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Theorem glbdm 16986
Description: Domain of the greatest lower bound function of a poset. (Contributed by NM, 6-Sep-2018.)
Hypotheses
Ref Expression
glbfval.b 𝐵 = (Base‘𝐾)
glbfval.l = (le‘𝐾)
glbfval.g 𝐺 = (glb‘𝐾)
glbfval.p (𝜓 ↔ (∀𝑦𝑠 𝑥 𝑦 ∧ ∀𝑧𝐵 (∀𝑦𝑠 𝑧 𝑦𝑧 𝑥)))
glbfval.k (𝜑𝐾𝑉)
Assertion
Ref Expression
glbdm (𝜑 → dom 𝐺 = {𝑠 ∈ 𝒫 𝐵 ∣ ∃!𝑥𝐵 𝜓})
Distinct variable groups:   𝑥,𝑠,𝑧,𝐵   𝑦,𝑠,𝐾,𝑥,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧,𝑠)   𝜓(𝑥,𝑦,𝑧,𝑠)   𝐵(𝑦)   𝐺(𝑥,𝑦,𝑧,𝑠)   (𝑥,𝑦,𝑧,𝑠)   𝑉(𝑥,𝑦,𝑧,𝑠)

Proof of Theorem glbdm
StepHypRef Expression
1 glbfval.b . . . 4 𝐵 = (Base‘𝐾)
2 glbfval.l . . . 4 = (le‘𝐾)
3 glbfval.g . . . 4 𝐺 = (glb‘𝐾)
4 glbfval.p . . . 4 (𝜓 ↔ (∀𝑦𝑠 𝑥 𝑦 ∧ ∀𝑧𝐵 (∀𝑦𝑠 𝑧 𝑦𝑧 𝑥)))
5 glbfval.k . . . 4 (𝜑𝐾𝑉)
61, 2, 3, 4, 5glbfval 16985 . . 3 (𝜑𝐺 = ((𝑠 ∈ 𝒫 𝐵 ↦ (𝑥𝐵 𝜓)) ↾ {𝑠 ∣ ∃!𝑥𝐵 𝜓}))
76dmeqd 5324 . 2 (𝜑 → dom 𝐺 = dom ((𝑠 ∈ 𝒫 𝐵 ↦ (𝑥𝐵 𝜓)) ↾ {𝑠 ∣ ∃!𝑥𝐵 𝜓}))
8 riotaex 6612 . . . . 5 (𝑥𝐵 𝜓) ∈ V
9 eqid 2621 . . . . 5 (𝑠 ∈ 𝒫 𝐵 ↦ (𝑥𝐵 𝜓)) = (𝑠 ∈ 𝒫 𝐵 ↦ (𝑥𝐵 𝜓))
108, 9dmmpti 6021 . . . 4 dom (𝑠 ∈ 𝒫 𝐵 ↦ (𝑥𝐵 𝜓)) = 𝒫 𝐵
1110ineq2i 3809 . . 3 ({𝑠 ∣ ∃!𝑥𝐵 𝜓} ∩ dom (𝑠 ∈ 𝒫 𝐵 ↦ (𝑥𝐵 𝜓))) = ({𝑠 ∣ ∃!𝑥𝐵 𝜓} ∩ 𝒫 𝐵)
12 dmres 5417 . . 3 dom ((𝑠 ∈ 𝒫 𝐵 ↦ (𝑥𝐵 𝜓)) ↾ {𝑠 ∣ ∃!𝑥𝐵 𝜓}) = ({𝑠 ∣ ∃!𝑥𝐵 𝜓} ∩ dom (𝑠 ∈ 𝒫 𝐵 ↦ (𝑥𝐵 𝜓)))
13 dfrab2 3901 . . 3 {𝑠 ∈ 𝒫 𝐵 ∣ ∃!𝑥𝐵 𝜓} = ({𝑠 ∣ ∃!𝑥𝐵 𝜓} ∩ 𝒫 𝐵)
1411, 12, 133eqtr4i 2653 . 2 dom ((𝑠 ∈ 𝒫 𝐵 ↦ (𝑥𝐵 𝜓)) ↾ {𝑠 ∣ ∃!𝑥𝐵 𝜓}) = {𝑠 ∈ 𝒫 𝐵 ∣ ∃!𝑥𝐵 𝜓}
157, 14syl6eq 2671 1 (𝜑 → dom 𝐺 = {𝑠 ∈ 𝒫 𝐵 ∣ ∃!𝑥𝐵 𝜓})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1482  wcel 1989  {cab 2607  wral 2911  ∃!wreu 2913  {crab 2915  cin 3571  𝒫 cpw 4156   class class class wbr 4651  cmpt 4727  dom cdm 5112  cres 5114  cfv 5886  crio 6607  Basecbs 15851  lecple 15942  glbcglb 16937
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-rep 4769  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-ral 2916  df-rex 2917  df-reu 2918  df-rab 2920  df-v 3200  df-sbc 3434  df-csb 3532  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-pw 4158  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-iun 4520  df-br 4652  df-opab 4711  df-mpt 4728  df-id 5022  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-f1 5891  df-fo 5892  df-f1o 5893  df-fv 5894  df-riota 6608  df-glb 16969
This theorem is referenced by:  glbeldm  16988  xrsclat  29665
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