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Theorem dibvalrel 31405
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 4873 . . 3  |-  Rel  (
( ( ( DIsoA `  K ) `  W
) `  X )  X.  { ( h  e.  ( ( LTrn `  K
) `  W )  |->  (  _I  |`  ( Base `  K ) ) ) } )
2 dibcl.h . . . . . . . 8  |-  H  =  ( LHyp `  K
)
3 eqid 2358 . . . . . . . 8  |-  ( (
DIsoA `  K ) `  W )  =  ( ( DIsoA `  K ) `  W )
4 dibcl.i . . . . . . . 8  |-  I  =  ( ( DIsoB `  K
) `  W )
52, 3, 4dibdiadm 31397 . . . . . . 7  |-  ( ( K  e.  V  /\  W  e.  H )  ->  dom  I  =  dom  ( ( DIsoA `  K
) `  W )
)
65eleq2d 2425 . . . . . 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 2358 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
9 eqid 2358 . . . . . 6  |-  ( (
LTrn `  K ) `  W )  =  ( ( LTrn `  K
) `  W )
10 eqid 2358 . . . . . 6  |-  ( h  e.  ( ( LTrn `  K ) `  W
)  |->  (  _I  |`  ( Base `  K ) ) )  =  ( h  e.  ( ( LTrn `  K ) `  W
)  |->  (  _I  |`  ( Base `  K ) ) )
118, 2, 9, 10, 3, 4dibval 31384 . . . . 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 4852 . . 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 4889 . . . 4  |-  Rel  (/)
16 ndmfv 5632 . . . . 5  |-  ( -.  X  e.  dom  I  ->  ( I `  X
)  =  (/) )
1716releqd 4852 . . . 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 1642    e. wcel 1710   (/)c0 3531   {csn 3716    e. cmpt 4156    _I cid 4383    X. cxp 4766   dom cdm 4768    |` cres 4770   Rel wrel 4773   ` cfv 5334   Basecbs 13239   LHypclh 30225   LTrncltrn 30342   DIsoAcdia 31270   DIsoBcdib 31380
This theorem is referenced by:  dibglbN  31408  dib2dim  31485  dih2dimbALTN  31487
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4210  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-reu 2626  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3907  df-iun 3986  df-br 4103  df-opab 4157  df-mpt 4158  df-id 4388  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-f1 5339  df-fo 5340  df-f1o 5341  df-fv 5342  df-dib 31381
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