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Theorem dibfna 31344
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 5539 . . . 4  |-  ( J `
 y )  e. 
_V
2 snex 4216 . . . 4  |-  { ( f  e.  ( (
LTrn `  K ) `  W )  |->  (  _I  |`  ( Base `  K
) ) ) }  e.  _V
31, 2xpex 4801 . . 3  |-  ( ( J `  y )  X.  { ( f  e.  ( ( LTrn `  K ) `  W
)  |->  (  _I  |`  ( Base `  K ) ) ) } )  e. 
_V
4 eqid 2283 . . 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 5372 . 2  |-  ( y  e.  dom  J  |->  ( ( J `  y
)  X.  { ( f  e.  ( (
LTrn `  K ) `  W )  |->  (  _I  |`  ( Base `  K
) ) ) } ) )  Fn  dom  J
6 eqid 2283 . . . 4  |-  ( Base `  K )  =  (
Base `  K )
7 dibfna.h . . . 4  |-  H  =  ( LHyp `  K
)
8 eqid 2283 . . . 4  |-  ( (
LTrn `  K ) `  W )  =  ( ( LTrn `  K
) `  W )
9 eqid 2283 . . . 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 31331 . . 3  |-  ( ( K  e.  V  /\  W  e.  H )  ->  I  =  ( y  e.  dom  J  |->  ( ( J `  y
)  X.  { ( f  e.  ( (
LTrn `  K ) `  W )  |->  (  _I  |`  ( Base `  K
) ) ) } ) ) )
1312fneq1d 5335 . 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 224 1  |-  ( ( K  e.  V  /\  W  e.  H )  ->  I  Fn  dom  J
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   {csn 3640    e. cmpt 4077    _I cid 4304    X. cxp 4687   dom cdm 4689    |` cres 4691    Fn wfn 5250   ` cfv 5255   Basecbs 13148   LHypclh 30173   LTrncltrn 30290   DIsoAcdia 31218   DIsoBcdib 31328
This theorem is referenced by:  dibdiadm  31345  dibfnN  31346  dibclN  31352
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-dib 31329
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