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Theorem mrcflem 13380
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 3179 . . . 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 13368 . . . . . 6  |-  ( C  e.  (Moore `  X
)  ->  X  e.  C )
54adantr 453 . . . . 5  |-  ( ( C  e.  (Moore `  X )  /\  x  e.  ~P X )  ->  X  e.  C )
6 elpwi 3538 . . . . . 6  |-  ( x  e.  ~P X  ->  x  C_  X )
76adantl 454 . . . . 5  |-  ( ( C  e.  (Moore `  X )  /\  x  e.  ~P X )  ->  x  C_  X )
8 sseq2 3121 . . . . . 6  |-  ( s  =  X  ->  (
x  C_  s  <->  x  C_  X
) )
98elrab 2860 . . . . 5  |-  ( X  e.  { s  e.  C  |  x  C_  s }  <->  ( X  e.  C  /\  x  C_  X ) )
105, 7, 9sylanbrc 648 . . . 4  |-  ( ( C  e.  (Moore `  X )  /\  x  e.  ~P X )  ->  X  e.  { s  e.  C  |  x  C_  s } )
11 ne0i 3368 . . . 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 13369 . . 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 1187 . 2  |-  ( ( C  e.  (Moore `  X )  /\  x  e.  ~P X )  ->  |^| { s  e.  C  |  x  C_  s }  e.  C )
15 eqid 2253 . 2  |-  ( x  e.  ~P X  |->  |^|
{ s  e.  C  |  x  C_  s } )  =  ( x  e.  ~P X  |->  |^|
{ s  e.  C  |  x  C_  s } )
1614, 15fmptd 5536 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 1621    =/= wne 2412   {crab 2512    C_ wss 3078   (/)c0 3362   ~Pcpw 3530   |^|cint 3760    e. cmpt 3974   -->wf 4588   ` cfv 4592  Moorecmre 13358
This theorem is referenced by:  fnmrc  13381  mrcfval  13382  mrcf  13383
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-rab 2516  df-v 2729  df-sbc 2922  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-int 3761  df-br 3921  df-opab 3975  df-mpt 3976  df-id 4202  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-fv 4608  df-mre 13361
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