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Theorem dibfna 35107
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 5801 . . . 4  |-  ( J `
 y )  e. 
_V
2 snex 4633 . . . 4  |-  { ( f  e.  ( (
LTrn `  K ) `  W )  |->  (  _I  |`  ( Base `  K
) ) ) }  e.  _V
31, 2xpex 6610 . . 3  |-  ( ( J `  y )  X.  { ( f  e.  ( ( LTrn `  K ) `  W
)  |->  (  _I  |`  ( Base `  K ) ) ) } )  e. 
_V
4 eqid 2451 . . 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 5639 . 2  |-  ( y  e.  dom  J  |->  ( ( J `  y
)  X.  { ( f  e.  ( (
LTrn `  K ) `  W )  |->  (  _I  |`  ( Base `  K
) ) ) } ) )  Fn  dom  J
6 eqid 2451 . . . 4  |-  ( Base `  K )  =  (
Base `  K )
7 dibfna.h . . . 4  |-  H  =  ( LHyp `  K
)
8 eqid 2451 . . . 4  |-  ( (
LTrn `  K ) `  W )  =  ( ( LTrn `  K
) `  W )
9 eqid 2451 . . . 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 35094 . . 3  |-  ( ( K  e.  V  /\  W  e.  H )  ->  I  =  ( y  e.  dom  J  |->  ( ( J `  y
)  X.  { ( f  e.  ( (
LTrn `  K ) `  W )  |->  (  _I  |`  ( Base `  K
) ) ) } ) ) )
1312fneq1d 5601 . 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 233 1  |-  ( ( K  e.  V  /\  W  e.  H )  ->  I  Fn  dom  J
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   {csn 3977    |-> cmpt 4450    _I cid 4731    X. cxp 4938   dom cdm 4940    |` cres 4942    Fn wfn 5513   ` cfv 5518   Basecbs 14278   LHypclh 33936   LTrncltrn 34053   DIsoAcdia 34981   DIsoBcdib 35091
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4503  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631  ax-un 6474
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3072  df-sbc 3287  df-csb 3389  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-op 3984  df-uni 4192  df-iun 4273  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4736  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fun 5520  df-fn 5521  df-f 5522  df-f1 5523  df-fo 5524  df-f1o 5525  df-fv 5526  df-dib 35092
This theorem is referenced by:  dibdiadm  35108  dibfnN  35109  dibclN  35115
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