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Theorem dicvalrelN 34824
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 4965 . . . 4  |-  Rel  { <. f ,  s >.  |  ( f  =  ( s `  ( iota_ g  e.  ( (
LTrn `  K ) `  W ) ( g `
 ( ( oc
`  K ) `  W ) )  =  X ) )  /\  s  e.  ( ( TEndo `  K ) `  W ) ) }
2 eqid 2471 . . . . . . . . . 10  |-  ( le
`  K )  =  ( le `  K
)
3 eqid 2471 . . . . . . . . . 10  |-  ( Atoms `  K )  =  (
Atoms `  K )
4 dicvalrel.h . . . . . . . . . 10  |-  H  =  ( LHyp `  K
)
5 dicvalrel.i . . . . . . . . . 10  |-  I  =  ( ( DIsoC `  K
) `  W )
62, 3, 4, 5dicdmN 34823 . . . . . . . . 9  |-  ( ( K  e.  V  /\  W  e.  H )  ->  dom  I  =  {
p  e.  ( Atoms `  K )  |  -.  p ( le `  K ) W }
)
76eleq2d 2534 . . . . . . . 8  |-  ( ( K  e.  V  /\  W  e.  H )  ->  ( X  e.  dom  I 
<->  X  e.  { p  e.  ( Atoms `  K )  |  -.  p ( le
`  K ) W } ) )
8 breq1 4398 . . . . . . . . . 10  |-  ( p  =  X  ->  (
p ( le `  K ) W  <->  X ( le `  K ) W ) )
98notbid 301 . . . . . . . . 9  |-  ( p  =  X  ->  ( -.  p ( le `  K ) W  <->  -.  X
( le `  K
) W ) )
109elrab 3184 . . . . . . . 8  |-  ( X  e.  { p  e.  ( Atoms `  K )  |  -.  p ( le
`  K ) W }  <->  ( X  e.  ( Atoms `  K )  /\  -.  X ( le
`  K ) W ) )
117, 10syl6bb 269 . . . . . . 7  |-  ( ( K  e.  V  /\  W  e.  H )  ->  ( X  e.  dom  I 
<->  ( X  e.  (
Atoms `  K )  /\  -.  X ( le `  K ) W ) ) )
1211biimpa 492 . . . . . 6  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  X  e.  dom  I )  ->  ( X  e.  ( Atoms `  K )  /\  -.  X ( le `  K ) W ) )
13 eqid 2471 . . . . . . 7  |-  ( ( oc `  K ) `
 W )  =  ( ( oc `  K ) `  W
)
14 eqid 2471 . . . . . . 7  |-  ( (
LTrn `  K ) `  W )  =  ( ( LTrn `  K
) `  W )
15 eqid 2471 . . . . . . 7  |-  ( (
TEndo `  K ) `  W )  =  ( ( TEndo `  K ) `  W )
162, 3, 4, 13, 14, 15, 5dicval 34815 . . . . . 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 478 . . . . 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 4924 . . . 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 241 . . 3  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  X  e.  dom  I )  ->  Rel  ( I `  X
) )
2019ex 441 . 2  |-  ( ( K  e.  V  /\  W  e.  H )  ->  ( X  e.  dom  I  ->  Rel  ( I `  X ) ) )
21 rel0 4963 . . 3  |-  Rel  (/)
22 ndmfv 5903 . . . 4  |-  ( -.  X  e.  dom  I  ->  ( I `  X
)  =  (/) )
2322releqd 4924 . . 3  |-  ( -.  X  e.  dom  I  ->  ( Rel  ( I `
 X )  <->  Rel  (/) ) )
2421, 23mpbiri 241 . 2  |-  ( -.  X  e.  dom  I  ->  Rel  ( I `  X ) )
2520, 24pm2.61d1 164 1  |-  ( ( K  e.  V  /\  W  e.  H )  ->  Rel  ( I `  X ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 376    = wceq 1452    e. wcel 1904   {crab 2760   (/)c0 3722   class class class wbr 4395   {copab 4453   dom cdm 4839   Rel wrel 4844   ` cfv 5589   iota_crio 6269   lecple 15275   occoc 15276   Atomscatm 32900   LHypclh 33620   LTrncltrn 33737   TEndoctendo 34390   DIsoCcdic 34811
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-dic 34812
This theorem is referenced by: (None)
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