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Theorem dibvalrel 32035
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 4986 . . 3  |-  Rel  (
( ( ( DIsoA `  K ) `  W
) `  X )  X.  { ( h  e.  ( ( LTrn `  K
) `  W )  |->  (  _I  |`  ( Base `  K ) ) ) } )
2 dibcl.h . . . . . . . 8  |-  H  =  ( LHyp `  K
)
3 eqid 2438 . . . . . . . 8  |-  ( (
DIsoA `  K ) `  W )  =  ( ( DIsoA `  K ) `  W )
4 dibcl.i . . . . . . . 8  |-  I  =  ( ( DIsoB `  K
) `  W )
52, 3, 4dibdiadm 32027 . . . . . . 7  |-  ( ( K  e.  V  /\  W  e.  H )  ->  dom  I  =  dom  ( ( DIsoA `  K
) `  W )
)
65eleq2d 2505 . . . . . 6  |-  ( ( K  e.  V  /\  W  e.  H )  ->  ( X  e.  dom  I 
<->  X  e.  dom  (
( DIsoA `  K ) `  W ) ) )
76biimpa 472 . . . . 5  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  X  e.  dom  I )  ->  X  e.  dom  ( ( DIsoA `  K ) `  W
) )
8 eqid 2438 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
9 eqid 2438 . . . . . 6  |-  ( (
LTrn `  K ) `  W )  =  ( ( LTrn `  K
) `  W )
10 eqid 2438 . . . . . 6  |-  ( h  e.  ( ( LTrn `  K ) `  W
)  |->  (  _I  |`  ( Base `  K ) ) )  =  ( h  e.  ( ( LTrn `  K ) `  W
)  |->  (  _I  |`  ( Base `  K ) ) )
118, 2, 9, 10, 3, 4dibval 32014 . . . . 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 458 . . . 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 4964 . . 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 226 . 2  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  X  e.  dom  I )  ->  Rel  ( I `  X
) )
15 rel0 5002 . . . 4  |-  Rel  (/)
16 ndmfv 5758 . . . . 5  |-  ( -.  X  e.  dom  I  ->  ( I `  X
)  =  (/) )
1716releqd 4964 . . . 4  |-  ( -.  X  e.  dom  I  ->  ( Rel  ( I `
 X )  <->  Rel  (/) ) )
1815, 17mpbiri 226 . . 3  |-  ( -.  X  e.  dom  I  ->  Rel  ( I `  X ) )
1918adantl 454 . 2  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  -.  X  e.  dom  I )  ->  Rel  ( I `  X
) )
2014, 19pm2.61dan 768 1  |-  ( ( K  e.  V  /\  W  e.  H )  ->  Rel  ( I `  X ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   (/)c0 3630   {csn 3816    e. cmpt 4269    _I cid 4496    X. cxp 4879   dom cdm 4881    |` cres 4883   Rel wrel 4886   ` cfv 5457   Basecbs 13474   LHypclh 30855   LTrncltrn 30972   DIsoAcdia 31900   DIsoBcdib 32010
This theorem is referenced by:  dibglbN  32038  dib2dim  32115  dih2dimbALTN  32117
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-dib 32011
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