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Theorem meeteu 17421
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 2778 . . . 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 17415 . . 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 253 . . . . . 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 474 . . . . . 6 ((𝜑 ∧ {𝑋, 𝑌} ∈ dom (glb‘𝐾)) → 𝐾𝑉)
12 simpr 479 . . . . . 6 ((𝜑 ∧ {𝑋, 𝑌} ∈ dom (glb‘𝐾)) → {𝑋, 𝑌} ∈ dom (glb‘𝐾))
138, 9, 2, 10, 11, 12glbeu 17393 . . . . 5 ((𝜑 ∧ {𝑋, 𝑌} ∈ dom (glb‘𝐾)) → ∃!𝑥𝐵 (∀𝑦 ∈ {𝑋, 𝑌}𝑥 𝑦 ∧ ∀𝑧𝐵 (∀𝑦 ∈ {𝑋, 𝑌}𝑧 𝑦𝑧 𝑥)))
1413ex 403 . . . 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 17419 . . . . . 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 579 . . . . 5 (𝜑 → ((∀𝑦 ∈ {𝑋, 𝑌}𝑥 𝑦 ∧ ∀𝑧𝐵 (∀𝑦 ∈ {𝑋, 𝑌}𝑧 𝑦𝑧 𝑥)) ↔ ((𝑥 𝑋𝑥 𝑌) ∧ ∀𝑧𝐵 ((𝑧 𝑋𝑧 𝑌) → 𝑧 𝑥))))
1716reubidv 3314 . . . 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 231 . . 3 (𝜑 → ({𝑋, 𝑌} ∈ dom (glb‘𝐾) → ∃!𝑥𝐵 ((𝑥 𝑋𝑥 𝑌) ∧ ∀𝑧𝐵 ((𝑧 𝑋𝑧 𝑌) → 𝑧 𝑥))))
197, 18sylbid 232 . 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 198  wa 386   = wceq 1601  wcel 2107  wral 3090  ∃!wreu 3092  {cpr 4400  cop 4404   class class class wbr 4888  dom cdm 5357  cfv 6137  Basecbs 16266  lecple 16356  glbcglb 17340  meetcmee 17342
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5008  ax-sep 5019  ax-nul 5027  ax-pow 5079  ax-pr 5140  ax-un 7228
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4674  df-iun 4757  df-br 4889  df-opab 4951  df-mpt 4968  df-id 5263  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-rn 5368  df-res 5369  df-ima 5370  df-iota 6101  df-fun 6139  df-fn 6140  df-f 6141  df-f1 6142  df-fo 6143  df-f1o 6144  df-fv 6145  df-riota 6885  df-oprab 6928  df-glb 17372  df-meet 17374
This theorem is referenced by:  meetlem  17422
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