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Theorem dibvalrel 31658
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 4950 . . 3  |-  Rel  (
( ( ( DIsoA `  K ) `  W
) `  X )  X.  { ( h  e.  ( ( LTrn `  K
) `  W )  |->  (  _I  |`  ( Base `  K ) ) ) } )
2 dibcl.h . . . . . . . 8  |-  H  =  ( LHyp `  K
)
3 eqid 2412 . . . . . . . 8  |-  ( (
DIsoA `  K ) `  W )  =  ( ( DIsoA `  K ) `  W )
4 dibcl.i . . . . . . . 8  |-  I  =  ( ( DIsoB `  K
) `  W )
52, 3, 4dibdiadm 31650 . . . . . . 7  |-  ( ( K  e.  V  /\  W  e.  H )  ->  dom  I  =  dom  ( ( DIsoA `  K
) `  W )
)
65eleq2d 2479 . . . . . 6  |-  ( ( K  e.  V  /\  W  e.  H )  ->  ( X  e.  dom  I 
<->  X  e.  dom  (
( DIsoA `  K ) `  W ) ) )
76biimpa 471 . . . . 5  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  X  e.  dom  I )  ->  X  e.  dom  ( ( DIsoA `  K ) `  W
) )
8 eqid 2412 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
9 eqid 2412 . . . . . 6  |-  ( (
LTrn `  K ) `  W )  =  ( ( LTrn `  K
) `  W )
10 eqid 2412 . . . . . 6  |-  ( h  e.  ( ( LTrn `  K ) `  W
)  |->  (  _I  |`  ( Base `  K ) ) )  =  ( h  e.  ( ( LTrn `  K ) `  W
)  |->  (  _I  |`  ( Base `  K ) ) )
118, 2, 9, 10, 3, 4dibval 31637 . . . . 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 457 . . . 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 4928 . . 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 225 . 2  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  X  e.  dom  I )  ->  Rel  ( I `  X
) )
15 rel0 4966 . . . 4  |-  Rel  (/)
16 ndmfv 5722 . . . . 5  |-  ( -.  X  e.  dom  I  ->  ( I `  X
)  =  (/) )
1716releqd 4928 . . . 4  |-  ( -.  X  e.  dom  I  ->  ( Rel  ( I `
 X )  <->  Rel  (/) ) )
1815, 17mpbiri 225 . . 3  |-  ( -.  X  e.  dom  I  ->  Rel  ( I `  X ) )
1918adantl 453 . 2  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  -.  X  e.  dom  I )  ->  Rel  ( I `  X
) )
2014, 19pm2.61dan 767 1  |-  ( ( K  e.  V  /\  W  e.  H )  ->  Rel  ( I `  X ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   (/)c0 3596   {csn 3782    e. cmpt 4234    _I cid 4461    X. cxp 4843   dom cdm 4845    |` cres 4847   Rel wrel 4850   ` cfv 5421   Basecbs 13432   LHypclh 30478   LTrncltrn 30595   DIsoAcdia 31523   DIsoBcdib 31633
This theorem is referenced by:  dibglbN  31661  dib2dim  31738  dih2dimbALTN  31740
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-reu 2681  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  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 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-dib 31634
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