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Theorem pats 35094
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 3364 . 2 (𝐾𝐷𝐾 ∈ V)
2 patoms.a . . 3 𝐴 = (Atoms‘𝐾)
3 fveq2 6333 . . . . . 6 (𝑝 = 𝐾 → (Base‘𝑝) = (Base‘𝐾))
4 patoms.b . . . . . 6 𝐵 = (Base‘𝐾)
53, 4syl6eqr 2823 . . . . 5 (𝑝 = 𝐾 → (Base‘𝑝) = 𝐵)
6 fveq2 6333 . . . . . . . 8 (𝑝 = 𝐾 → ( ⋖ ‘𝑝) = ( ⋖ ‘𝐾))
7 patoms.c . . . . . . . 8 𝐶 = ( ⋖ ‘𝐾)
86, 7syl6eqr 2823 . . . . . . 7 (𝑝 = 𝐾 → ( ⋖ ‘𝑝) = 𝐶)
98breqd 4798 . . . . . 6 (𝑝 = 𝐾 → ((0.‘𝑝)( ⋖ ‘𝑝)𝑥 ↔ (0.‘𝑝)𝐶𝑥))
10 fveq2 6333 . . . . . . . 8 (𝑝 = 𝐾 → (0.‘𝑝) = (0.‘𝐾))
11 patoms.z . . . . . . . 8 0 = (0.‘𝐾)
1210, 11syl6eqr 2823 . . . . . . 7 (𝑝 = 𝐾 → (0.‘𝑝) = 0 )
1312breq1d 4797 . . . . . 6 (𝑝 = 𝐾 → ((0.‘𝑝)𝐶𝑥0 𝐶𝑥))
149, 13bitrd 268 . . . . 5 (𝑝 = 𝐾 → ((0.‘𝑝)( ⋖ ‘𝑝)𝑥0 𝐶𝑥))
155, 14rabeqbidv 3345 . . . 4 (𝑝 = 𝐾 → {𝑥 ∈ (Base‘𝑝) ∣ (0.‘𝑝)( ⋖ ‘𝑝)𝑥} = {𝑥𝐵0 𝐶𝑥})
16 df-ats 35076 . . . 4 Atoms = (𝑝 ∈ V ↦ {𝑥 ∈ (Base‘𝑝) ∣ (0.‘𝑝)( ⋖ ‘𝑝)𝑥})
174fvexi 6345 . . . . 5 𝐵 ∈ V
1817rabex 4947 . . . 4 {𝑥𝐵0 𝐶𝑥} ∈ V
1915, 16, 18fvmpt 6426 . . 3 (𝐾 ∈ V → (Atoms‘𝐾) = {𝑥𝐵0 𝐶𝑥})
202, 19syl5eq 2817 . 2 (𝐾 ∈ V → 𝐴 = {𝑥𝐵0 𝐶𝑥})
211, 20syl 17 1 (𝐾𝐷𝐴 = {𝑥𝐵0 𝐶𝑥})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1631  wcel 2145  {crab 3065  Vcvv 3351   class class class wbr 4787  cfv 6030  Basecbs 16064  0.cp0 17245  ccvr 35071  Atomscatm 35072
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pr 5035
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-iota 5993  df-fun 6032  df-fv 6038  df-ats 35076
This theorem is referenced by:  isat  35095
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