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Theorem meeteu 17071
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 2651 . . . 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 17065 . . 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 251 . . . . . 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 476 . . . . . 6 ((𝜑 ∧ {𝑋, 𝑌} ∈ dom (glb‘𝐾)) → {𝑋, 𝑌} ∈ dom (glb‘𝐾))
138, 9, 2, 10, 11, 12glbeu 17043 . . . . 5 ((𝜑 ∧ {𝑋, 𝑌} ∈ dom (glb‘𝐾)) → ∃!𝑥𝐵 (∀𝑦 ∈ {𝑋, 𝑌}𝑥 𝑦 ∧ ∀𝑧𝐵 (∀𝑦 ∈ {𝑋, 𝑌}𝑧 𝑦𝑧 𝑥)))
1413ex 449 . . . 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 17069 . . . . . 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 694 . . . . 5 (𝜑 → ((∀𝑦 ∈ {𝑋, 𝑌}𝑥 𝑦 ∧ ∀𝑧𝐵 (∀𝑦 ∈ {𝑋, 𝑌}𝑧 𝑦𝑧 𝑥)) ↔ ((𝑥 𝑋𝑥 𝑌) ∧ ∀𝑧𝐵 ((𝑧 𝑋𝑧 𝑌) → 𝑧 𝑥))))
1716reubidv 3156 . . . 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 229 . . 3 (𝜑 → ({𝑋, 𝑌} ∈ dom (glb‘𝐾) → ∃!𝑥𝐵 ((𝑥 𝑋𝑥 𝑌) ∧ ∀𝑧𝐵 ((𝑧 𝑋𝑧 𝑌) → 𝑧 𝑥))))
197, 18sylbid 230 . 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 196  wa 383   = wceq 1523  wcel 2030  wral 2941  ∃!wreu 2943  {cpr 4212  cop 4216   class class class wbr 4685  dom cdm 5143  cfv 5926  Basecbs 15904  lecple 15995  glbcglb 16990  meetcmee 16992
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-oprab 6694  df-glb 17022  df-meet 17024
This theorem is referenced by:  meetlem  17072
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