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Theorem dibvalrel 34440
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 4962 . . 3  |-  Rel  (
( ( ( DIsoA `  K ) `  W
) `  X )  X.  { ( h  e.  ( ( LTrn `  K
) `  W )  |->  (  _I  |`  ( Base `  K ) ) ) } )
2 dibcl.h . . . . . . . 8  |-  H  =  ( LHyp `  K
)
3 eqid 2429 . . . . . . . 8  |-  ( (
DIsoA `  K ) `  W )  =  ( ( DIsoA `  K ) `  W )
4 dibcl.i . . . . . . . 8  |-  I  =  ( ( DIsoB `  K
) `  W )
52, 3, 4dibdiadm 34432 . . . . . . 7  |-  ( ( K  e.  V  /\  W  e.  H )  ->  dom  I  =  dom  ( ( DIsoA `  K
) `  W )
)
65eleq2d 2499 . . . . . 6  |-  ( ( K  e.  V  /\  W  e.  H )  ->  ( X  e.  dom  I 
<->  X  e.  dom  (
( DIsoA `  K ) `  W ) ) )
76biimpa 486 . . . . 5  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  X  e.  dom  I )  ->  X  e.  dom  ( ( DIsoA `  K ) `  W
) )
8 eqid 2429 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
9 eqid 2429 . . . . . 6  |-  ( (
LTrn `  K ) `  W )  =  ( ( LTrn `  K
) `  W )
10 eqid 2429 . . . . . 6  |-  ( h  e.  ( ( LTrn `  K ) `  W
)  |->  (  _I  |`  ( Base `  K ) ) )  =  ( h  e.  ( ( LTrn `  K ) `  W
)  |->  (  _I  |`  ( Base `  K ) ) )
118, 2, 9, 10, 3, 4dibval 34419 . . . . 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 472 . . . 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 4939 . . 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 236 . 2  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  X  e.  dom  I )  ->  Rel  ( I `  X
) )
15 rel0 4978 . . . 4  |-  Rel  (/)
16 ndmfv 5905 . . . . 5  |-  ( -.  X  e.  dom  I  ->  ( I `  X
)  =  (/) )
1716releqd 4939 . . . 4  |-  ( -.  X  e.  dom  I  ->  ( Rel  ( I `
 X )  <->  Rel  (/) ) )
1815, 17mpbiri 236 . . 3  |-  ( -.  X  e.  dom  I  ->  Rel  ( I `  X ) )
1918adantl 467 . 2  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  -.  X  e.  dom  I )  ->  Rel  ( I `  X
) )
2014, 19pm2.61dan 798 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 1870   (/)c0 3767   {csn 4002    |-> cmpt 4484    _I cid 4764    X. cxp 4852   dom cdm 4854    |` cres 4856   Rel wrel 4859   ` cfv 5601   Basecbs 15084   LHypclh 33258   LTrncltrn 33375   DIsoAcdia 34305   DIsoBcdib 34415
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 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597
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 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-reu 2789  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-dib 34416
This theorem is referenced by:  dibglbN  34443  dib2dim  34520  dih2dimbALTN  34522
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