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Theorem diafn 31149
Description: Functionality and domain of the partial isomorphism A. (Contributed by NM, 26-Nov-2013.)
Hypotheses
Ref Expression
diafn.b  |-  B  =  ( Base `  K
)
diafn.l  |-  .<_  =  ( le `  K )
diafn.h  |-  H  =  ( LHyp `  K
)
diafn.i  |-  I  =  ( ( DIsoA `  K
) `  W )
Assertion
Ref Expression
diafn  |-  ( ( K  e.  V  /\  W  e.  H )  ->  I  Fn  { x  e.  B  |  x  .<_  W } )
Distinct variable groups:    x,  .<_    x, B    x, K    x, W
Allowed substitution hints:    H( x)    I( x)    V( x)

Proof of Theorem diafn
Dummy variables  y 
f are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fvex 5682 . . . 4  |-  ( (
LTrn `  K ) `  W )  e.  _V
21rabex 4295 . . 3  |-  { f  e.  ( ( LTrn `  K ) `  W
)  |  ( ( ( trL `  K
) `  W ) `  f )  .<_  y }  e.  _V
3 eqid 2387 . . 3  |-  ( y  e.  { x  e.  B  |  x  .<_  W }  |->  { f  e.  ( ( LTrn `  K
) `  W )  |  ( ( ( trL `  K ) `
 W ) `  f )  .<_  y } )  =  ( y  e.  { x  e.  B  |  x  .<_  W }  |->  { f  e.  ( ( LTrn `  K
) `  W )  |  ( ( ( trL `  K ) `
 W ) `  f )  .<_  y } )
42, 3fnmpti 5513 . 2  |-  ( y  e.  { x  e.  B  |  x  .<_  W }  |->  { f  e.  ( ( LTrn `  K
) `  W )  |  ( ( ( trL `  K ) `
 W ) `  f )  .<_  y } )  Fn  { x  e.  B  |  x  .<_  W }
5 diafn.b . . . 4  |-  B  =  ( Base `  K
)
6 diafn.l . . . 4  |-  .<_  =  ( le `  K )
7 diafn.h . . . 4  |-  H  =  ( LHyp `  K
)
8 eqid 2387 . . . 4  |-  ( (
LTrn `  K ) `  W )  =  ( ( LTrn `  K
) `  W )
9 eqid 2387 . . . 4  |-  ( ( trL `  K ) `
 W )  =  ( ( trL `  K
) `  W )
10 diafn.i . . . 4  |-  I  =  ( ( DIsoA `  K
) `  W )
115, 6, 7, 8, 9, 10diafval 31146 . . 3  |-  ( ( K  e.  V  /\  W  e.  H )  ->  I  =  ( y  e.  { x  e.  B  |  x  .<_  W }  |->  { f  e.  ( ( LTrn `  K
) `  W )  |  ( ( ( trL `  K ) `
 W ) `  f )  .<_  y } ) )
1211fneq1d 5476 . 2  |-  ( ( K  e.  V  /\  W  e.  H )  ->  ( I  Fn  {
x  e.  B  |  x  .<_  W }  <->  ( y  e.  { x  e.  B  |  x  .<_  W }  |->  { f  e.  ( ( LTrn `  K
) `  W )  |  ( ( ( trL `  K ) `
 W ) `  f )  .<_  y } )  Fn  { x  e.  B  |  x  .<_  W } ) )
134, 12mpbiri 225 1  |-  ( ( K  e.  V  /\  W  e.  H )  ->  I  Fn  { x  e.  B  |  x  .<_  W } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   {crab 2653   class class class wbr 4153    e. cmpt 4207    Fn wfn 5389   ` cfv 5394   Basecbs 13396   lecple 13463   LHypclh 30098   LTrncltrn 30215   trLctrl 30272   DIsoAcdia 31143
This theorem is referenced by:  diadm  31150  diaelrnN  31160  diaf11N  31164
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 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pr 4344
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 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-reu 2656  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-disoa 31144
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