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Theorem pats 35111
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 3118 . 2  |-  ( K  e.  D  ->  K  e.  _V )
2 patoms.a . . 3  |-  A  =  ( Atoms `  K )
3 fveq2 5872 . . . . . 6  |-  ( p  =  K  ->  ( Base `  p )  =  ( Base `  K
) )
4 patoms.b . . . . . 6  |-  B  =  ( Base `  K
)
53, 4syl6eqr 2516 . . . . 5  |-  ( p  =  K  ->  ( Base `  p )  =  B )
6 fveq2 5872 . . . . . . . 8  |-  ( p  =  K  ->  (  <o  `  p )  =  (  <o  `  K )
)
7 patoms.c . . . . . . . 8  |-  C  =  (  <o  `  K )
86, 7syl6eqr 2516 . . . . . . 7  |-  ( p  =  K  ->  (  <o  `  p )  =  C )
98breqd 4467 . . . . . 6  |-  ( p  =  K  ->  (
( 0. `  p
) (  <o  `  p
) x  <->  ( 0. `  p ) C x ) )
10 fveq2 5872 . . . . . . . 8  |-  ( p  =  K  ->  ( 0. `  p )  =  ( 0. `  K
) )
11 patoms.z . . . . . . . 8  |-  .0.  =  ( 0. `  K )
1210, 11syl6eqr 2516 . . . . . . 7  |-  ( p  =  K  ->  ( 0. `  p )  =  .0.  )
1312breq1d 4466 . . . . . 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 3104 . . . 4  |-  ( p  =  K  ->  { x  e.  ( Base `  p
)  |  ( 0.
`  p ) ( 
<o  `  p ) x }  =  { x  e.  B  |  .0.  C x } )
16 df-ats 35093 . . . 4  |-  Atoms  =  ( p  e.  _V  |->  { x  e.  ( Base `  p )  |  ( 0. `  p ) (  <o  `  p )
x } )
17 fvex 5882 . . . . . 6  |-  ( Base `  K )  e.  _V
184, 17eqeltri 2541 . . . . 5  |-  B  e. 
_V
1918rabex 4607 . . . 4  |-  { x  e.  B  |  .0.  C x }  e.  _V
2015, 16, 19fvmpt 5956 . . 3  |-  ( K  e.  _V  ->  ( Atoms `  K )  =  { x  e.  B  |  .0.  C x }
)
212, 20syl5eq 2510 . 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 1395    e. wcel 1819   {crab 2811   _Vcvv 3109   class class class wbr 4456   ` cfv 5594   Basecbs 14643   0.cp0 15793    <o ccvr 35088   Atomscatm 35089
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-iota 5557  df-fun 5596  df-fv 5602  df-ats 35093
This theorem is referenced by:  isat  35112
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