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Theorem pats 33590
 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 = ( ⋖ ‘𝐾)
patoms.a 𝐴 = (Atoms‘𝐾)
Assertion
Ref Expression
pats (𝐾𝐷𝐴 = {𝑥𝐵0 𝐶𝑥})
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 3185 . 2 (𝐾𝐷𝐾 ∈ V)
2 patoms.a . . 3 𝐴 = (Atoms‘𝐾)
3 fveq2 6103 . . . . . 6 (𝑝 = 𝐾 → (Base‘𝑝) = (Base‘𝐾))
4 patoms.b . . . . . 6 𝐵 = (Base‘𝐾)
53, 4syl6eqr 2662 . . . . 5 (𝑝 = 𝐾 → (Base‘𝑝) = 𝐵)
6 fveq2 6103 . . . . . . . 8 (𝑝 = 𝐾 → ( ⋖ ‘𝑝) = ( ⋖ ‘𝐾))
7 patoms.c . . . . . . . 8 𝐶 = ( ⋖ ‘𝐾)
86, 7syl6eqr 2662 . . . . . . 7 (𝑝 = 𝐾 → ( ⋖ ‘𝑝) = 𝐶)
98breqd 4594 . . . . . 6 (𝑝 = 𝐾 → ((0.‘𝑝)( ⋖ ‘𝑝)𝑥 ↔ (0.‘𝑝)𝐶𝑥))
10 fveq2 6103 . . . . . . . 8 (𝑝 = 𝐾 → (0.‘𝑝) = (0.‘𝐾))
11 patoms.z . . . . . . . 8 0 = (0.‘𝐾)
1210, 11syl6eqr 2662 . . . . . . 7 (𝑝 = 𝐾 → (0.‘𝑝) = 0 )
1312breq1d 4593 . . . . . 6 (𝑝 = 𝐾 → ((0.‘𝑝)𝐶𝑥0 𝐶𝑥))
149, 13bitrd 267 . . . . 5 (𝑝 = 𝐾 → ((0.‘𝑝)( ⋖ ‘𝑝)𝑥0 𝐶𝑥))
155, 14rabeqbidv 3168 . . . 4 (𝑝 = 𝐾 → {𝑥 ∈ (Base‘𝑝) ∣ (0.‘𝑝)( ⋖ ‘𝑝)𝑥} = {𝑥𝐵0 𝐶𝑥})
16 df-ats 33572 . . . 4 Atoms = (𝑝 ∈ V ↦ {𝑥 ∈ (Base‘𝑝) ∣ (0.‘𝑝)( ⋖ ‘𝑝)𝑥})
17 fvex 6113 . . . . . 6 (Base‘𝐾) ∈ V
184, 17eqeltri 2684 . . . . 5 𝐵 ∈ V
1918rabex 4740 . . . 4 {𝑥𝐵0 𝐶𝑥} ∈ V
2015, 16, 19fvmpt 6191 . . 3 (𝐾 ∈ V → (Atoms‘𝐾) = {𝑥𝐵0 𝐶𝑥})
212, 20syl5eq 2656 . 2 (𝐾 ∈ V → 𝐴 = {𝑥𝐵0 𝐶𝑥})
221, 21syl 17 1 (𝐾𝐷𝐴 = {𝑥𝐵0 𝐶𝑥})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  {crab 2900  Vcvv 3173   class class class wbr 4583  ‘cfv 5804  Basecbs 15695  0.cp0 16860   ⋖ ccvr 33567  Atomscatm 33568 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-ats 33572 This theorem is referenced by:  isat  33591
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