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Theorem pats 30145
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 2966 . 2  |-  ( K  e.  D  ->  K  e.  _V )
2 patoms.a . . 3  |-  A  =  ( Atoms `  K )
3 fveq2 5730 . . . . . 6  |-  ( p  =  K  ->  ( Base `  p )  =  ( Base `  K
) )
4 patoms.b . . . . . 6  |-  B  =  ( Base `  K
)
53, 4syl6eqr 2488 . . . . 5  |-  ( p  =  K  ->  ( Base `  p )  =  B )
6 fveq2 5730 . . . . . . . 8  |-  ( p  =  K  ->  (  <o  `  p )  =  (  <o  `  K )
)
7 patoms.c . . . . . . . 8  |-  C  =  (  <o  `  K )
86, 7syl6eqr 2488 . . . . . . 7  |-  ( p  =  K  ->  (  <o  `  p )  =  C )
98breqd 4225 . . . . . 6  |-  ( p  =  K  ->  (
( 0. `  p
) (  <o  `  p
) x  <->  ( 0. `  p ) C x ) )
10 fveq2 5730 . . . . . . . 8  |-  ( p  =  K  ->  ( 0. `  p )  =  ( 0. `  K
) )
11 patoms.z . . . . . . . 8  |-  .0.  =  ( 0. `  K )
1210, 11syl6eqr 2488 . . . . . . 7  |-  ( p  =  K  ->  ( 0. `  p )  =  .0.  )
1312breq1d 4224 . . . . . 6  |-  ( p  =  K  ->  (
( 0. `  p
) C x  <->  .0.  C x ) )
149, 13bitrd 246 . . . . 5  |-  ( p  =  K  ->  (
( 0. `  p
) (  <o  `  p
) x  <->  .0.  C x ) )
155, 14rabeqbidv 2953 . . . 4  |-  ( p  =  K  ->  { x  e.  ( Base `  p
)  |  ( 0.
`  p ) ( 
<o  `  p ) x }  =  { x  e.  B  |  .0.  C x } )
16 df-ats 30127 . . . 4  |-  Atoms  =  ( p  e.  _V  |->  { x  e.  ( Base `  p )  |  ( 0. `  p ) (  <o  `  p )
x } )
17 fvex 5744 . . . . . 6  |-  ( Base `  K )  e.  _V
184, 17eqeltri 2508 . . . . 5  |-  B  e. 
_V
1918rabex 4356 . . . 4  |-  { x  e.  B  |  .0.  C x }  e.  _V
2015, 16, 19fvmpt 5808 . . 3  |-  ( K  e.  _V  ->  ( Atoms `  K )  =  { x  e.  B  |  .0.  C x }
)
212, 20syl5eq 2482 . 2  |-  ( K  e.  _V  ->  A  =  { x  e.  B  |  .0.  C x }
)
221, 21syl 16 1  |-  ( K  e.  D  ->  A  =  { x  e.  B  |  .0.  C x }
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653    e. wcel 1726   {crab 2711   _Vcvv 2958   class class class wbr 4214   ` cfv 5456   Basecbs 13471   0.cp0 14468    <o ccvr 30122   Atomscatm 30123
This theorem is referenced by:  isat  30146
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405
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-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-iota 5420  df-fun 5458  df-fv 5464  df-ats 30127
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