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Theorem joineu 17232
Description: Uniqueness of join of elements in the domain. (Contributed by NM, 12-Sep-2018.)
Hypotheses
Ref Expression
joinval2.b 𝐵 = (Base‘𝐾)
joinval2.l = (le‘𝐾)
joinval2.j = (join‘𝐾)
joinval2.k (𝜑𝐾𝑉)
joinval2.x (𝜑𝑋𝐵)
joinval2.y (𝜑𝑌𝐵)
joinlem.e (𝜑 → ⟨𝑋, 𝑌⟩ ∈ dom )
Assertion
Ref Expression
joineu (𝜑 → ∃!𝑥𝐵 ((𝑋 𝑥𝑌 𝑥) ∧ ∀𝑧𝐵 ((𝑋 𝑧𝑌 𝑧) → 𝑥 𝑧)))
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 joineu
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 joinlem.e . 2 (𝜑 → ⟨𝑋, 𝑌⟩ ∈ dom )
2 eqid 2761 . . . 4 (lub‘𝐾) = (lub‘𝐾)
3 joinval2.j . . . 4 = (join‘𝐾)
4 joinval2.k . . . 4 (𝜑𝐾𝑉)
5 joinval2.x . . . 4 (𝜑𝑋𝐵)
6 joinval2.y . . . 4 (𝜑𝑌𝐵)
72, 3, 4, 5, 6joindef 17226 . . 3 (𝜑 → (⟨𝑋, 𝑌⟩ ∈ dom ↔ {𝑋, 𝑌} ∈ dom (lub‘𝐾)))
8 joinval2.b . . . . . 6 𝐵 = (Base‘𝐾)
9 joinval2.l . . . . . 6 = (le‘𝐾)
10 biid 251 . . . . . 6 ((∀𝑦 ∈ {𝑋, 𝑌}𝑦 𝑥 ∧ ∀𝑧𝐵 (∀𝑦 ∈ {𝑋, 𝑌}𝑦 𝑧𝑥 𝑧)) ↔ (∀𝑦 ∈ {𝑋, 𝑌}𝑦 𝑥 ∧ ∀𝑧𝐵 (∀𝑦 ∈ {𝑋, 𝑌}𝑦 𝑧𝑥 𝑧)))
114adantr 472 . . . . . 6 ((𝜑 ∧ {𝑋, 𝑌} ∈ dom (lub‘𝐾)) → 𝐾𝑉)
12 simpr 479 . . . . . 6 ((𝜑 ∧ {𝑋, 𝑌} ∈ dom (lub‘𝐾)) → {𝑋, 𝑌} ∈ dom (lub‘𝐾))
138, 9, 2, 10, 11, 12lubeu 17205 . . . . 5 ((𝜑 ∧ {𝑋, 𝑌} ∈ dom (lub‘𝐾)) → ∃!𝑥𝐵 (∀𝑦 ∈ {𝑋, 𝑌}𝑦 𝑥 ∧ ∀𝑧𝐵 (∀𝑦 ∈ {𝑋, 𝑌}𝑦 𝑧𝑥 𝑧)))
1413ex 449 . . . 4 (𝜑 → ({𝑋, 𝑌} ∈ dom (lub‘𝐾) → ∃!𝑥𝐵 (∀𝑦 ∈ {𝑋, 𝑌}𝑦 𝑥 ∧ ∀𝑧𝐵 (∀𝑦 ∈ {𝑋, 𝑌}𝑦 𝑧𝑥 𝑧))))
158, 9, 3, 4, 5, 6joinval2lem 17230 . . . . . 6 ((𝑋𝐵𝑌𝐵) → ((∀𝑦 ∈ {𝑋, 𝑌}𝑦 𝑥 ∧ ∀𝑧𝐵 (∀𝑦 ∈ {𝑋, 𝑌}𝑦 𝑧𝑥 𝑧)) ↔ ((𝑋 𝑥𝑌 𝑥) ∧ ∀𝑧𝐵 ((𝑋 𝑧𝑌 𝑧) → 𝑥 𝑧))))
165, 6, 15syl2anc 696 . . . . 5 (𝜑 → ((∀𝑦 ∈ {𝑋, 𝑌}𝑦 𝑥 ∧ ∀𝑧𝐵 (∀𝑦 ∈ {𝑋, 𝑌}𝑦 𝑧𝑥 𝑧)) ↔ ((𝑋 𝑥𝑌 𝑥) ∧ ∀𝑧𝐵 ((𝑋 𝑧𝑌 𝑧) → 𝑥 𝑧))))
1716reubidv 3266 . . . 4 (𝜑 → (∃!𝑥𝐵 (∀𝑦 ∈ {𝑋, 𝑌}𝑦 𝑥 ∧ ∀𝑧𝐵 (∀𝑦 ∈ {𝑋, 𝑌}𝑦 𝑧𝑥 𝑧)) ↔ ∃!𝑥𝐵 ((𝑋 𝑥𝑌 𝑥) ∧ ∀𝑧𝐵 ((𝑋 𝑧𝑌 𝑧) → 𝑥 𝑧))))
1814, 17sylibd 229 . . 3 (𝜑 → ({𝑋, 𝑌} ∈ dom (lub‘𝐾) → ∃!𝑥𝐵 ((𝑋 𝑥𝑌 𝑥) ∧ ∀𝑧𝐵 ((𝑋 𝑧𝑌 𝑧) → 𝑥 𝑧))))
197, 18sylbid 230 . 2 (𝜑 → (⟨𝑋, 𝑌⟩ ∈ dom → ∃!𝑥𝐵 ((𝑋 𝑥𝑌 𝑥) ∧ ∀𝑧𝐵 ((𝑋 𝑧𝑌 𝑧) → 𝑥 𝑧))))
201, 19mpd 15 1 (𝜑 → ∃!𝑥𝐵 ((𝑋 𝑥𝑌 𝑥) ∧ ∀𝑧𝐵 ((𝑋 𝑧𝑌 𝑧) → 𝑥 𝑧)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1632  wcel 2140  wral 3051  ∃!wreu 3053  {cpr 4324  cop 4328   class class class wbr 4805  dom cdm 5267  cfv 6050  Basecbs 16080  lecple 16171  lubclub 17164  joincjn 17166
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-8 2142  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-rep 4924  ax-sep 4934  ax-nul 4942  ax-pow 4993  ax-pr 5056  ax-un 7116
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-eu 2612  df-mo 2613  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ne 2934  df-ral 3056  df-rex 3057  df-reu 3058  df-rab 3060  df-v 3343  df-sbc 3578  df-csb 3676  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-nul 4060  df-if 4232  df-pw 4305  df-sn 4323  df-pr 4325  df-op 4329  df-uni 4590  df-iun 4675  df-br 4806  df-opab 4866  df-mpt 4883  df-id 5175  df-xp 5273  df-rel 5274  df-cnv 5275  df-co 5276  df-dm 5277  df-rn 5278  df-res 5279  df-ima 5280  df-iota 6013  df-fun 6052  df-fn 6053  df-f 6054  df-f1 6055  df-fo 6056  df-f1o 6057  df-fv 6058  df-riota 6776  df-oprab 6819  df-lub 17196  df-join 17198
This theorem is referenced by:  joinlem  17233
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