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Theorem mrcflem 13502
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 445 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  x  e.  ~P X )  ->  C  e.  (Moore `  X
) )
2 ssrab2 3259 . . . 4  |-  { s  e.  C  |  x 
C_  s }  C_  C
32a1i 12 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  x  e.  ~P X )  ->  { s  e.  C  |  x  C_  s } 
C_  C )
4 mre1cl 13490 . . . . . 6  |-  ( C  e.  (Moore `  X
)  ->  X  e.  C )
54adantr 453 . . . . 5  |-  ( ( C  e.  (Moore `  X )  /\  x  e.  ~P X )  ->  X  e.  C )
6 elpwi 3634 . . . . . 6  |-  ( x  e.  ~P X  ->  x  C_  X )
76adantl 454 . . . . 5  |-  ( ( C  e.  (Moore `  X )  /\  x  e.  ~P X )  ->  x  C_  X )
8 sseq2 3201 . . . . . 6  |-  ( s  =  X  ->  (
x  C_  s  <->  x  C_  X
) )
98elrab 2924 . . . . 5  |-  ( X  e.  { s  e.  C  |  x  C_  s }  <->  ( X  e.  C  /\  x  C_  X ) )
105, 7, 9sylanbrc 647 . . . 4  |-  ( ( C  e.  (Moore `  X )  /\  x  e.  ~P X )  ->  X  e.  { s  e.  C  |  x  C_  s } )
11 ne0i 3462 . . . 4  |-  ( X  e.  { s  e.  C  |  x  C_  s }  ->  { s  e.  C  |  x 
C_  s }  =/=  (/) )
1210, 11syl 17 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  x  e.  ~P X )  ->  { s  e.  C  |  x  C_  s }  =/=  (/) )
13 mreintcl 13491 . . 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 2284 . 2  |-  ( x  e.  ~P X  |->  |^|
{ s  e.  C  |  x  C_  s } )  =  ( x  e.  ~P X  |->  |^|
{ s  e.  C  |  x  C_  s } )
1614, 15fmptd 5645 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 6    /\ wa 360    e. wcel 1685    =/= wne 2447   {crab 2548    C_ wss 3153   (/)c0 3456   ~Pcpw 3626   |^|cint 3863    e. cmpt 4078   -->wf 5217   ` cfv 5221  Moorecmre 13478
This theorem is referenced by:  fnmrc  13503  mrcfval  13504  mrcf  13505
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-rab 2553  df-v 2791  df-sbc 2993  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-int 3864  df-br 4025  df-opab 4079  df-mpt 4080  df-id 4308  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-fv 5229  df-mre 13482
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