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Theorem pats 30097
Description: The set of atoms in a poset. (Contributed by NM, 18-Sep-2011.)
Hypotheses
Ref Expression
patoms.b  |-  B  =  ( Base `  K
)
patoms.z  |-  .0.  =  ( 0. `  K )
patoms.c  |-  C  =  (  <o  `  K )
patoms.a  |-  A  =  ( Atoms `  K )
Assertion
Ref Expression
pats  |-  ( K  e.  D  ->  A  =  { x  e.  B  |  .0.  C x }
)
Distinct variable groups:    x, B    x, K
Allowed substitution hints:    A( x)    C( x)    D( x)    .0. ( x)

Proof of Theorem pats
Dummy variable  p is distinct from all other variables.
StepHypRef Expression
1 elex 2809 . 2  |-  ( K  e.  D  ->  K  e.  _V )
2 patoms.a . . 3  |-  A  =  ( Atoms `  K )
3 fveq2 5541 . . . . . 6  |-  ( p  =  K  ->  ( Base `  p )  =  ( Base `  K
) )
4 patoms.b . . . . . 6  |-  B  =  ( Base `  K
)
53, 4syl6eqr 2346 . . . . 5  |-  ( p  =  K  ->  ( Base `  p )  =  B )
6 fveq2 5541 . . . . . . . 8  |-  ( p  =  K  ->  (  <o  `  p )  =  (  <o  `  K )
)
7 patoms.c . . . . . . . 8  |-  C  =  (  <o  `  K )
86, 7syl6eqr 2346 . . . . . . 7  |-  ( p  =  K  ->  (  <o  `  p )  =  C )
98breqd 4050 . . . . . 6  |-  ( p  =  K  ->  (
( 0. `  p
) (  <o  `  p
) x  <->  ( 0. `  p ) C x ) )
10 fveq2 5541 . . . . . . . 8  |-  ( p  =  K  ->  ( 0. `  p )  =  ( 0. `  K
) )
11 patoms.z . . . . . . . 8  |-  .0.  =  ( 0. `  K )
1210, 11syl6eqr 2346 . . . . . . 7  |-  ( p  =  K  ->  ( 0. `  p )  =  .0.  )
1312breq1d 4049 . . . . . 6  |-  ( p  =  K  ->  (
( 0. `  p
) C x  <->  .0.  C x ) )
149, 13bitrd 244 . . . . 5  |-  ( p  =  K  ->  (
( 0. `  p
) (  <o  `  p
) x  <->  .0.  C x ) )
155, 14rabeqbidv 2796 . . . 4  |-  ( p  =  K  ->  { x  e.  ( Base `  p
)  |  ( 0.
`  p ) ( 
<o  `  p ) x }  =  { x  e.  B  |  .0.  C x } )
16 df-ats 30079 . . . 4  |-  Atoms  =  ( p  e.  _V  |->  { x  e.  ( Base `  p )  |  ( 0. `  p ) (  <o  `  p )
x } )
17 fvex 5555 . . . . . 6  |-  ( Base `  K )  e.  _V
184, 17eqeltri 2366 . . . . 5  |-  B  e. 
_V
1918rabex 4181 . . . 4  |-  { x  e.  B  |  .0.  C x }  e.  _V
2015, 16, 19fvmpt 5618 . . 3  |-  ( K  e.  _V  ->  ( Atoms `  K )  =  { x  e.  B  |  .0.  C x }
)
212, 20syl5eq 2340 . 2  |-  ( K  e.  _V  ->  A  =  { x  e.  B  |  .0.  C x }
)
221, 21syl 15 1  |-  ( K  e.  D  ->  A  =  { x  e.  B  |  .0.  C x }
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    e. wcel 1696   {crab 2560   _Vcvv 2801   class class class wbr 4039   ` cfv 5271   Basecbs 13164   0.cp0 14159    <o ccvr 30074   Atomscatm 30075
This theorem is referenced by:  isat  30098
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-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
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-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  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-iota 5235  df-fun 5273  df-fv 5279  df-ats 30079
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