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Theorem mrcflem 13470
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 3233 . . . 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 13458 . . . . . 6  |-  ( C  e.  (Moore `  X
)  ->  X  e.  C )
54adantr 453 . . . . 5  |-  ( ( C  e.  (Moore `  X )  /\  x  e.  ~P X )  ->  X  e.  C )
6 elpwi 3607 . . . . . 6  |-  ( x  e.  ~P X  ->  x  C_  X )
76adantl 454 . . . . 5  |-  ( ( C  e.  (Moore `  X )  /\  x  e.  ~P X )  ->  x  C_  X )
8 sseq2 3175 . . . . . 6  |-  ( s  =  X  ->  (
x  C_  s  <->  x  C_  X
) )
98elrab 2898 . . . . 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 3436 . . . 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 13459 . . 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 2258 . 2  |-  ( x  e.  ~P X  |->  |^|
{ s  e.  C  |  x  C_  s } )  =  ( x  e.  ~P X  |->  |^|
{ s  e.  C  |  x  C_  s } )
1614, 15fmptd 5618 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 2421   {crab 2522    C_ wss 3127   (/)c0 3430   ~Pcpw 3599   |^|cint 3836    e. cmpt 4051   -->wf 4669   ` cfv 4673  Moorecmre 13446
This theorem is referenced by:  fnmrc  13471  mrcfval  13472  mrcf  13473
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 1927  ax-ext 2239  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484
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 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-ral 2523  df-rex 2524  df-rab 2527  df-v 2765  df-sbc 2967  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-op 3623  df-uni 3802  df-int 3837  df-br 3998  df-opab 4052  df-mpt 4053  df-id 4281  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-fv 4689  df-mre 13450
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