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Theorem dicvalrelN 31375
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 4812 . . . 4  |-  Rel  { <. f ,  s >.  |  ( f  =  ( s `  ( iota_ g  e.  ( (
LTrn `  K ) `  W ) ( g `
 ( ( oc
`  K ) `  W ) )  =  X ) )  /\  s  e.  ( ( TEndo `  K ) `  W ) ) }
2 eqid 2283 . . . . . . . . . 10  |-  ( le
`  K )  =  ( le `  K
)
3 eqid 2283 . . . . . . . . . 10  |-  ( Atoms `  K )  =  (
Atoms `  K )
4 dicvalrel.h . . . . . . . . . 10  |-  H  =  ( LHyp `  K
)
5 dicvalrel.i . . . . . . . . . 10  |-  I  =  ( ( DIsoC `  K
) `  W )
62, 3, 4, 5dicdmN 31374 . . . . . . . . 9  |-  ( ( K  e.  V  /\  W  e.  H )  ->  dom  I  =  {
p  e.  ( Atoms `  K )  |  -.  p ( le `  K ) W }
)
76eleq2d 2350 . . . . . . . 8  |-  ( ( K  e.  V  /\  W  e.  H )  ->  ( X  e.  dom  I 
<->  X  e.  { p  e.  ( Atoms `  K )  |  -.  p ( le
`  K ) W } ) )
8 breq1 4026 . . . . . . . . . 10  |-  ( p  =  X  ->  (
p ( le `  K ) W  <->  X ( le `  K ) W ) )
98notbid 285 . . . . . . . . 9  |-  ( p  =  X  ->  ( -.  p ( le `  K ) W  <->  -.  X
( le `  K
) W ) )
109elrab 2923 . . . . . . . 8  |-  ( X  e.  { p  e.  ( Atoms `  K )  |  -.  p ( le
`  K ) W }  <->  ( X  e.  ( Atoms `  K )  /\  -.  X ( le
`  K ) W ) )
117, 10syl6bb 252 . . . . . . 7  |-  ( ( K  e.  V  /\  W  e.  H )  ->  ( X  e.  dom  I 
<->  ( X  e.  (
Atoms `  K )  /\  -.  X ( le `  K ) W ) ) )
1211biimpa 470 . . . . . 6  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  X  e.  dom  I )  ->  ( X  e.  ( Atoms `  K )  /\  -.  X ( le `  K ) W ) )
13 eqid 2283 . . . . . . 7  |-  ( ( oc `  K ) `
 W )  =  ( ( oc `  K ) `  W
)
14 eqid 2283 . . . . . . 7  |-  ( (
LTrn `  K ) `  W )  =  ( ( LTrn `  K
) `  W )
15 eqid 2283 . . . . . . 7  |-  ( (
TEndo `  K ) `  W )  =  ( ( TEndo `  K ) `  W )
162, 3, 4, 13, 14, 15, 5dicval 31366 . . . . . 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 456 . . . . 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 4773 . . . 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 224 . . 3  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  X  e.  dom  I )  ->  Rel  ( I `  X
) )
2019ex 423 . 2  |-  ( ( K  e.  V  /\  W  e.  H )  ->  ( X  e.  dom  I  ->  Rel  ( I `  X ) ) )
21 rel0 4810 . . 3  |-  Rel  (/)
22 ndmfv 5552 . . . 4  |-  ( -.  X  e.  dom  I  ->  ( I `  X
)  =  (/) )
2322releqd 4773 . . 3  |-  ( -.  X  e.  dom  I  ->  ( Rel  ( I `
 X )  <->  Rel  (/) ) )
2421, 23mpbiri 224 . 2  |-  ( -.  X  e.  dom  I  ->  Rel  ( I `  X ) )
2520, 24pm2.61d1 151 1  |-  ( ( K  e.  V  /\  W  e.  H )  ->  Rel  ( I `  X ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   {crab 2547   (/)c0 3455   class class class wbr 4023   {copab 4076   dom cdm 4689   Rel wrel 4694   ` cfv 5255   iota_crio 6297   lecple 13215   occoc 13216   Atomscatm 29453   LHypclh 30173   LTrncltrn 30290   TEndoctendo 30941   DIsoCcdic 31362
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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