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Theorem pats 34391
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 3207 . 2 (𝐾𝐷𝐾 ∈ V)
2 patoms.a . . 3 𝐴 = (Atoms‘𝐾)
3 fveq2 6178 . . . . . 6 (𝑝 = 𝐾 → (Base‘𝑝) = (Base‘𝐾))
4 patoms.b . . . . . 6 𝐵 = (Base‘𝐾)
53, 4syl6eqr 2672 . . . . 5 (𝑝 = 𝐾 → (Base‘𝑝) = 𝐵)
6 fveq2 6178 . . . . . . . 8 (𝑝 = 𝐾 → ( ⋖ ‘𝑝) = ( ⋖ ‘𝐾))
7 patoms.c . . . . . . . 8 𝐶 = ( ⋖ ‘𝐾)
86, 7syl6eqr 2672 . . . . . . 7 (𝑝 = 𝐾 → ( ⋖ ‘𝑝) = 𝐶)
98breqd 4655 . . . . . 6 (𝑝 = 𝐾 → ((0.‘𝑝)( ⋖ ‘𝑝)𝑥 ↔ (0.‘𝑝)𝐶𝑥))
10 fveq2 6178 . . . . . . . 8 (𝑝 = 𝐾 → (0.‘𝑝) = (0.‘𝐾))
11 patoms.z . . . . . . . 8 0 = (0.‘𝐾)
1210, 11syl6eqr 2672 . . . . . . 7 (𝑝 = 𝐾 → (0.‘𝑝) = 0 )
1312breq1d 4654 . . . . . 6 (𝑝 = 𝐾 → ((0.‘𝑝)𝐶𝑥0 𝐶𝑥))
149, 13bitrd 268 . . . . 5 (𝑝 = 𝐾 → ((0.‘𝑝)( ⋖ ‘𝑝)𝑥0 𝐶𝑥))
155, 14rabeqbidv 3190 . . . 4 (𝑝 = 𝐾 → {𝑥 ∈ (Base‘𝑝) ∣ (0.‘𝑝)( ⋖ ‘𝑝)𝑥} = {𝑥𝐵0 𝐶𝑥})
16 df-ats 34373 . . . 4 Atoms = (𝑝 ∈ V ↦ {𝑥 ∈ (Base‘𝑝) ∣ (0.‘𝑝)( ⋖ ‘𝑝)𝑥})
17 fvex 6188 . . . . . 6 (Base‘𝐾) ∈ V
184, 17eqeltri 2695 . . . . 5 𝐵 ∈ V
1918rabex 4804 . . . 4 {𝑥𝐵0 𝐶𝑥} ∈ V
2015, 16, 19fvmpt 6269 . . 3 (𝐾 ∈ V → (Atoms‘𝐾) = {𝑥𝐵0 𝐶𝑥})
212, 20syl5eq 2666 . 2 (𝐾 ∈ V → 𝐴 = {𝑥𝐵0 𝐶𝑥})
221, 21syl 17 1 (𝐾𝐷𝐴 = {𝑥𝐵0 𝐶𝑥})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1481  wcel 1988  {crab 2913  Vcvv 3195   class class class wbr 4644  cfv 5876  Basecbs 15838  0.cp0 17018  ccvr 34368  Atomscatm 34369
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pr 4897
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-iota 5839  df-fun 5878  df-fv 5884  df-ats 34373
This theorem is referenced by:  isat  34392
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