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Theorem dibfna 31853
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 5734 . . . 4  |-  ( J `
 y )  e. 
_V
2 snex 4397 . . . 4  |-  { ( f  e.  ( (
LTrn `  K ) `  W )  |->  (  _I  |`  ( Base `  K
) ) ) }  e.  _V
31, 2xpex 4982 . . 3  |-  ( ( J `  y )  X.  { ( f  e.  ( ( LTrn `  K ) `  W
)  |->  (  _I  |`  ( Base `  K ) ) ) } )  e. 
_V
4 eqid 2435 . . 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 5565 . 2  |-  ( y  e.  dom  J  |->  ( ( J `  y
)  X.  { ( f  e.  ( (
LTrn `  K ) `  W )  |->  (  _I  |`  ( Base `  K
) ) ) } ) )  Fn  dom  J
6 eqid 2435 . . . 4  |-  ( Base `  K )  =  (
Base `  K )
7 dibfna.h . . . 4  |-  H  =  ( LHyp `  K
)
8 eqid 2435 . . . 4  |-  ( (
LTrn `  K ) `  W )  =  ( ( LTrn `  K
) `  W )
9 eqid 2435 . . . 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 31840 . . 3  |-  ( ( K  e.  V  /\  W  e.  H )  ->  I  =  ( y  e.  dom  J  |->  ( ( J `  y
)  X.  { ( f  e.  ( (
LTrn `  K ) `  W )  |->  (  _I  |`  ( Base `  K
) ) ) } ) ) )
1312fneq1d 5528 . 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 1652    e. wcel 1725   {csn 3806    e. cmpt 4258    _I cid 4485    X. cxp 4868   dom cdm 4870    |` cres 4872    Fn wfn 5441   ` cfv 5446   Basecbs 13459   LHypclh 30682   LTrncltrn 30799   DIsoAcdia 31727   DIsoBcdib 31837
This theorem is referenced by:  dibdiadm  31854  dibfnN  31855  dibclN  31861
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-dib 31838
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