Theorem joineu 17010

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 2622	. . . 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 17004	. . 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 ∧ ∀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 481	. . . . . 6 ⊢ ((𝜑 ∧ {𝑋, 𝑌} ∈ dom (lub‘𝐾)) → 𝐾 ∈ 𝑉)
12		simpr 477	. . . . . 6 ⊢ ((𝜑 ∧ {𝑋, 𝑌} ∈ dom (lub‘𝐾)) → {𝑋, 𝑌} ∈ dom (lub‘𝐾))
13	8, 9, 2, 10, 11, 12	lubeu 16983	. . . . 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 450	. . . 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 17008	. . . . . 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 693	. . . . 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 3126	. . . 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 229	. . 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 230	. 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 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∀wral 2912  ∃!wreu 2914  {cpr 4179  ⟨cop 4183   class class class wbr 4653  dom cdm 5114  ‘cfv 5888  Basecbs 15857  lecple 15948  lubclub 16942  joincjn 16944

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-oprab 6654  df-lub 16974  df-join 16976

This theorem is referenced by:  joinlem  17011