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Theorem dibfna 31271
Description: Functionality and domain of the partial isomorphism B. (Contributed by NM, 17-Jan-2014.)
Hypotheses
Ref Expression
dibfna.h  |-  H  =  ( LHyp `  K
)
dibfna.j  |-  J  =  ( ( DIsoA `  K
) `  W )
dibfna.i  |-  I  =  ( ( DIsoB `  K
) `  W )
Assertion
Ref Expression
dibfna  |-  ( ( K  e.  V  /\  W  e.  H )  ->  I  Fn  dom  J
)

Proof of Theorem dibfna
Dummy variables  y 
f are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fvex 5684 . . . 4  |-  ( J `
 y )  e. 
_V
2 snex 4348 . . . 4  |-  { ( f  e.  ( (
LTrn `  K ) `  W )  |->  (  _I  |`  ( Base `  K
) ) ) }  e.  _V
31, 2xpex 4932 . . 3  |-  ( ( J `  y )  X.  { ( f  e.  ( ( LTrn `  K ) `  W
)  |->  (  _I  |`  ( Base `  K ) ) ) } )  e. 
_V
4 eqid 2389 . . 3  |-  ( y  e.  dom  J  |->  ( ( J `  y
)  X.  { ( f  e.  ( (
LTrn `  K ) `  W )  |->  (  _I  |`  ( Base `  K
) ) ) } ) )  =  ( y  e.  dom  J  |->  ( ( J `  y )  X.  {
( f  e.  ( ( LTrn `  K
) `  W )  |->  (  _I  |`  ( Base `  K ) ) ) } ) )
53, 4fnmpti 5515 . 2  |-  ( y  e.  dom  J  |->  ( ( J `  y
)  X.  { ( f  e.  ( (
LTrn `  K ) `  W )  |->  (  _I  |`  ( Base `  K
) ) ) } ) )  Fn  dom  J
6 eqid 2389 . . . 4  |-  ( Base `  K )  =  (
Base `  K )
7 dibfna.h . . . 4  |-  H  =  ( LHyp `  K
)
8 eqid 2389 . . . 4  |-  ( (
LTrn `  K ) `  W )  =  ( ( LTrn `  K
) `  W )
9 eqid 2389 . . . 4  |-  ( f  e.  ( ( LTrn `  K ) `  W
)  |->  (  _I  |`  ( Base `  K ) ) )  =  ( f  e.  ( ( LTrn `  K ) `  W
)  |->  (  _I  |`  ( Base `  K ) ) )
10 dibfna.j . . . 4  |-  J  =  ( ( DIsoA `  K
) `  W )
11 dibfna.i . . . 4  |-  I  =  ( ( DIsoB `  K
) `  W )
126, 7, 8, 9, 10, 11dibfval 31258 . . 3  |-  ( ( K  e.  V  /\  W  e.  H )  ->  I  =  ( y  e.  dom  J  |->  ( ( J `  y
)  X.  { ( f  e.  ( (
LTrn `  K ) `  W )  |->  (  _I  |`  ( Base `  K
) ) ) } ) ) )
1312fneq1d 5478 . 2  |-  ( ( K  e.  V  /\  W  e.  H )  ->  ( I  Fn  dom  J  <-> 
( y  e.  dom  J 
|->  ( ( J `  y )  X.  {
( f  e.  ( ( LTrn `  K
) `  W )  |->  (  _I  |`  ( Base `  K ) ) ) } ) )  Fn  dom  J ) )
145, 13mpbiri 225 1  |-  ( ( K  e.  V  /\  W  e.  H )  ->  I  Fn  dom  J
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   {csn 3759    e. cmpt 4209    _I cid 4436    X. cxp 4818   dom cdm 4820    |` cres 4822    Fn wfn 5391   ` cfv 5396   Basecbs 13398   LHypclh 30100   LTrncltrn 30217   DIsoAcdia 31145   DIsoBcdib 31255
This theorem is referenced by:  dibdiadm  31272  dibfnN  31273  dibclN  31279
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-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643
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 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-reu 2658  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-dib 31256
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