MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  mrcun Unicode version

Theorem mrcun 13540
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 955 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  X  /\  V  C_  X )  ->  C  e.  (Moore `  X )
)
2 mre1cl 13512 . . . . . . 7  |-  ( C  e.  (Moore `  X
)  ->  X  e.  C )
3 elpw2g 4190 . . . . . . 7  |-  ( X  e.  C  ->  ( U  e.  ~P X  <->  U 
C_  X ) )
42, 3syl 15 . . . . . 6  |-  ( C  e.  (Moore `  X
)  ->  ( U  e.  ~P X  <->  U  C_  X
) )
54biimpar 471 . . . . 5  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  X )  ->  U  e.  ~P X )
653adant3 975 . . . 4  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  X  /\  V  C_  X )  ->  U  e.  ~P X )
7 elpw2g 4190 . . . . . . 7  |-  ( X  e.  C  ->  ( V  e.  ~P X  <->  V 
C_  X ) )
82, 7syl 15 . . . . . 6  |-  ( C  e.  (Moore `  X
)  ->  ( V  e.  ~P X  <->  V  C_  X
) )
98biimpar 471 . . . . 5  |-  ( ( C  e.  (Moore `  X )  /\  V  C_  X )  ->  V  e.  ~P X )
1093adant2 974 . . . 4  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  X  /\  V  C_  X )  ->  V  e.  ~P X )
11 prssi 3787 . . . 4  |-  ( ( U  e.  ~P X  /\  V  e.  ~P X )  ->  { U ,  V }  C_  ~P X )
126, 10, 11syl2anc 642 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  X  /\  V  C_  X )  ->  { U ,  V }  C_  ~P X )
13 mrcfval.f . . . 4  |-  F  =  (mrCls `  C )
1413mrcuni 13539 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  { U ,  V }  C_ 
~P X )  -> 
( F `  U. { U ,  V }
)  =  ( F `
 U. ( F
" { U ,  V } ) ) )
151, 12, 14syl2anc 642 . 2  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  X  /\  V  C_  X )  ->  ( F `  U. { U ,  V } )  =  ( F `  U. ( F " { U ,  V } ) ) )
16 uniprg 3858 . . . 4  |-  ( ( U  e.  ~P X  /\  V  e.  ~P X )  ->  U. { U ,  V }  =  ( U  u.  V ) )
176, 10, 16syl2anc 642 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  X  /\  V  C_  X )  ->  U. { U ,  V }  =  ( U  u.  V ) )
1817fveq2d 5545 . 2  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  X  /\  V  C_  X )  ->  ( F `  U. { U ,  V } )  =  ( F `  ( U  u.  V )
) )
1913mrcf 13527 . . . . . . . 8  |-  ( C  e.  (Moore `  X
)  ->  F : ~P X --> C )
20 ffn 5405 . . . . . . . 8  |-  ( F : ~P X --> C  ->  F  Fn  ~P X
)
2119, 20syl 15 . . . . . . 7  |-  ( C  e.  (Moore `  X
)  ->  F  Fn  ~P X )
22213ad2ant1 976 . . . . . 6  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  X  /\  V  C_  X )  ->  F  Fn  ~P X )
23 fnimapr 5599 . . . . . 6  |-  ( ( F  Fn  ~P X  /\  U  e.  ~P X  /\  V  e.  ~P X )  ->  ( F " { U ,  V } )  =  {
( F `  U
) ,  ( F `
 V ) } )
2422, 6, 10, 23syl3anc 1182 . . . . 5  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  X  /\  V  C_  X )  ->  ( F " { U ,  V } )  =  {
( F `  U
) ,  ( F `
 V ) } )
2524unieqd 3854 . . . 4  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  X  /\  V  C_  X )  ->  U. ( F " { U ,  V } )  =  U. { ( F `  U ) ,  ( F `  V ) } )
26 fvex 5555 . . . . 5  |-  ( F `
 U )  e. 
_V
27 fvex 5555 . . . . 5  |-  ( F `
 V )  e. 
_V
2826, 27unipr 3857 . . . 4  |-  U. {
( F `  U
) ,  ( F `
 V ) }  =  ( ( F `
 U )  u.  ( F `  V
) )
2925, 28syl6eq 2344 . . 3  |-  ( ( C  e.  (Moore `  X )  /\  U  C_  X  /\  V  C_  X )  ->  U. ( F " { U ,  V } )  =  ( ( F `  U
)  u.  ( F `
 V ) ) )
3029fveq2d 5545 . 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 2336 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 176    /\ w3a 934    = wceq 1632    e. wcel 1696    u. cun 3163    C_ wss 3165   ~Pcpw 3638   {cpr 3654   U.cuni 3843   "cima 4708    Fn wfn 5266   -->wf 5267   ` cfv 5271  Moorecmre 13500  mrClscmrc 13501
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279  df-mre 13504  df-mrc 13505
  Copyright terms: Public domain W3C validator