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Theorem pats 32927
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 2979 . 2  |-  ( K  e.  D  ->  K  e.  _V )
2 patoms.a . . 3  |-  A  =  ( Atoms `  K )
3 fveq2 5689 . . . . . 6  |-  ( p  =  K  ->  ( Base `  p )  =  ( Base `  K
) )
4 patoms.b . . . . . 6  |-  B  =  ( Base `  K
)
53, 4syl6eqr 2491 . . . . 5  |-  ( p  =  K  ->  ( Base `  p )  =  B )
6 fveq2 5689 . . . . . . . 8  |-  ( p  =  K  ->  (  <o  `  p )  =  (  <o  `  K )
)
7 patoms.c . . . . . . . 8  |-  C  =  (  <o  `  K )
86, 7syl6eqr 2491 . . . . . . 7  |-  ( p  =  K  ->  (  <o  `  p )  =  C )
98breqd 4301 . . . . . 6  |-  ( p  =  K  ->  (
( 0. `  p
) (  <o  `  p
) x  <->  ( 0. `  p ) C x ) )
10 fveq2 5689 . . . . . . . 8  |-  ( p  =  K  ->  ( 0. `  p )  =  ( 0. `  K
) )
11 patoms.z . . . . . . . 8  |-  .0.  =  ( 0. `  K )
1210, 11syl6eqr 2491 . . . . . . 7  |-  ( p  =  K  ->  ( 0. `  p )  =  .0.  )
1312breq1d 4300 . . . . . 6  |-  ( p  =  K  ->  (
( 0. `  p
) C x  <->  .0.  C x ) )
149, 13bitrd 253 . . . . 5  |-  ( p  =  K  ->  (
( 0. `  p
) (  <o  `  p
) x  <->  .0.  C x ) )
155, 14rabeqbidv 2965 . . . 4  |-  ( p  =  K  ->  { x  e.  ( Base `  p
)  |  ( 0.
`  p ) ( 
<o  `  p ) x }  =  { x  e.  B  |  .0.  C x } )
16 df-ats 32909 . . . 4  |-  Atoms  =  ( p  e.  _V  |->  { x  e.  ( Base `  p )  |  ( 0. `  p ) (  <o  `  p )
x } )
17 fvex 5699 . . . . . 6  |-  ( Base `  K )  e.  _V
184, 17eqeltri 2511 . . . . 5  |-  B  e. 
_V
1918rabex 4441 . . . 4  |-  { x  e.  B  |  .0.  C x }  e.  _V
2015, 16, 19fvmpt 5772 . . 3  |-  ( K  e.  _V  ->  ( Atoms `  K )  =  { x  e.  B  |  .0.  C x }
)
212, 20syl5eq 2485 . 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 setvar class
Syntax hints:    -> wi 4    = wceq 1369    e. wcel 1756   {crab 2717   _Vcvv 2970   class class class wbr 4290   ` cfv 5416   Basecbs 14172   0.cp0 15205    <o ccvr 32904   Atomscatm 32905
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-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-sep 4411  ax-nul 4419  ax-pr 4529
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 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-sbc 3185  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-nul 3636  df-if 3790  df-sn 3876  df-pr 3878  df-op 3882  df-uni 4090  df-br 4291  df-opab 4349  df-mpt 4350  df-id 4634  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-iota 5379  df-fun 5418  df-fv 5424  df-ats 32909
This theorem is referenced by:  isat  32928
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