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Theorem dicvalrelN 34491
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 4971 . . . 4  |-  Rel  { <. f ,  s >.  |  ( f  =  ( s `  ( iota_ g  e.  ( (
LTrn `  K ) `  W ) ( g `
 ( ( oc
`  K ) `  W ) )  =  X ) )  /\  s  e.  ( ( TEndo `  K ) `  W ) ) }
2 eqid 2420 . . . . . . . . . 10  |-  ( le
`  K )  =  ( le `  K
)
3 eqid 2420 . . . . . . . . . 10  |-  ( Atoms `  K )  =  (
Atoms `  K )
4 dicvalrel.h . . . . . . . . . 10  |-  H  =  ( LHyp `  K
)
5 dicvalrel.i . . . . . . . . . 10  |-  I  =  ( ( DIsoC `  K
) `  W )
62, 3, 4, 5dicdmN 34490 . . . . . . . . 9  |-  ( ( K  e.  V  /\  W  e.  H )  ->  dom  I  =  {
p  e.  ( Atoms `  K )  |  -.  p ( le `  K ) W }
)
76eleq2d 2490 . . . . . . . 8  |-  ( ( K  e.  V  /\  W  e.  H )  ->  ( X  e.  dom  I 
<->  X  e.  { p  e.  ( Atoms `  K )  |  -.  p ( le
`  K ) W } ) )
8 breq1 4420 . . . . . . . . . 10  |-  ( p  =  X  ->  (
p ( le `  K ) W  <->  X ( le `  K ) W ) )
98notbid 295 . . . . . . . . 9  |-  ( p  =  X  ->  ( -.  p ( le `  K ) W  <->  -.  X
( le `  K
) W ) )
109elrab 3226 . . . . . . . 8  |-  ( X  e.  { p  e.  ( Atoms `  K )  |  -.  p ( le
`  K ) W }  <->  ( X  e.  ( Atoms `  K )  /\  -.  X ( le
`  K ) W ) )
117, 10syl6bb 264 . . . . . . 7  |-  ( ( K  e.  V  /\  W  e.  H )  ->  ( X  e.  dom  I 
<->  ( X  e.  (
Atoms `  K )  /\  -.  X ( le `  K ) W ) ) )
1211biimpa 486 . . . . . 6  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  X  e.  dom  I )  ->  ( X  e.  ( Atoms `  K )  /\  -.  X ( le `  K ) W ) )
13 eqid 2420 . . . . . . 7  |-  ( ( oc `  K ) `
 W )  =  ( ( oc `  K ) `  W
)
14 eqid 2420 . . . . . . 7  |-  ( (
LTrn `  K ) `  W )  =  ( ( LTrn `  K
) `  W )
15 eqid 2420 . . . . . . 7  |-  ( (
TEndo `  K ) `  W )  =  ( ( TEndo `  K ) `  W )
162, 3, 4, 13, 14, 15, 5dicval 34482 . . . . . 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 472 . . . . 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 4930 . . . 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 236 . . 3  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  X  e.  dom  I )  ->  Rel  ( I `  X
) )
2019ex 435 . 2  |-  ( ( K  e.  V  /\  W  e.  H )  ->  ( X  e.  dom  I  ->  Rel  ( I `  X ) ) )
21 rel0 4969 . . 3  |-  Rel  (/)
22 ndmfv 5896 . . . 4  |-  ( -.  X  e.  dom  I  ->  ( I `  X
)  =  (/) )
2322releqd 4930 . . 3  |-  ( -.  X  e.  dom  I  ->  ( Rel  ( I `
 X )  <->  Rel  (/) ) )
2421, 23mpbiri 236 . 2  |-  ( -.  X  e.  dom  I  ->  Rel  ( I `  X ) )
2520, 24pm2.61d1 162 1  |-  ( ( K  e.  V  /\  W  e.  H )  ->  Rel  ( I `  X ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1867   {crab 2777   (/)c0 3758   class class class wbr 4417   {copab 4474   dom cdm 4845   Rel wrel 4850   ` cfv 5592   iota_crio 6257   lecple 15149   occoc 15150   Atomscatm 32567   LHypclh 33287   LTrncltrn 33404   TEndoctendo 34057   DIsoCcdic 34478
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-rep 4529  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-reu 2780  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-iun 4295  df-br 4418  df-opab 4476  df-mpt 4477  df-id 4760  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-riota 6258  df-dic 34479
This theorem is referenced by: (None)
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