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Theorem dicfnN 35110
Description: Functionality and domain of the partial isomorphism C. (Contributed by NM, 8-Mar-2014.) (New usage is discouraged.)
Hypotheses
Ref Expression
dicfn.l  |-  .<_  =  ( le `  K )
dicfn.a  |-  A  =  ( Atoms `  K )
dicfn.h  |-  H  =  ( LHyp `  K
)
dicfn.i  |-  I  =  ( ( DIsoC `  K
) `  W )
Assertion
Ref Expression
dicfnN  |-  ( ( K  e.  V  /\  W  e.  H )  ->  I  Fn  { p  e.  A  |  -.  p  .<_  W } )
Distinct variable groups:    .<_ , p    A, p    K, p    W, p
Allowed substitution hints:    H( p)    I( p)    V( p)

Proof of Theorem dicfnN
Dummy variables  q 
f  s  u are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 breq1 4379 . . . . . . 7  |-  ( p  =  q  ->  (
p  .<_  W  <->  q  .<_  W ) )
21notbid 294 . . . . . 6  |-  ( p  =  q  ->  ( -.  p  .<_  W  <->  -.  q  .<_  W ) )
32elrab 3200 . . . . 5  |-  ( q  e.  { p  e.  A  |  -.  p  .<_  W }  <->  ( q  e.  A  /\  -.  q  .<_  W ) )
4 dicfn.l . . . . . . 7  |-  .<_  =  ( le `  K )
5 dicfn.a . . . . . . 7  |-  A  =  ( Atoms `  K )
6 dicfn.h . . . . . . 7  |-  H  =  ( LHyp `  K
)
7 eqid 2450 . . . . . . 7  |-  ( ( oc `  K ) `
 W )  =  ( ( oc `  K ) `  W
)
8 eqid 2450 . . . . . . 7  |-  ( (
LTrn `  K ) `  W )  =  ( ( LTrn `  K
) `  W )
9 eqid 2450 . . . . . . 7  |-  ( (
TEndo `  K ) `  W )  =  ( ( TEndo `  K ) `  W )
10 dicfn.i . . . . . . 7  |-  I  =  ( ( DIsoC `  K
) `  W )
114, 5, 6, 7, 8, 9, 10dicval 35103 . . . . . 6  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( q  e.  A  /\  -.  q  .<_  W ) )  -> 
( I `  q
)  =  { <. f ,  s >.  |  ( f  =  ( s `
 ( iota_ u  e.  ( ( LTrn `  K
) `  W )
( u `  (
( oc `  K
) `  W )
)  =  q ) )  /\  s  e.  ( ( TEndo `  K
) `  W )
) } )
12 fvex 5785 . . . . . 6  |-  ( I `
 q )  e. 
_V
1311, 12syl6eqelr 2545 . . . . 5  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( q  e.  A  /\  -.  q  .<_  W ) )  ->  { <. f ,  s
>.  |  ( f  =  ( s `  ( iota_ u  e.  ( ( LTrn `  K
) `  W )
( u `  (
( oc `  K
) `  W )
)  =  q ) )  /\  s  e.  ( ( TEndo `  K
) `  W )
) }  e.  _V )
143, 13sylan2b 475 . . . 4  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  q  e.  { p  e.  A  |  -.  p  .<_  W }
)  ->  { <. f ,  s >.  |  ( f  =  ( s `
 ( iota_ u  e.  ( ( LTrn `  K
) `  W )
( u `  (
( oc `  K
) `  W )
)  =  q ) )  /\  s  e.  ( ( TEndo `  K
) `  W )
) }  e.  _V )
1514ralrimiva 2881 . . 3  |-  ( ( K  e.  V  /\  W  e.  H )  ->  A. q  e.  {
p  e.  A  |  -.  p  .<_  W }  { <. f ,  s
>.  |  ( f  =  ( s `  ( iota_ u  e.  ( ( LTrn `  K
) `  W )
( u `  (
( oc `  K
) `  W )
)  =  q ) )  /\  s  e.  ( ( TEndo `  K
) `  W )
) }  e.  _V )
16 eqid 2450 . . . 4  |-  ( q  e.  { p  e.  A  |  -.  p  .<_  W }  |->  { <. f ,  s >.  |  ( f  =  ( s `
 ( iota_ u  e.  ( ( LTrn `  K
) `  W )
( u `  (
( oc `  K
) `  W )
)  =  q ) )  /\  s  e.  ( ( TEndo `  K
) `  W )
) } )  =  ( q  e.  {
p  e.  A  |  -.  p  .<_  W }  |->  { <. f ,  s
>.  |  ( f  =  ( s `  ( iota_ u  e.  ( ( LTrn `  K
) `  W )
( u `  (
( oc `  K
) `  W )
)  =  q ) )  /\  s  e.  ( ( TEndo `  K
) `  W )
) } )
1716fnmpt 5621 . . 3  |-  ( A. q  e.  { p  e.  A  |  -.  p  .<_  W }  { <. f ,  s >.  |  ( f  =  ( s `  ( iota_ u  e.  ( (
LTrn `  K ) `  W ) ( u `
 ( ( oc
`  K ) `  W ) )  =  q ) )  /\  s  e.  ( ( TEndo `  K ) `  W ) ) }  e.  _V  ->  (
q  e.  { p  e.  A  |  -.  p  .<_  W }  |->  {
<. f ,  s >.  |  ( f  =  ( s `  ( iota_ u  e.  ( (
LTrn `  K ) `  W ) ( u `
 ( ( oc
`  K ) `  W ) )  =  q ) )  /\  s  e.  ( ( TEndo `  K ) `  W ) ) } )  Fn  { p  e.  A  |  -.  p  .<_  W } )
1815, 17syl 16 . 2  |-  ( ( K  e.  V  /\  W  e.  H )  ->  ( q  e.  {
p  e.  A  |  -.  p  .<_  W }  |->  { <. f ,  s
>.  |  ( f  =  ( s `  ( iota_ u  e.  ( ( LTrn `  K
) `  W )
( u `  (
( oc `  K
) `  W )
)  =  q ) )  /\  s  e.  ( ( TEndo `  K
) `  W )
) } )  Fn 
{ p  e.  A  |  -.  p  .<_  W }
)
194, 5, 6, 7, 8, 9, 10dicfval 35102 . . 3  |-  ( ( K  e.  V  /\  W  e.  H )  ->  I  =  ( q  e.  { p  e.  A  |  -.  p  .<_  W }  |->  { <. f ,  s >.  |  ( f  =  ( s `
 ( iota_ u  e.  ( ( LTrn `  K
) `  W )
( u `  (
( oc `  K
) `  W )
)  =  q ) )  /\  s  e.  ( ( TEndo `  K
) `  W )
) } ) )
2019fneq1d 5585 . 2  |-  ( ( K  e.  V  /\  W  e.  H )  ->  ( I  Fn  {
p  e.  A  |  -.  p  .<_  W }  <->  ( q  e.  { p  e.  A  |  -.  p  .<_  W }  |->  {
<. f ,  s >.  |  ( f  =  ( s `  ( iota_ u  e.  ( (
LTrn `  K ) `  W ) ( u `
 ( ( oc
`  K ) `  W ) )  =  q ) )  /\  s  e.  ( ( TEndo `  K ) `  W ) ) } )  Fn  { p  e.  A  |  -.  p  .<_  W } ) )
2118, 20mpbird 232 1  |-  ( ( K  e.  V  /\  W  e.  H )  ->  I  Fn  { p  e.  A  |  -.  p  .<_  W } )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1757   A.wral 2792   {crab 2796   _Vcvv 3054   class class class wbr 4376   {copab 4433    |-> cmpt 4434    Fn wfn 5497   ` cfv 5502   iota_crio 6136   lecple 14333   occoc 14334   Atomscatm 33190   LHypclh 33910   LTrncltrn 34027   TEndoctendo 34678   DIsoCcdic 35099
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 1709  ax-7 1729  ax-8 1759  ax-9 1761  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2429  ax-rep 4487  ax-sep 4497  ax-nul 4505  ax-pow 4554  ax-pr 4615  ax-un 6458
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 1702  df-eu 2263  df-mo 2264  df-clab 2436  df-cleq 2442  df-clel 2445  df-nfc 2598  df-ne 2643  df-ral 2797  df-rex 2798  df-reu 2799  df-rab 2801  df-v 3056  df-sbc 3271  df-csb 3373  df-dif 3415  df-un 3417  df-in 3419  df-ss 3426  df-nul 3722  df-if 3876  df-pw 3946  df-sn 3962  df-pr 3964  df-op 3968  df-uni 4176  df-iun 4257  df-br 4377  df-opab 4435  df-mpt 4436  df-id 4720  df-xp 4930  df-rel 4931  df-cnv 4932  df-co 4933  df-dm 4934  df-rn 4935  df-res 4936  df-ima 4937  df-iota 5465  df-fun 5504  df-fn 5505  df-f 5506  df-f1 5507  df-fo 5508  df-f1o 5509  df-fv 5510  df-riota 6137  df-dic 35100
This theorem is referenced by:  dicdmN  35111
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