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Theorem mrcun 14894
Description: Idempotence of closure under a pair union. (Contributed by Stefan O'Rear, 31-Jan-2015.)
Hypothesis
Ref Expression
mrcfval.f  |-  F  =  (mrCls `  C )
Assertion
Ref Expression
mrcun  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  X  /\  V  C_  X )  ->  ( F `  ( U  u.  V ) )  =  ( F `  (
( F `  U
)  u.  ( F `
 V ) ) ) )

Proof of Theorem mrcun
StepHypRef Expression
1 simp1 996 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  X  /\  V  C_  X )  ->  C  e.  (Moore `  X )
)
2 mre1cl 14866 . . . . . . 7  |-  ( C  e.  (Moore `  X
)  ->  X  e.  C )
3 elpw2g 4616 . . . . . . 7  |-  ( X  e.  C  ->  ( U  e.  ~P X  <->  U 
C_  X ) )
42, 3syl 16 . . . . . 6  |-  ( C  e.  (Moore `  X
)  ->  ( U  e.  ~P X  <->  U  C_  X
) )
54biimpar 485 . . . . 5  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  X )  ->  U  e.  ~P X )
653adant3 1016 . . . 4  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  X  /\  V  C_  X )  ->  U  e.  ~P X )
7 elpw2g 4616 . . . . . . 7  |-  ( X  e.  C  ->  ( V  e.  ~P X  <->  V 
C_  X ) )
82, 7syl 16 . . . . . 6  |-  ( C  e.  (Moore `  X
)  ->  ( V  e.  ~P X  <->  V  C_  X
) )
98biimpar 485 . . . . 5  |-  ( ( C  e.  (Moore `  X )  /\  V  C_  X )  ->  V  e.  ~P X )
1093adant2 1015 . . . 4  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  X  /\  V  C_  X )  ->  V  e.  ~P X )
11 prssi 4189 . . . 4  |-  ( ( U  e.  ~P X  /\  V  e.  ~P X )  ->  { U ,  V }  C_  ~P X )
126, 10, 11syl2anc 661 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  X  /\  V  C_  X )  ->  { U ,  V }  C_  ~P X )
13 mrcfval.f . . . 4  |-  F  =  (mrCls `  C )
1413mrcuni 14893 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  { U ,  V }  C_ 
~P X )  -> 
( F `  U. { U ,  V }
)  =  ( F `
 U. ( F
" { U ,  V } ) ) )
151, 12, 14syl2anc 661 . 2  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  X  /\  V  C_  X )  ->  ( F `  U. { U ,  V } )  =  ( F `  U. ( F " { U ,  V } ) ) )
16 uniprg 4265 . . . 4  |-  ( ( U  e.  ~P X  /\  V  e.  ~P X )  ->  U. { U ,  V }  =  ( U  u.  V ) )
176, 10, 16syl2anc 661 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  X  /\  V  C_  X )  ->  U. { U ,  V }  =  ( U  u.  V ) )
1817fveq2d 5876 . 2  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  X  /\  V  C_  X )  ->  ( F `  U. { U ,  V } )  =  ( F `  ( U  u.  V )
) )
1913mrcf 14881 . . . . . . . 8  |-  ( C  e.  (Moore `  X
)  ->  F : ~P X --> C )
20 ffn 5737 . . . . . . . 8  |-  ( F : ~P X --> C  ->  F  Fn  ~P X
)
2119, 20syl 16 . . . . . . 7  |-  ( C  e.  (Moore `  X
)  ->  F  Fn  ~P X )
22213ad2ant1 1017 . . . . . 6  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  X  /\  V  C_  X )  ->  F  Fn  ~P X )
23 fnimapr 5938 . . . . . 6  |-  ( ( F  Fn  ~P X  /\  U  e.  ~P X  /\  V  e.  ~P X )  ->  ( F " { U ,  V } )  =  {
( F `  U
) ,  ( F `
 V ) } )
2422, 6, 10, 23syl3anc 1228 . . . . 5  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  X  /\  V  C_  X )  ->  ( F " { U ,  V } )  =  {
( F `  U
) ,  ( F `
 V ) } )
2524unieqd 4261 . . . 4  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  X  /\  V  C_  X )  ->  U. ( F " { U ,  V } )  =  U. { ( F `  U ) ,  ( F `  V ) } )
26 fvex 5882 . . . . 5  |-  ( F `
 U )  e. 
_V
27 fvex 5882 . . . . 5  |-  ( F `
 V )  e. 
_V
2826, 27unipr 4264 . . . 4  |-  U. {
( F `  U
) ,  ( F `
 V ) }  =  ( ( F `
 U )  u.  ( F `  V
) )
2925, 28syl6eq 2524 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  X  /\  V  C_  X )  ->  U. ( F " { U ,  V } )  =  ( ( F `  U
)  u.  ( F `
 V ) ) )
3029fveq2d 5876 . 2  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  X  /\  V  C_  X )  ->  ( F `  U. ( F
" { U ,  V } ) )  =  ( F `  (
( F `  U
)  u.  ( F `
 V ) ) ) )
3115, 18, 303eqtr3d 2516 1  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  X  /\  V  C_  X )  ->  ( F `  ( U  u.  V ) )  =  ( F `  (
( F `  U
)  u.  ( F `
 V ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ w3a 973    = wceq 1379    e. wcel 1767    u. cun 3479    C_ wss 3481   ~Pcpw 4016   {cpr 4035   U.cuni 4251   "cima 5008    Fn wfn 5589   -->wf 5590   ` cfv 5594  Moorecmre 14854  mrClscmrc 14855
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-int 4289  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-fv 5602  df-mre 14858  df-mrc 14859
This theorem is referenced by: (None)
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