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

Theorem dicvalrelN 31983
Description: The value of partial isomorphism C is a relation. (Contributed by NM, 8-Mar-2014.) (New usage is discouraged.)
Hypotheses
Ref Expression
dicvalrel.h  |-  H  =  ( LHyp `  K
)
dicvalrel.i  |-  I  =  ( ( DIsoC `  K
) `  W )
Assertion
Ref Expression
dicvalrelN  |-  ( ( K  e.  V  /\  W  e.  H )  ->  Rel  ( I `  X ) )

Proof of Theorem dicvalrelN
Dummy variables  f 
g  p  s are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relopab 5001 . . . 4  |-  Rel  { <. f ,  s >.  |  ( f  =  ( s `  ( iota_ g  e.  ( (
LTrn `  K ) `  W ) ( g `
 ( ( oc
`  K ) `  W ) )  =  X ) )  /\  s  e.  ( ( TEndo `  K ) `  W ) ) }
2 eqid 2436 . . . . . . . . . 10  |-  ( le
`  K )  =  ( le `  K
)
3 eqid 2436 . . . . . . . . . 10  |-  ( Atoms `  K )  =  (
Atoms `  K )
4 dicvalrel.h . . . . . . . . . 10  |-  H  =  ( LHyp `  K
)
5 dicvalrel.i . . . . . . . . . 10  |-  I  =  ( ( DIsoC `  K
) `  W )
62, 3, 4, 5dicdmN 31982 . . . . . . . . 9  |-  ( ( K  e.  V  /\  W  e.  H )  ->  dom  I  =  {
p  e.  ( Atoms `  K )  |  -.  p ( le `  K ) W }
)
76eleq2d 2503 . . . . . . . 8  |-  ( ( K  e.  V  /\  W  e.  H )  ->  ( X  e.  dom  I 
<->  X  e.  { p  e.  ( Atoms `  K )  |  -.  p ( le
`  K ) W } ) )
8 breq1 4215 . . . . . . . . . 10  |-  ( p  =  X  ->  (
p ( le `  K ) W  <->  X ( le `  K ) W ) )
98notbid 286 . . . . . . . . 9  |-  ( p  =  X  ->  ( -.  p ( le `  K ) W  <->  -.  X
( le `  K
) W ) )
109elrab 3092 . . . . . . . 8  |-  ( X  e.  { p  e.  ( Atoms `  K )  |  -.  p ( le
`  K ) W }  <->  ( X  e.  ( Atoms `  K )  /\  -.  X ( le
`  K ) W ) )
117, 10syl6bb 253 . . . . . . 7  |-  ( ( K  e.  V  /\  W  e.  H )  ->  ( X  e.  dom  I 
<->  ( X  e.  (
Atoms `  K )  /\  -.  X ( le `  K ) W ) ) )
1211biimpa 471 . . . . . 6  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  X  e.  dom  I )  ->  ( X  e.  ( Atoms `  K )  /\  -.  X ( le `  K ) W ) )
13 eqid 2436 . . . . . . 7  |-  ( ( oc `  K ) `
 W )  =  ( ( oc `  K ) `  W
)
14 eqid 2436 . . . . . . 7  |-  ( (
LTrn `  K ) `  W )  =  ( ( LTrn `  K
) `  W )
15 eqid 2436 . . . . . . 7  |-  ( (
TEndo `  K ) `  W )  =  ( ( TEndo `  K ) `  W )
162, 3, 4, 13, 14, 15, 5dicval 31974 . . . . . 6  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( X  e.  ( Atoms `  K )  /\  -.  X ( le
`  K ) W ) )  ->  (
I `  X )  =  { <. f ,  s
>.  |  ( f  =  ( s `  ( iota_ g  e.  ( ( LTrn `  K
) `  W )
( g `  (
( oc `  K
) `  W )
)  =  X ) )  /\  s  e.  ( ( TEndo `  K
) `  W )
) } )
1712, 16syldan 457 . . . . 5  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  X  e.  dom  I )  ->  (
I `  X )  =  { <. f ,  s
>.  |  ( f  =  ( s `  ( iota_ g  e.  ( ( LTrn `  K
) `  W )
( g `  (
( oc `  K
) `  W )
)  =  X ) )  /\  s  e.  ( ( TEndo `  K
) `  W )
) } )
1817releqd 4961 . . . 4  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  X  e.  dom  I )  ->  ( Rel  ( I `  X
)  <->  Rel  { <. f ,  s >.  |  ( f  =  ( s `
 ( iota_ g  e.  ( ( LTrn `  K
) `  W )
( g `  (
( oc `  K
) `  W )
)  =  X ) )  /\  s  e.  ( ( TEndo `  K
) `  W )
) } ) )
191, 18mpbiri 225 . . 3  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  X  e.  dom  I )  ->  Rel  ( I `  X
) )
2019ex 424 . 2  |-  ( ( K  e.  V  /\  W  e.  H )  ->  ( X  e.  dom  I  ->  Rel  ( I `  X ) ) )
21 rel0 4999 . . 3  |-  Rel  (/)
22 ndmfv 5755 . . . 4  |-  ( -.  X  e.  dom  I  ->  ( I `  X
)  =  (/) )
2322releqd 4961 . . 3  |-  ( -.  X  e.  dom  I  ->  ( Rel  ( I `
 X )  <->  Rel  (/) ) )
2421, 23mpbiri 225 . 2  |-  ( -.  X  e.  dom  I  ->  Rel  ( I `  X ) )
2520, 24pm2.61d1 153 1  |-  ( ( K  e.  V  /\  W  e.  H )  ->  Rel  ( I `  X ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   {crab 2709   (/)c0 3628   class class class wbr 4212   {copab 4265   dom cdm 4878   Rel wrel 4883   ` cfv 5454   iota_crio 6542   lecple 13536   occoc 13537   Atomscatm 30061   LHypclh 30781   LTrncltrn 30898   TEndoctendo 31549   DIsoCcdic 31970
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-riota 6549  df-dic 31971
  Copyright terms: Public domain W3C validator