Theorem joineu 18015

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	⊢ (𝜑 → 𝐾 ∈ 𝑉)
joinval2.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
joinval2.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)
joinlem.e	⊢ (𝜑 → ⟨𝑋, 𝑌⟩ ∈ dom ∨ )

Assertion

Ref	Expression
joineu	⊢ (𝜑 → ∃!𝑥 ∈ 𝐵 ((𝑋 ≤ 𝑥 ∧ 𝑌 ≤ 𝑥) ∧ ∀𝑧 ∈ 𝐵 ((𝑋 ≤ 𝑧 ∧ 𝑌 ≤ 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.

Step	Hyp	Ref	Expression
1		joinlem.e	. 2 ⊢ (𝜑 → ⟨𝑋, 𝑌⟩ ∈ dom ∨ )
2		eqid 2738	. . . 4 ⊢ (lub‘𝐾) = (lub‘𝐾)
3		joinval2.j	. . . 4 ⊢ ∨ = (join‘𝐾)
4		joinval2.k	. . . 4 ⊢ (𝜑 → 𝐾 ∈ 𝑉)
5		joinval2.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
6		joinval2.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
7	2, 3, 4, 5, 6	joindef 18009	. . 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 260	. . . . . 6 ⊢ ((∀𝑦 ∈ {𝑋, 𝑌}𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ {𝑋, 𝑌}𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧)) ↔ (∀𝑦 ∈ {𝑋, 𝑌}𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ {𝑋, 𝑌}𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧)))
11	4	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ {𝑋, 𝑌} ∈ dom (lub‘𝐾)) → 𝐾 ∈ 𝑉)
12		simpr 484	. . . . . 6 ⊢ ((𝜑 ∧ {𝑋, 𝑌} ∈ dom (lub‘𝐾)) → {𝑋, 𝑌} ∈ dom (lub‘𝐾))
13	8, 9, 2, 10, 11, 12	lubeu 17988	. . . . 5 ⊢ ((𝜑 ∧ {𝑋, 𝑌} ∈ dom (lub‘𝐾)) → ∃!𝑥 ∈ 𝐵 (∀𝑦 ∈ {𝑋, 𝑌}𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ {𝑋, 𝑌}𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧)))
14	13	ex 412	. . . 4 ⊢ (𝜑 → ({𝑋, 𝑌} ∈ dom (lub‘𝐾) → ∃!𝑥 ∈ 𝐵 (∀𝑦 ∈ {𝑋, 𝑌}𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ {𝑋, 𝑌}𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧))))
15	8, 9, 3, 4, 5, 6	joinval2lem 18013	. . . . . 6 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((∀𝑦 ∈ {𝑋, 𝑌}𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ {𝑋, 𝑌}𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧)) ↔ ((𝑋 ≤ 𝑥 ∧ 𝑌 ≤ 𝑥) ∧ ∀𝑧 ∈ 𝐵 ((𝑋 ≤ 𝑧 ∧ 𝑌 ≤ 𝑧) → 𝑥 ≤ 𝑧))))
16	5, 6, 15	syl2anc 583	. . . . 5 ⊢ (𝜑 → ((∀𝑦 ∈ {𝑋, 𝑌}𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ {𝑋, 𝑌}𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧)) ↔ ((𝑋 ≤ 𝑥 ∧ 𝑌 ≤ 𝑥) ∧ ∀𝑧 ∈ 𝐵 ((𝑋 ≤ 𝑧 ∧ 𝑌 ≤ 𝑧) → 𝑥 ≤ 𝑧))))
17	16	reubidv 3315	. . . 4 ⊢ (𝜑 → (∃!𝑥 ∈ 𝐵 (∀𝑦 ∈ {𝑋, 𝑌}𝑦 ≤ 𝑥 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ {𝑋, 𝑌}𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧)) ↔ ∃!𝑥 ∈ 𝐵 ((𝑋 ≤ 𝑥 ∧ 𝑌 ≤ 𝑥) ∧ ∀𝑧 ∈ 𝐵 ((𝑋 ≤ 𝑧 ∧ 𝑌 ≤ 𝑧) → 𝑥 ≤ 𝑧))))
18	14, 17	sylibd 238	. . 3 ⊢ (𝜑 → ({𝑋, 𝑌} ∈ dom (lub‘𝐾) → ∃!𝑥 ∈ 𝐵 ((𝑋 ≤ 𝑥 ∧ 𝑌 ≤ 𝑥) ∧ ∀𝑧 ∈ 𝐵 ((𝑋 ≤ 𝑧 ∧ 𝑌 ≤ 𝑧) → 𝑥 ≤ 𝑧))))
19	7, 18	sylbid 239	. 2 ⊢ (𝜑 → (⟨𝑋, 𝑌⟩ ∈ dom ∨ → ∃!𝑥 ∈ 𝐵 ((𝑋 ≤ 𝑥 ∧ 𝑌 ≤ 𝑥) ∧ ∀𝑧 ∈ 𝐵 ((𝑋 ≤ 𝑧 ∧ 𝑌 ≤ 𝑧) → 𝑥 ≤ 𝑧))))
20	1, 19	mpd 15	1 ⊢ (𝜑 → ∃!𝑥 ∈ 𝐵 ((𝑋 ≤ 𝑥 ∧ 𝑌 ≤ 𝑥) ∧ ∀𝑧 ∈ 𝐵 ((𝑋 ≤ 𝑧 ∧ 𝑌 ≤ 𝑧) → 𝑥 ≤ 𝑧)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539   ∈ wcel 2108  ∀wral 3063  ∃!wreu 3065  {cpr 4560  ⟨cop 4564   class class class wbr 5070  dom cdm 5580  ‘cfv 6418  Basecbs 16840  lecple 16895  lubclub 17942  joincjn 17944

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-oprab 7259  df-lub 17979  df-join 17981

This theorem is referenced by:  joinlem  18016