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Theorem meeteu 18441
Description: Uniqueness of meet of elements in the domain. (Contributed by NM, 12-Sep-2018.)
Hypotheses
Ref Expression
meetval2.b 𝐵 = (Base‘𝐾)
meetval2.l = (le‘𝐾)
meetval2.m = (meet‘𝐾)
meetval2.k (𝜑𝐾𝑉)
meetval2.x (𝜑𝑋𝐵)
meetval2.y (𝜑𝑌𝐵)
meetlem.e (𝜑 → ⟨𝑋, 𝑌⟩ ∈ dom )
Assertion
Ref Expression
meeteu (𝜑 → ∃!𝑥𝐵 ((𝑥 𝑋𝑥 𝑌) ∧ ∀𝑧𝐵 ((𝑧 𝑋𝑧 𝑌) → 𝑧 𝑥)))
Distinct variable groups:   𝑥,𝑧,𝐵   𝑥, ,𝑧   𝑥,𝐾,𝑧   𝑥,𝑋,𝑧   𝑥,𝑌,𝑧   𝜑,𝑥
Allowed substitution hints:   𝜑(𝑧)   (𝑥,𝑧)   𝑉(𝑥,𝑧)

Proof of Theorem meeteu
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 meetlem.e . 2 (𝜑 → ⟨𝑋, 𝑌⟩ ∈ dom )
2 eqid 2737 . . . 4 (glb‘𝐾) = (glb‘𝐾)
3 meetval2.m . . . 4 = (meet‘𝐾)
4 meetval2.k . . . 4 (𝜑𝐾𝑉)
5 meetval2.x . . . 4 (𝜑𝑋𝐵)
6 meetval2.y . . . 4 (𝜑𝑌𝐵)
72, 3, 4, 5, 6meetdef 18435 . . 3 (𝜑 → (⟨𝑋, 𝑌⟩ ∈ dom ↔ {𝑋, 𝑌} ∈ dom (glb‘𝐾)))
8 meetval2.b . . . . . 6 𝐵 = (Base‘𝐾)
9 meetval2.l . . . . . 6 = (le‘𝐾)
10 biid 261 . . . . . 6 ((∀𝑦 ∈ {𝑋, 𝑌}𝑥 𝑦 ∧ ∀𝑧𝐵 (∀𝑦 ∈ {𝑋, 𝑌}𝑧 𝑦𝑧 𝑥)) ↔ (∀𝑦 ∈ {𝑋, 𝑌}𝑥 𝑦 ∧ ∀𝑧𝐵 (∀𝑦 ∈ {𝑋, 𝑌}𝑧 𝑦𝑧 𝑥)))
114adantr 480 . . . . . 6 ((𝜑 ∧ {𝑋, 𝑌} ∈ dom (glb‘𝐾)) → 𝐾𝑉)
12 simpr 484 . . . . . 6 ((𝜑 ∧ {𝑋, 𝑌} ∈ dom (glb‘𝐾)) → {𝑋, 𝑌} ∈ dom (glb‘𝐾))
138, 9, 2, 10, 11, 12glbeu 18413 . . . . 5 ((𝜑 ∧ {𝑋, 𝑌} ∈ dom (glb‘𝐾)) → ∃!𝑥𝐵 (∀𝑦 ∈ {𝑋, 𝑌}𝑥 𝑦 ∧ ∀𝑧𝐵 (∀𝑦 ∈ {𝑋, 𝑌}𝑧 𝑦𝑧 𝑥)))
1413ex 412 . . . 4 (𝜑 → ({𝑋, 𝑌} ∈ dom (glb‘𝐾) → ∃!𝑥𝐵 (∀𝑦 ∈ {𝑋, 𝑌}𝑥 𝑦 ∧ ∀𝑧𝐵 (∀𝑦 ∈ {𝑋, 𝑌}𝑧 𝑦𝑧 𝑥))))
158, 9, 3, 4, 5, 6meetval2lem 18439 . . . . . 6 ((𝑋𝐵𝑌𝐵) → ((∀𝑦 ∈ {𝑋, 𝑌}𝑥 𝑦 ∧ ∀𝑧𝐵 (∀𝑦 ∈ {𝑋, 𝑌}𝑧 𝑦𝑧 𝑥)) ↔ ((𝑥 𝑋𝑥 𝑌) ∧ ∀𝑧𝐵 ((𝑧 𝑋𝑧 𝑌) → 𝑧 𝑥))))
165, 6, 15syl2anc 584 . . . . 5 (𝜑 → ((∀𝑦 ∈ {𝑋, 𝑌}𝑥 𝑦 ∧ ∀𝑧𝐵 (∀𝑦 ∈ {𝑋, 𝑌}𝑧 𝑦𝑧 𝑥)) ↔ ((𝑥 𝑋𝑥 𝑌) ∧ ∀𝑧𝐵 ((𝑧 𝑋𝑧 𝑌) → 𝑧 𝑥))))
1716reubidv 3398 . . . 4 (𝜑 → (∃!𝑥𝐵 (∀𝑦 ∈ {𝑋, 𝑌}𝑥 𝑦 ∧ ∀𝑧𝐵 (∀𝑦 ∈ {𝑋, 𝑌}𝑧 𝑦𝑧 𝑥)) ↔ ∃!𝑥𝐵 ((𝑥 𝑋𝑥 𝑌) ∧ ∀𝑧𝐵 ((𝑧 𝑋𝑧 𝑌) → 𝑧 𝑥))))
1814, 17sylibd 239 . . 3 (𝜑 → ({𝑋, 𝑌} ∈ dom (glb‘𝐾) → ∃!𝑥𝐵 ((𝑥 𝑋𝑥 𝑌) ∧ ∀𝑧𝐵 ((𝑧 𝑋𝑧 𝑌) → 𝑧 𝑥))))
197, 18sylbid 240 . 2 (𝜑 → (⟨𝑋, 𝑌⟩ ∈ dom → ∃!𝑥𝐵 ((𝑥 𝑋𝑥 𝑌) ∧ ∀𝑧𝐵 ((𝑧 𝑋𝑧 𝑌) → 𝑧 𝑥))))
201, 19mpd 15 1 (𝜑 → ∃!𝑥𝐵 ((𝑥 𝑋𝑥 𝑌) ∧ ∀𝑧𝐵 ((𝑧 𝑋𝑧 𝑌) → 𝑧 𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  wral 3061  ∃!wreu 3378  {cpr 4628  cop 4632   class class class wbr 5143  dom cdm 5685  cfv 6561  Basecbs 17247  lecple 17304  glbcglb 18356  meetcmee 18358
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-oprab 7435  df-glb 18392  df-meet 18394
This theorem is referenced by:  meetlem  18442
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