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Theorem pats 38751
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 3489 . 2 (𝐾 ∈ 𝐷 β†’ 𝐾 ∈ V)
2 patoms.a . . 3 𝐴 = (Atomsβ€˜πΎ)
3 fveq2 6891 . . . . . 6 (𝑝 = 𝐾 β†’ (Baseβ€˜π‘) = (Baseβ€˜πΎ))
4 patoms.b . . . . . 6 𝐡 = (Baseβ€˜πΎ)
53, 4eqtr4di 2786 . . . . 5 (𝑝 = 𝐾 β†’ (Baseβ€˜π‘) = 𝐡)
6 fveq2 6891 . . . . . . . 8 (𝑝 = 𝐾 β†’ ( β‹– β€˜π‘) = ( β‹– β€˜πΎ))
7 patoms.c . . . . . . . 8 𝐢 = ( β‹– β€˜πΎ)
86, 7eqtr4di 2786 . . . . . . 7 (𝑝 = 𝐾 β†’ ( β‹– β€˜π‘) = 𝐢)
98breqd 5153 . . . . . 6 (𝑝 = 𝐾 β†’ ((0.β€˜π‘)( β‹– β€˜π‘)π‘₯ ↔ (0.β€˜π‘)𝐢π‘₯))
10 fveq2 6891 . . . . . . . 8 (𝑝 = 𝐾 β†’ (0.β€˜π‘) = (0.β€˜πΎ))
11 patoms.z . . . . . . . 8 0 = (0.β€˜πΎ)
1210, 11eqtr4di 2786 . . . . . . 7 (𝑝 = 𝐾 β†’ (0.β€˜π‘) = 0 )
1312breq1d 5152 . . . . . 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 3445 . . . 4 (𝑝 = 𝐾 β†’ {π‘₯ ∈ (Baseβ€˜π‘) ∣ (0.β€˜π‘)( β‹– β€˜π‘)π‘₯} = {π‘₯ ∈ 𝐡 ∣ 0 𝐢π‘₯})
16 df-ats 38733 . . . 4 Atoms = (𝑝 ∈ V ↦ {π‘₯ ∈ (Baseβ€˜π‘) ∣ (0.β€˜π‘)( β‹– β€˜π‘)π‘₯})
174fvexi 6905 . . . . 5 𝐡 ∈ V
1817rabex 5328 . . . 4 {π‘₯ ∈ 𝐡 ∣ 0 𝐢π‘₯} ∈ V
1915, 16, 18fvmpt 6999 . . 3 (𝐾 ∈ V β†’ (Atomsβ€˜πΎ) = {π‘₯ ∈ 𝐡 ∣ 0 𝐢π‘₯})
202, 19eqtrid 2780 . 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 1534   ∈ wcel 2099  {crab 3428  Vcvv 3470   class class class wbr 5142  β€˜cfv 6542  Basecbs 17173  0.cp0 18408   β‹– ccvr 38728  Atomscatm 38729
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5293  ax-nul 5300  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2937  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3472  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-iota 6494  df-fun 6544  df-fv 6550  df-ats 38733
This theorem is referenced by:  isat  38752
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