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Theorem dicfnN 31373
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 4026 . . . . . . 7  |-  ( p  =  q  ->  (
p  .<_  W  <->  q  .<_  W ) )
21notbid 285 . . . . . 6  |-  ( p  =  q  ->  ( -.  p  .<_  W  <->  -.  q  .<_  W ) )
32elrab 2923 . . . . 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 2283 . . . . . . 7  |-  ( ( oc `  K ) `
 W )  =  ( ( oc `  K ) `  W
)
8 eqid 2283 . . . . . . 7  |-  ( (
LTrn `  K ) `  W )  =  ( ( LTrn `  K
) `  W )
9 eqid 2283 . . . . . . 7  |-  ( (
TEndo `  K ) `  W )  =  ( ( TEndo `  K ) `  W )
10 dicfn.i . . . . . . 7  |-  I  =  ( ( DIsoC `  K
) `  W )
114, 5, 6, 7, 8, 9, 10dicval 31366 . . . . . 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 5539 . . . . . 6  |-  ( I `
 q )  e. 
_V
1311, 12syl6eqelr 2372 . . . . 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 2626 . . 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 2283 . . . 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 5370 . . 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 31365 . . 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 5335 . 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 1623    e. wcel 1684   A.wral 2543   {crab 2547   _Vcvv 2788   class class class wbr 4023   {copab 4076    e. cmpt 4077    Fn wfn 5250   ` cfv 5255   iota_crio 6297   lecple 13215   occoc 13216   Atomscatm 29453   LHypclh 30173   LTrncltrn 30290   TEndoctendo 30941   DIsoCcdic 31362
This theorem is referenced by:  dicdmN  31374
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-riota 6304  df-dic 31363
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