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Theorem dicfnN 31918
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 4207 . . . . . . 7  |-  ( p  =  q  ->  (
p  .<_  W  <->  q  .<_  W ) )
21notbid 286 . . . . . 6  |-  ( p  =  q  ->  ( -.  p  .<_  W  <->  -.  q  .<_  W ) )
32elrab 3084 . . . . 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 2435 . . . . . . 7  |-  ( ( oc `  K ) `
 W )  =  ( ( oc `  K ) `  W
)
8 eqid 2435 . . . . . . 7  |-  ( (
LTrn `  K ) `  W )  =  ( ( LTrn `  K
) `  W )
9 eqid 2435 . . . . . . 7  |-  ( (
TEndo `  K ) `  W )  =  ( ( TEndo `  K ) `  W )
10 dicfn.i . . . . . . 7  |-  I  =  ( ( DIsoC `  K
) `  W )
114, 5, 6, 7, 8, 9, 10dicval 31911 . . . . . 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 5734 . . . . . 6  |-  ( I `
 q )  e. 
_V
1311, 12syl6eqelr 2524 . . . . 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 462 . . . 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 2781 . . 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 2435 . . . 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 5563 . . 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 31910 . . 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 5528 . 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 224 1  |-  ( ( K  e.  V  /\  W  e.  H )  ->  I  Fn  { p  e.  A  |  -.  p  .<_  W } )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2697   {crab 2701   _Vcvv 2948   class class class wbr 4204   {copab 4257    e. cmpt 4258    Fn wfn 5441   ` cfv 5446   iota_crio 6534   lecple 13528   occoc 13529   Atomscatm 29998   LHypclh 30718   LTrncltrn 30835   TEndoctendo 31486   DIsoCcdic 31907
This theorem is referenced by:  dicdmN  31919
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-riota 6541  df-dic 31908
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