Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dicfnN Unicode version

Theorem dicfnN 31300
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 4158 . . . . . . 7  |-  ( p  =  q  ->  (
p  .<_  W  <->  q  .<_  W ) )
21notbid 286 . . . . . 6  |-  ( p  =  q  ->  ( -.  p  .<_  W  <->  -.  q  .<_  W ) )
32elrab 3037 . . . . 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 2389 . . . . . . 7  |-  ( ( oc `  K ) `
 W )  =  ( ( oc `  K ) `  W
)
8 eqid 2389 . . . . . . 7  |-  ( (
LTrn `  K ) `  W )  =  ( ( LTrn `  K
) `  W )
9 eqid 2389 . . . . . . 7  |-  ( (
TEndo `  K ) `  W )  =  ( ( TEndo `  K ) `  W )
10 dicfn.i . . . . . . 7  |-  I  =  ( ( DIsoC `  K
) `  W )
114, 5, 6, 7, 8, 9, 10dicval 31293 . . . . . 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 5684 . . . . . 6  |-  ( I `
 q )  e. 
_V
1311, 12syl6eqelr 2478 . . . . 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 462 . . . 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 2734 . . 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 2389 . . . 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 5513 . . 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 16 . 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 31292 . . 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 5478 . 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 224 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 359    = wceq 1649    e. wcel 1717   A.wral 2651   {crab 2655   _Vcvv 2901   class class class wbr 4155   {copab 4208    e. cmpt 4209    Fn wfn 5391   ` cfv 5396   iota_crio 6480   lecple 13465   occoc 13466   Atomscatm 29380   LHypclh 30100   LTrncltrn 30217   TEndoctendo 30868   DIsoCcdic 31289
This theorem is referenced by:  dicdmN  31301
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-reu 2658  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-riota 6487  df-dic 31290
  Copyright terms: Public domain W3C validator