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Theorem diafn 31846
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 5555 . . . 4  |-  ( (
LTrn `  K ) `  W )  e.  _V
21rabex 4181 . . 3  |-  { f  e.  ( ( LTrn `  K ) `  W
)  |  ( ( ( trL `  K
) `  W ) `  f )  .<_  y }  e.  _V
3 eqid 2296 . . 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 5388 . 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 2296 . . . 4  |-  ( (
LTrn `  K ) `  W )  =  ( ( LTrn `  K
) `  W )
9 eqid 2296 . . . 4  |-  ( ( trL `  K ) `
 W )  =  ( ( trL `  K
) `  W )
10 diafn.i . . . 4  |-  I  =  ( ( DIsoA `  K
) `  W )
115, 6, 7, 8, 9, 10diafval 31843 . . 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 5351 . 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 224 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 358    = wceq 1632    e. wcel 1696   {crab 2560   class class class wbr 4039    e. cmpt 4093    Fn wfn 5266   ` cfv 5271   Basecbs 13164   lecple 13231   LHypclh 30795   LTrncltrn 30912   trLctrl 30969   DIsoAcdia 31840
This theorem is referenced by:  diadm  31847  diaelrnN  31857  diaf11N  31861
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-disoa 31841
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