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Theorem dicvalrelN 36000
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 5129 . . . 4  |-  Rel  { <. f ,  s >.  |  ( f  =  ( s `  ( iota_ g  e.  ( (
LTrn `  K ) `  W ) ( g `
 ( ( oc
`  K ) `  W ) )  =  X ) )  /\  s  e.  ( ( TEndo `  K ) `  W ) ) }
2 eqid 2467 . . . . . . . . . 10  |-  ( le
`  K )  =  ( le `  K
)
3 eqid 2467 . . . . . . . . . 10  |-  ( Atoms `  K )  =  (
Atoms `  K )
4 dicvalrel.h . . . . . . . . . 10  |-  H  =  ( LHyp `  K
)
5 dicvalrel.i . . . . . . . . . 10  |-  I  =  ( ( DIsoC `  K
) `  W )
62, 3, 4, 5dicdmN 35999 . . . . . . . . 9  |-  ( ( K  e.  V  /\  W  e.  H )  ->  dom  I  =  {
p  e.  ( Atoms `  K )  |  -.  p ( le `  K ) W }
)
76eleq2d 2537 . . . . . . . 8  |-  ( ( K  e.  V  /\  W  e.  H )  ->  ( X  e.  dom  I 
<->  X  e.  { p  e.  ( Atoms `  K )  |  -.  p ( le
`  K ) W } ) )
8 breq1 4450 . . . . . . . . . 10  |-  ( p  =  X  ->  (
p ( le `  K ) W  <->  X ( le `  K ) W ) )
98notbid 294 . . . . . . . . 9  |-  ( p  =  X  ->  ( -.  p ( le `  K ) W  <->  -.  X
( le `  K
) W ) )
109elrab 3261 . . . . . . . 8  |-  ( X  e.  { p  e.  ( Atoms `  K )  |  -.  p ( le
`  K ) W }  <->  ( X  e.  ( Atoms `  K )  /\  -.  X ( le
`  K ) W ) )
117, 10syl6bb 261 . . . . . . 7  |-  ( ( K  e.  V  /\  W  e.  H )  ->  ( X  e.  dom  I 
<->  ( X  e.  (
Atoms `  K )  /\  -.  X ( le `  K ) W ) ) )
1211biimpa 484 . . . . . 6  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  X  e.  dom  I )  ->  ( X  e.  ( Atoms `  K )  /\  -.  X ( le `  K ) W ) )
13 eqid 2467 . . . . . . 7  |-  ( ( oc `  K ) `
 W )  =  ( ( oc `  K ) `  W
)
14 eqid 2467 . . . . . . 7  |-  ( (
LTrn `  K ) `  W )  =  ( ( LTrn `  K
) `  W )
15 eqid 2467 . . . . . . 7  |-  ( (
TEndo `  K ) `  W )  =  ( ( TEndo `  K ) `  W )
162, 3, 4, 13, 14, 15, 5dicval 35991 . . . . . 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 470 . . . . 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 5087 . . . 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 233 . . 3  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  X  e.  dom  I )  ->  Rel  ( I `  X
) )
2019ex 434 . 2  |-  ( ( K  e.  V  /\  W  e.  H )  ->  ( X  e.  dom  I  ->  Rel  ( I `  X ) ) )
21 rel0 5127 . . 3  |-  Rel  (/)
22 ndmfv 5890 . . . 4  |-  ( -.  X  e.  dom  I  ->  ( I `  X
)  =  (/) )
2322releqd 5087 . . 3  |-  ( -.  X  e.  dom  I  ->  ( Rel  ( I `
 X )  <->  Rel  (/) ) )
2421, 23mpbiri 233 . 2  |-  ( -.  X  e.  dom  I  ->  Rel  ( I `  X ) )
2520, 24pm2.61d1 159 1  |-  ( ( K  e.  V  /\  W  e.  H )  ->  Rel  ( I `  X ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   {crab 2818   (/)c0 3785   class class class wbr 4447   {copab 4504   dom cdm 4999   Rel wrel 5004   ` cfv 5588   iota_crio 6244   lecple 14562   occoc 14563   Atomscatm 34078   LHypclh 34798   LTrncltrn 34915   TEndoctendo 35566   DIsoCcdic 35987
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6245  df-dic 35988
This theorem is referenced by: (None)
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