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Theorem pats 37793
Description: The set of atoms in a poset. (Contributed by NM, 18-Sep-2011.)
Hypotheses
Ref Expression
patoms.b 𝐡 = (Baseβ€˜πΎ)
patoms.z 0 = (0.β€˜πΎ)
patoms.c 𝐢 = ( β‹– β€˜πΎ)
patoms.a 𝐴 = (Atomsβ€˜πΎ)
Assertion
Ref Expression
pats (𝐾 ∈ 𝐷 β†’ 𝐴 = {π‘₯ ∈ 𝐡 ∣ 0 𝐢π‘₯})
Distinct variable groups:   π‘₯,𝐡   π‘₯,𝐾
Allowed substitution hints:   𝐴(π‘₯)   𝐢(π‘₯)   𝐷(π‘₯)   0 (π‘₯)

Proof of Theorem pats
Dummy variable 𝑝 is distinct from all other variables.
StepHypRef Expression
1 elex 3462 . 2 (𝐾 ∈ 𝐷 β†’ 𝐾 ∈ V)
2 patoms.a . . 3 𝐴 = (Atomsβ€˜πΎ)
3 fveq2 6843 . . . . . 6 (𝑝 = 𝐾 β†’ (Baseβ€˜π‘) = (Baseβ€˜πΎ))
4 patoms.b . . . . . 6 𝐡 = (Baseβ€˜πΎ)
53, 4eqtr4di 2791 . . . . 5 (𝑝 = 𝐾 β†’ (Baseβ€˜π‘) = 𝐡)
6 fveq2 6843 . . . . . . . 8 (𝑝 = 𝐾 β†’ ( β‹– β€˜π‘) = ( β‹– β€˜πΎ))
7 patoms.c . . . . . . . 8 𝐢 = ( β‹– β€˜πΎ)
86, 7eqtr4di 2791 . . . . . . 7 (𝑝 = 𝐾 β†’ ( β‹– β€˜π‘) = 𝐢)
98breqd 5117 . . . . . 6 (𝑝 = 𝐾 β†’ ((0.β€˜π‘)( β‹– β€˜π‘)π‘₯ ↔ (0.β€˜π‘)𝐢π‘₯))
10 fveq2 6843 . . . . . . . 8 (𝑝 = 𝐾 β†’ (0.β€˜π‘) = (0.β€˜πΎ))
11 patoms.z . . . . . . . 8 0 = (0.β€˜πΎ)
1210, 11eqtr4di 2791 . . . . . . 7 (𝑝 = 𝐾 β†’ (0.β€˜π‘) = 0 )
1312breq1d 5116 . . . . . 6 (𝑝 = 𝐾 β†’ ((0.β€˜π‘)𝐢π‘₯ ↔ 0 𝐢π‘₯))
149, 13bitrd 279 . . . . 5 (𝑝 = 𝐾 β†’ ((0.β€˜π‘)( β‹– β€˜π‘)π‘₯ ↔ 0 𝐢π‘₯))
155, 14rabeqbidv 3423 . . . 4 (𝑝 = 𝐾 β†’ {π‘₯ ∈ (Baseβ€˜π‘) ∣ (0.β€˜π‘)( β‹– β€˜π‘)π‘₯} = {π‘₯ ∈ 𝐡 ∣ 0 𝐢π‘₯})
16 df-ats 37775 . . . 4 Atoms = (𝑝 ∈ V ↦ {π‘₯ ∈ (Baseβ€˜π‘) ∣ (0.β€˜π‘)( β‹– β€˜π‘)π‘₯})
174fvexi 6857 . . . . 5 𝐡 ∈ V
1817rabex 5290 . . . 4 {π‘₯ ∈ 𝐡 ∣ 0 𝐢π‘₯} ∈ V
1915, 16, 18fvmpt 6949 . . 3 (𝐾 ∈ V β†’ (Atomsβ€˜πΎ) = {π‘₯ ∈ 𝐡 ∣ 0 𝐢π‘₯})
202, 19eqtrid 2785 . 2 (𝐾 ∈ V β†’ 𝐴 = {π‘₯ ∈ 𝐡 ∣ 0 𝐢π‘₯})
211, 20syl 17 1 (𝐾 ∈ 𝐷 β†’ 𝐴 = {π‘₯ ∈ 𝐡 ∣ 0 𝐢π‘₯})
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1542   ∈ wcel 2107  {crab 3406  Vcvv 3444   class class class wbr 5106  β€˜cfv 6497  Basecbs 17088  0.cp0 18317   β‹– ccvr 37770  Atomscatm 37771
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-iota 6449  df-fun 6499  df-fv 6505  df-ats 37775
This theorem is referenced by:  isat  37794
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