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Theorem meetcl 17626
 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 2801 . . 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 17625 . 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 17624 . . . 4 (𝜑 → (⟨𝑋, 𝑌⟩ ∈ dom ↔ {𝑋, 𝑌} ∈ dom (glb‘𝐾)))
108, 9mpbid 235 . . 3 (𝜑 → {𝑋, 𝑌} ∈ dom (glb‘𝐾))
117, 1, 3, 10glbcl 17604 . 2 (𝜑 → ((glb‘𝐾)‘{𝑋, 𝑌}) ∈ 𝐵)
126, 11eqeltrd 2893 1 (𝜑 → (𝑋 𝑌) ∈ 𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2112  {cpr 4530  ⟨cop 4534  dom cdm 5523  ‘cfv 6328  (class class class)co 7139  Basecbs 16479  glbcglb 17549  meetcmee 17551 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-riota 7097  df-ov 7142  df-oprab 7143  df-glb 17581  df-meet 17583 This theorem is referenced by:  meetle  17634  latlem  17655
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