K  e.  V  /\  W  e.  H )  ->  A. q  e.  {
p  e.  A  |  -.  p  .<_  W }  { <. f ,  s
>.  |  ( f  =  ( s `  ( iota_ u  e.  ( ( LTrn `  K
) `  W )
( u `  (
( oc `  K
) `  W )
)  =  q ) )  /\  s  e.  ( ( TEndo `  K
) `  W )
) }  e.  _V )
16 eqid 2296 . . . 4  |-  ( q  e.  { p  e.  A  |  -.  p  .<_  W }  |->  { <. f ,  s >.  |  ( f  =  ( s `
 ( iota_ u  e.  ( ( LTrn `  K
) `  W )
( u `  (
( oc `  K
) `  W )
)  =  q ) )  /\  s  e.  ( ( TEndo `  K
) `  W )
) } )  =  ( q  e.  {
p  e.  A  |  -.  p  .<_  W }  |->  { <. f ,  s
>.  |  ( f  =  ( s `  ( iota_ u  e.  ( ( LTrn `  K
) `  W )
( u `  (
( oc `  K
) `  W )
)  =  q ) )  /\  s  e.  ( ( TEndo `  K
) `  W )
) } )
1716fnmpt 5386 . . 3  |-  ( A. q  e.  { p  e.  A  |  -.  p  .<_  W }  { <. f ,  s >.  |  ( f  =  ( s `  ( iota_ u  e.  ( (
LTrn `  K ) `  W ) ( u `
 ( ( oc
`  K ) `  W ) )  =  q ) )  /\  s  e.  ( ( TEndo `  K ) `  W ) ) }  e.  _V  ->  (
q  e.  { p  e.  A  |  -.  p  .<_  W }  |->  {
<. f ,  s >.  |  ( f  =  ( s `  ( iota_ u  e.  ( (
LTrn `  K ) `  W ) ( u `
 ( ( oc
`  K ) `  W ) )  =  q ) )  /\  s  e.  ( ( TEndo `  K ) `  W ) ) } )  Fn  { p  e.  A  |  -.  p  .<_  W } )
1815, 17syl 15 . 2  |-  ( ( K  e.  V  /\  W  e.  H )  ->  ( q  e.  {
p  e.  A  |  -.  p  .<_  W }  |->  { <. f ,  s
>.  |  ( f  =  ( s `  ( iota_ u  e.  ( ( LTrn `  K
) `  W )
( u `  (
( oc `  K
) `  W )
)  =  q ) )  /\  s  e.  ( ( TEndo `  K
) `  W )
) } )  Fn 
{ p  e.  A  |  -.  p  .<_  W }
)
194, 5, 6, 7, 8, 9, 10dicfval 31987 . . 3  |-  ( ( K  e.  V  /\  W  e.  H )  ->  I  =  ( q  e.  { p  e.  A  |  -.  p  .<_  W }  |->  { <. f ,  s >.  |  ( f  =  ( s `
 ( iota_ u  e.  ( ( LTrn `  K
) `  W )
( u `  (
( oc `  K
) `  W )
)  =  q ) )  /\  s  e.  ( ( TEndo `  K
) `  W )
) } ) )
2019fneq1d 5351 . 2  |-  ( ( K  e.  V  /\  W  e.  H )  ->  ( I  Fn  {
p  e.  A  |  -.  p  .<_  W }  <->  ( q  e.  { p  e.  A  |  -.  p  .<_  W }  |->  {
<. f ,  s >.  |  ( f  =  ( s `  ( iota_ u  e.  ( (
LTrn `  K ) `  W ) ( u `
 ( ( oc
`  K ) `  W ) )  =  q ) )  /\  s  e.  ( ( TEndo `  K ) `  W ) ) } )  Fn  { p  e.  A  |  -.  p  .<_  W } ) )
2118, 20mpbird 223 1  |-  ( ( K  e.  V  /\  W  e.  H )  ->  I  Fn  { p  e.  A  |  -.  p  .<_  W } )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   {crab 2560   _Vcvv 2801   class class class wbr 4039   {copab 4092    e. cmpt 4093    Fn wfn 5266   ` cfv 5271   iota_crio 6313   lecple 13231   occoc 13232   Atomscatm 30075   LHypclh 30795   LTrncltrn 30912   TEndoctendo 31563   DIsoCcdic 31984
This theorem is referenced by:  dicdmN  31996
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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