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Theorem dibvalrel 31353
Description: The value of partial isomorphism B is a relation. (Contributed by NM, 8-Mar-2014.)
Hypotheses
Ref Expression
dibcl.h  |-  H  =  ( LHyp `  K
)
dibcl.i  |-  I  =  ( ( DIsoB `  K
) `  W )
Assertion
Ref Expression
dibvalrel  |-  ( ( K  e.  V  /\  W  e.  H )  ->  Rel  ( I `  X ) )

Proof of Theorem dibvalrel
Dummy variable  h is distinct from all other variables.
StepHypRef Expression
1 relxp 4794 . . 3  |-  Rel  (
( ( ( DIsoA `  K ) `  W
) `  X )  X.  { ( h  e.  ( ( LTrn `  K
) `  W )  |->  (  _I  |`  ( Base `  K ) ) ) } )
2 dibcl.h . . . . . . . 8  |-  H  =  ( LHyp `  K
)
3 eqid 2283 . . . . . . . 8  |-  ( (
DIsoA `  K ) `  W )  =  ( ( DIsoA `  K ) `  W )
4 dibcl.i . . . . . . . 8  |-  I  =  ( ( DIsoB `  K
) `  W )
52, 3, 4dibdiadm 31345 . . . . . . 7  |-  ( ( K  e.  V  /\  W  e.  H )  ->  dom  I  =  dom  ( ( DIsoA `  K
) `  W )
)
65eleq2d 2350 . . . . . 6  |-  ( ( K  e.  V  /\  W  e.  H )  ->  ( X  e.  dom  I 
<->  X  e.  dom  (
( DIsoA `  K ) `  W ) ) )
76biimpa 470 . . . . 5  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  X  e.  dom  I )  ->  X  e.  dom  ( ( DIsoA `  K ) `  W
) )
8 eqid 2283 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
9 eqid 2283 . . . . . 6  |-  ( (
LTrn `  K ) `  W )  =  ( ( LTrn `  K
) `  W )
10 eqid 2283 . . . . . 6  |-  ( h  e.  ( ( LTrn `  K ) `  W
)  |->  (  _I  |`  ( Base `  K ) ) )  =  ( h  e.  ( ( LTrn `  K ) `  W
)  |->  (  _I  |`  ( Base `  K ) ) )
118, 2, 9, 10, 3, 4dibval 31332 . . . . 5  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  X  e.  dom  ( ( DIsoA `  K
) `  W )
)  ->  ( I `  X )  =  ( ( ( ( DIsoA `  K ) `  W
) `  X )  X.  { ( h  e.  ( ( LTrn `  K
) `  W )  |->  (  _I  |`  ( Base `  K ) ) ) } ) )
127, 11syldan 456 . . . 4  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  X  e.  dom  I )  ->  (
I `  X )  =  ( ( ( ( DIsoA `  K ) `  W ) `  X
)  X.  { ( h  e.  ( (
LTrn `  K ) `  W )  |->  (  _I  |`  ( Base `  K
) ) ) } ) )
1312releqd 4773 . . 3  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  X  e.  dom  I )  ->  ( Rel  ( I `  X
)  <->  Rel  ( ( ( ( DIsoA `  K ) `  W ) `  X
)  X.  { ( h  e.  ( (
LTrn `  K ) `  W )  |->  (  _I  |`  ( Base `  K
) ) ) } ) ) )
141, 13mpbiri 224 . 2  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  X  e.  dom  I )  ->  Rel  ( I `  X
) )
15 rel0 4810 . . . 4  |-  Rel  (/)
16 ndmfv 5552 . . . . 5  |-  ( -.  X  e.  dom  I  ->  ( I `  X
)  =  (/) )
1716releqd 4773 . . . 4  |-  ( -.  X  e.  dom  I  ->  ( Rel  ( I `
 X )  <->  Rel  (/) ) )
1815, 17mpbiri 224 . . 3  |-  ( -.  X  e.  dom  I  ->  Rel  ( I `  X ) )
1918adantl 452 . 2  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  -.  X  e.  dom  I )  ->  Rel  ( I `  X
) )
2014, 19pm2.61dan 766 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   (/)c0 3455   {csn 3640    e. cmpt 4077    _I cid 4304    X. cxp 4687   dom cdm 4689    |` cres 4691   Rel wrel 4694   ` cfv 5255   Basecbs 13148   LHypclh 30173   LTrncltrn 30290   DIsoAcdia 31218   DIsoBcdib 31328
This theorem is referenced by:  dibglbN  31356  dib2dim  31433  dih2dimbALTN  31435
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-dib 31329
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