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Theorem mrcflem 13831
Description: The domain and range of the function expression for Moore closures. (Contributed by Stefan O'Rear, 31-Jan-2015.)
Assertion
Ref Expression
mrcflem  |-  ( C  e.  (Moore `  X
)  ->  ( x  e.  ~P X  |->  |^| { s  e.  C  |  x 
C_  s } ) : ~P X --> C )
Distinct variable groups:    x, s, C    x, X, s

Proof of Theorem mrcflem
StepHypRef Expression
1 simpl 444 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  x  e.  ~P X )  ->  C  e.  (Moore `  X
) )
2 ssrab2 3428 . . . 4  |-  { s  e.  C  |  x 
C_  s }  C_  C
32a1i 11 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  x  e.  ~P X )  ->  { s  e.  C  |  x  C_  s } 
C_  C )
4 mre1cl 13819 . . . . . 6  |-  ( C  e.  (Moore `  X
)  ->  X  e.  C )
54adantr 452 . . . . 5  |-  ( ( C  e.  (Moore `  X )  /\  x  e.  ~P X )  ->  X  e.  C )
6 elpwi 3807 . . . . . 6  |-  ( x  e.  ~P X  ->  x  C_  X )
76adantl 453 . . . . 5  |-  ( ( C  e.  (Moore `  X )  /\  x  e.  ~P X )  ->  x  C_  X )
8 sseq2 3370 . . . . . 6  |-  ( s  =  X  ->  (
x  C_  s  <->  x  C_  X
) )
98elrab 3092 . . . . 5  |-  ( X  e.  { s  e.  C  |  x  C_  s }  <->  ( X  e.  C  /\  x  C_  X ) )
105, 7, 9sylanbrc 646 . . . 4  |-  ( ( C  e.  (Moore `  X )  /\  x  e.  ~P X )  ->  X  e.  { s  e.  C  |  x  C_  s } )
11 ne0i 3634 . . . 4  |-  ( X  e.  { s  e.  C  |  x  C_  s }  ->  { s  e.  C  |  x 
C_  s }  =/=  (/) )
1210, 11syl 16 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  x  e.  ~P X )  ->  { s  e.  C  |  x  C_  s }  =/=  (/) )
13 mreintcl 13820 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  {
s  e.  C  |  x  C_  s }  C_  C  /\  { s  e.  C  |  x  C_  s }  =/=  (/) )  ->  |^| { s  e.  C  |  x  C_  s }  e.  C )
141, 3, 12, 13syl3anc 1184 . 2  |-  ( ( C  e.  (Moore `  X )  /\  x  e.  ~P X )  ->  |^| { s  e.  C  |  x  C_  s }  e.  C )
15 eqid 2436 . 2  |-  ( x  e.  ~P X  |->  |^|
{ s  e.  C  |  x  C_  s } )  =  ( x  e.  ~P X  |->  |^|
{ s  e.  C  |  x  C_  s } )
1614, 15fmptd 5893 1  |-  ( C  e.  (Moore `  X
)  ->  ( x  e.  ~P X  |->  |^| { s  e.  C  |  x 
C_  s } ) : ~P X --> C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    e. wcel 1725    =/= wne 2599   {crab 2709    C_ wss 3320   (/)c0 3628   ~Pcpw 3799   |^|cint 4050    e. cmpt 4266   -->wf 5450   ` cfv 5454  Moorecmre 13807
This theorem is referenced by:  fnmrc  13832  mrcfval  13833  mrcf  13834
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 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-int 4051  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-fv 5462  df-mre 13811
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