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Theorem mrcflem 13833
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 3430 . . . 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 13821 . . . . . 6  |-  ( C  e.  (Moore `  X
)  ->  X  e.  C )
54adantr 453 . . . . 5  |-  ( ( C  e.  (Moore `  X )  /\  x  e.  ~P X )  ->  X  e.  C )
6 elpwi 3809 . . . . . 6  |-  ( x  e.  ~P X  ->  x  C_  X )
76adantl 454 . . . . 5  |-  ( ( C  e.  (Moore `  X )  /\  x  e.  ~P X )  ->  x  C_  X )
8 sseq2 3372 . . . . . 6  |-  ( s  =  X  ->  (
x  C_  s  <->  x  C_  X
) )
98elrab 3094 . . . . 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 3636 . . . 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 13822 . . 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 1185 . 2  |-  ( ( C  e.  (Moore `  X )  /\  x  e.  ~P X )  ->  |^| { s  e.  C  |  x  C_  s }  e.  C )
15 eqid 2438 . 2  |-  ( x  e.  ~P X  |->  |^|
{ s  e.  C  |  x  C_  s } )  =  ( x  e.  ~P X  |->  |^|
{ s  e.  C  |  x  C_  s } )
1614, 15fmptd 5895 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 360    e. wcel 1726    =/= wne 2601   {crab 2711    C_ wss 3322   (/)c0 3630   ~Pcpw 3801   |^|cint 4052    e. cmpt 4268   -->wf 5452   ` cfv 5456  Moorecmre 13809
This theorem is referenced by:  fnmrc  13834  mrcfval  13835  mrcf  13836
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-int 4053  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-fv 5464  df-mre 13813
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