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Theorem mrcun 14560
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 988 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  X  /\  V  C_  X )  ->  C  e.  (Moore `  X )
)
2 mre1cl 14532 . . . . . . 7  |-  ( C  e.  (Moore `  X
)  ->  X  e.  C )
3 elpw2g 4455 . . . . . . 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 1008 . . . 4  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  X  /\  V  C_  X )  ->  U  e.  ~P X )
7 elpw2g 4455 . . . . . . 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 1007 . . . 4  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  X  /\  V  C_  X )  ->  V  e.  ~P X )
11 prssi 4029 . . . 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 14559 . . 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 4105 . . . 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 5695 . 2  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  X  /\  V  C_  X )  ->  ( F `  U. { U ,  V } )  =  ( F `  ( U  u.  V )
) )
1913mrcf 14547 . . . . . . . 8  |-  ( C  e.  (Moore `  X
)  ->  F : ~P X --> C )
20 ffn 5559 . . . . . . . 8  |-  ( F : ~P X --> C  ->  F  Fn  ~P X
)
2119, 20syl 16 . . . . . . 7  |-  ( C  e.  (Moore `  X
)  ->  F  Fn  ~P X )
22213ad2ant1 1009 . . . . . 6  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  X  /\  V  C_  X )  ->  F  Fn  ~P X )
23 fnimapr 5755 . . . . . 6  |-  ( ( F  Fn  ~P X  /\  U  e.  ~P X  /\  V  e.  ~P X )  ->  ( F " { U ,  V } )  =  {
( F `  U
) ,  ( F `
 V ) } )
2422, 6, 10, 23syl3anc 1218 . . . . 5  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  X  /\  V  C_  X )  ->  ( F " { U ,  V } )  =  {
( F `  U
) ,  ( F `
 V ) } )
2524unieqd 4101 . . . 4  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  X  /\  V  C_  X )  ->  U. ( F " { U ,  V } )  =  U. { ( F `  U ) ,  ( F `  V ) } )
26 fvex 5701 . . . . 5  |-  ( F `
 U )  e. 
_V
27 fvex 5701 . . . . 5  |-  ( F `
 V )  e. 
_V
2826, 27unipr 4104 . . . 4  |-  U. {
( F `  U
) ,  ( F `
 V ) }  =  ( ( F `
 U )  u.  ( F `  V
) )
2925, 28syl6eq 2491 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  X  /\  V  C_  X )  ->  U. ( F " { U ,  V } )  =  ( ( F `  U
)  u.  ( F `
 V ) ) )
3029fveq2d 5695 . 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 2483 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 965    = wceq 1369    e. wcel 1756    u. cun 3326    C_ wss 3328   ~Pcpw 3860   {cpr 3879   U.cuni 4091   "cima 4843    Fn wfn 5413   -->wf 5414   ` cfv 5418  Moorecmre 14520  mrClscmrc 14521
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-int 4129  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-fv 5426  df-mre 14524  df-mrc 14525
This theorem is referenced by: (None)
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