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Theorem meetcl 18450
Description: Closure of meet of elements in the domain. (Contributed by NM, 12-Sep-2018.)
Hypotheses
Ref Expression
meetcl.b 𝐵 = (Base‘𝐾)
meetcl.m = (meet‘𝐾)
meetcl.k (𝜑𝐾𝑉)
meetcl.x (𝜑𝑋𝐵)
meetcl.y (𝜑𝑌𝐵)
meetcl.e (𝜑 → ⟨𝑋, 𝑌⟩ ∈ dom )
Assertion
Ref Expression
meetcl (𝜑 → (𝑋 𝑌) ∈ 𝐵)

Proof of Theorem meetcl
StepHypRef Expression
1 eqid 2735 . . 3 (glb‘𝐾) = (glb‘𝐾)
2 meetcl.m . . 3 = (meet‘𝐾)
3 meetcl.k . . 3 (𝜑𝐾𝑉)
4 meetcl.x . . 3 (𝜑𝑋𝐵)
5 meetcl.y . . 3 (𝜑𝑌𝐵)
61, 2, 3, 4, 5meetval 18449 . 2 (𝜑 → (𝑋 𝑌) = ((glb‘𝐾)‘{𝑋, 𝑌}))
7 meetcl.b . . 3 𝐵 = (Base‘𝐾)
8 meetcl.e . . . 4 (𝜑 → ⟨𝑋, 𝑌⟩ ∈ dom )
91, 2, 3, 4, 5meetdef 18448 . . . 4 (𝜑 → (⟨𝑋, 𝑌⟩ ∈ dom ↔ {𝑋, 𝑌} ∈ dom (glb‘𝐾)))
108, 9mpbid 232 . . 3 (𝜑 → {𝑋, 𝑌} ∈ dom (glb‘𝐾))
117, 1, 3, 10glbcl 18428 . 2 (𝜑 → ((glb‘𝐾)‘{𝑋, 𝑌}) ∈ 𝐵)
126, 11eqeltrd 2839 1 (𝜑 → (𝑋 𝑌) ∈ 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2106  {cpr 4633  cop 4637  dom cdm 5689  cfv 6563  (class class class)co 7431  Basecbs 17245  glbcglb 18368  meetcmee 18370
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-glb 18405  df-meet 18407
This theorem is referenced by:  meetle  18458  latlem  18495
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