Theorem meeteu 18482

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	⊢ (𝜑 → 𝐾 ∈ 𝑉)
meetval2.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
meetval2.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)
meetlem.e	⊢ (𝜑 → ⟨𝑋, 𝑌⟩ ∈ dom ∧ )

Assertion

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

Step	Hyp	Ref	Expression
1		meetlem.e	. 2 ⊢ (𝜑 → ⟨𝑋, 𝑌⟩ ∈ dom ∧ )
2		eqid 2760	. . . 4 ⊢ (glb‘𝐾) = (glb‘𝐾)
3		meetval2.m	. . . 4 ⊢ ∧ = (meet‘𝐾)
4		meetval2.k	. . . 4 ⊢ (𝜑 → 𝐾 ∈ 𝑉)
5		meetval2.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
6		meetval2.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
7	2, 3, 4, 5, 6	meetdef 18476	. . 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 264	. . . . . 6 ⊢ ((∀𝑦 ∈ {𝑋, 𝑌}𝑥 ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ {𝑋, 𝑌}𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥)) ↔ (∀𝑦 ∈ {𝑋, 𝑌}𝑥 ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ {𝑋, 𝑌}𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥)))
11	4	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ {𝑋, 𝑌} ∈ dom (glb‘𝐾)) → 𝐾 ∈ 𝑉)
12		simpr 490	. . . . . 6 ⊢ ((𝜑 ∧ {𝑋, 𝑌} ∈ dom (glb‘𝐾)) → {𝑋, 𝑌} ∈ dom (glb‘𝐾))
13	8, 9, 2, 10, 11, 12	glbeu 18454	. . . . 5 ⊢ ((𝜑 ∧ {𝑋, 𝑌} ∈ dom (glb‘𝐾)) → ∃!𝑥 ∈ 𝐵 (∀𝑦 ∈ {𝑋, 𝑌}𝑥 ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ {𝑋, 𝑌}𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥)))
14	13	ex 418	. . . 4 ⊢ (𝜑 → ({𝑋, 𝑌} ∈ dom (glb‘𝐾) → ∃!𝑥 ∈ 𝐵 (∀𝑦 ∈ {𝑋, 𝑌}𝑥 ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ {𝑋, 𝑌}𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥))))
15	8, 9, 3, 4, 5, 6	meetval2lem 18480	. . . . . 6 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((∀𝑦 ∈ {𝑋, 𝑌}𝑥 ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ {𝑋, 𝑌}𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥)) ↔ ((𝑥 ≤ 𝑋 ∧ 𝑥 ≤ 𝑌) ∧ ∀𝑧 ∈ 𝐵 ((𝑧 ≤ 𝑋 ∧ 𝑧 ≤ 𝑌) → 𝑧 ≤ 𝑥))))
16	5, 6, 15	syl2anc 596	. . . . 5 ⊢ (𝜑 → ((∀𝑦 ∈ {𝑋, 𝑌}𝑥 ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ {𝑋, 𝑌}𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥)) ↔ ((𝑥 ≤ 𝑋 ∧ 𝑥 ≤ 𝑌) ∧ ∀𝑧 ∈ 𝐵 ((𝑧 ≤ 𝑋 ∧ 𝑧 ≤ 𝑌) → 𝑧 ≤ 𝑥))))
17	16	reubidv 3381	. . . 4 ⊢ (𝜑 → (∃!𝑥 ∈ 𝐵 (∀𝑦 ∈ {𝑋, 𝑌}𝑥 ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ {𝑋, 𝑌}𝑧 ≤ 𝑦 → 𝑧 ≤ 𝑥)) ↔ ∃!𝑥 ∈ 𝐵 ((𝑥 ≤ 𝑋 ∧ 𝑥 ≤ 𝑌) ∧ ∀𝑧 ∈ 𝐵 ((𝑧 ≤ 𝑋 ∧ 𝑧 ≤ 𝑌) → 𝑧 ≤ 𝑥))))
18	14, 17	sylibd 242	. . 3 ⊢ (𝜑 → ({𝑋, 𝑌} ∈ dom (glb‘𝐾) → ∃!𝑥 ∈ 𝐵 ((𝑥 ≤ 𝑋 ∧ 𝑥 ≤ 𝑌) ∧ ∀𝑧 ∈ 𝐵 ((𝑧 ≤ 𝑋 ∧ 𝑧 ≤ 𝑌) → 𝑧 ≤ 𝑥))))
19	7, 18	sylbid 243	. 2 ⊢ (𝜑 → (⟨𝑋, 𝑌⟩ ∈ dom ∧ → ∃!𝑥 ∈ 𝐵 ((𝑥 ≤ 𝑋 ∧ 𝑥 ≤ 𝑌) ∧ ∀𝑧 ∈ 𝐵 ((𝑧 ≤ 𝑋 ∧ 𝑧 ≤ 𝑌) → 𝑧 ≤ 𝑥))))
20	1, 19	mpd 16	1 ⊢ (𝜑 → ∃!𝑥 ∈ 𝐵 ((𝑥 ≤ 𝑋 ∧ 𝑥 ≤ 𝑌) ∧ ∀𝑧 ∈ 𝐵 ((𝑧 ≤ 𝑋 ∧ 𝑧 ≤ 𝑌) → 𝑧 ≤ 𝑥)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3076  ∃!wreu 3363  {cpr 4586  ⟨cop 4590   class class class wbr 5103  dom cdm 5655  ‘cfv 6533  Basecbs 17301  lecple 17349  glbcglb 18398  meetcmee 18400

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7736

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7370  df-oprab 7417  df-glb 18433  df-meet 18435

This theorem is used by:  meetlem  18483