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Theorem dibfna 35951
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 5874 . . . 4  |-  ( J `
 y )  e. 
_V
2 snex 4688 . . . 4  |-  { ( f  e.  ( (
LTrn `  K ) `  W )  |->  (  _I  |`  ( Base `  K
) ) ) }  e.  _V
31, 2xpex 6711 . . 3  |-  ( ( J `  y )  X.  { ( f  e.  ( ( LTrn `  K ) `  W
)  |->  (  _I  |`  ( Base `  K ) ) ) } )  e. 
_V
4 eqid 2467 . . 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 5707 . 2  |-  ( y  e.  dom  J  |->  ( ( J `  y
)  X.  { ( f  e.  ( (
LTrn `  K ) `  W )  |->  (  _I  |`  ( Base `  K
) ) ) } ) )  Fn  dom  J
6 eqid 2467 . . . 4  |-  ( Base `  K )  =  (
Base `  K )
7 dibfna.h . . . 4  |-  H  =  ( LHyp `  K
)
8 eqid 2467 . . . 4  |-  ( (
LTrn `  K ) `  W )  =  ( ( LTrn `  K
) `  W )
9 eqid 2467 . . . 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 35938 . . 3  |-  ( ( K  e.  V  /\  W  e.  H )  ->  I  =  ( y  e.  dom  J  |->  ( ( J `  y
)  X.  { ( f  e.  ( (
LTrn `  K ) `  W )  |->  (  _I  |`  ( Base `  K
) ) ) } ) ) )
1312fneq1d 5669 . 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 1379    e. wcel 1767   {csn 4027    |-> cmpt 4505    _I cid 4790    X. cxp 4997   dom cdm 4999    |` cres 5001    Fn wfn 5581   ` cfv 5586   Basecbs 14486   LHypclh 34780   LTrncltrn 34897   DIsoAcdia 35825   DIsoBcdib 35935
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-dib 35936
This theorem is referenced by:  dibdiadm  35952  dibfnN  35953  dibclN  35959
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