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Theorem mrcun 13852
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 958 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  X  /\  V  C_  X )  ->  C  e.  (Moore `  X )
)
2 mre1cl 13824 . . . . . . 7  |-  ( C  e.  (Moore `  X
)  ->  X  e.  C )
3 elpw2g 4366 . . . . . . 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 473 . . . . 5  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  X )  ->  U  e.  ~P X )
653adant3 978 . . . 4  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  X  /\  V  C_  X )  ->  U  e.  ~P X )
7 elpw2g 4366 . . . . . . 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 473 . . . . 5  |-  ( ( C  e.  (Moore `  X )  /\  V  C_  X )  ->  V  e.  ~P X )
1093adant2 977 . . . 4  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  X  /\  V  C_  X )  ->  V  e.  ~P X )
11 prssi 3956 . . . 4  |-  ( ( U  e.  ~P X  /\  V  e.  ~P X )  ->  { U ,  V }  C_  ~P X )
126, 10, 11syl2anc 644 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  X  /\  V  C_  X )  ->  { U ,  V }  C_  ~P X )
13 mrcfval.f . . . 4  |-  F  =  (mrCls `  C )
1413mrcuni 13851 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  { U ,  V }  C_ 
~P X )  -> 
( F `  U. { U ,  V }
)  =  ( F `
 U. ( F
" { U ,  V } ) ) )
151, 12, 14syl2anc 644 . 2  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  X  /\  V  C_  X )  ->  ( F `  U. { U ,  V } )  =  ( F `  U. ( F " { U ,  V } ) ) )
16 uniprg 4032 . . . 4  |-  ( ( U  e.  ~P X  /\  V  e.  ~P X )  ->  U. { U ,  V }  =  ( U  u.  V ) )
176, 10, 16syl2anc 644 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  X  /\  V  C_  X )  ->  U. { U ,  V }  =  ( U  u.  V ) )
1817fveq2d 5735 . 2  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  X  /\  V  C_  X )  ->  ( F `  U. { U ,  V } )  =  ( F `  ( U  u.  V )
) )
1913mrcf 13839 . . . . . . . 8  |-  ( C  e.  (Moore `  X
)  ->  F : ~P X --> C )
20 ffn 5594 . . . . . . . 8  |-  ( F : ~P X --> C  ->  F  Fn  ~P X
)
2119, 20syl 16 . . . . . . 7  |-  ( C  e.  (Moore `  X
)  ->  F  Fn  ~P X )
22213ad2ant1 979 . . . . . 6  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  X  /\  V  C_  X )  ->  F  Fn  ~P X )
23 fnimapr 5790 . . . . . 6  |-  ( ( F  Fn  ~P X  /\  U  e.  ~P X  /\  V  e.  ~P X )  ->  ( F " { U ,  V } )  =  {
( F `  U
) ,  ( F `
 V ) } )
2422, 6, 10, 23syl3anc 1185 . . . . 5  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  X  /\  V  C_  X )  ->  ( F " { U ,  V } )  =  {
( F `  U
) ,  ( F `
 V ) } )
2524unieqd 4028 . . . 4  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  X  /\  V  C_  X )  ->  U. ( F " { U ,  V } )  =  U. { ( F `  U ) ,  ( F `  V ) } )
26 fvex 5745 . . . . 5  |-  ( F `
 U )  e. 
_V
27 fvex 5745 . . . . 5  |-  ( F `
 V )  e. 
_V
2826, 27unipr 4031 . . . 4  |-  U. {
( F `  U
) ,  ( F `
 V ) }  =  ( ( F `
 U )  u.  ( F `  V
) )
2925, 28syl6eq 2486 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  X  /\  V  C_  X )  ->  U. ( F " { U ,  V } )  =  ( ( F `  U
)  u.  ( F `
 V ) ) )
3029fveq2d 5735 . 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 2478 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 set class
Syntax hints:    -> wi 4    <-> wb 178    /\ w3a 937    = wceq 1653    e. wcel 1726    u. cun 3320    C_ wss 3322   ~Pcpw 3801   {cpr 3817   U.cuni 4017   "cima 4884    Fn wfn 5452   -->wf 5453   ` cfv 5457  Moorecmre 13812  mrClscmrc 13813
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
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-csb 3254  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-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-fv 5465  df-mre 13816  df-mrc 13817
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