Theorem joindm2 48922

Description: The join of any two elements always exists iff all unordered pairs have LUB. (Contributed by Zhi Wang, 25-Sep-2024.)

Hypotheses

Ref	Expression
joindm2.b	⊢ 𝐵 = (Base‘𝐾)
joindm2.k	⊢ (𝜑 → 𝐾 ∈ 𝑉)
joindm2.u	⊢ 𝑈 = (lub‘𝐾)
joindm2.j	⊢ ∨ = (join‘𝐾)

Assertion

Ref	Expression
joindm2	⊢ (𝜑 → (dom ∨ = (𝐵 × 𝐵) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 {𝑥, 𝑦} ∈ dom 𝑈))

Distinct variable groups:   𝑥,𝑦, ∨   𝑥,𝐵,𝑦   𝜑,𝑥,𝑦

Allowed substitution hints:   𝑈(𝑥,𝑦)   𝐾(𝑥,𝑦)   𝑉(𝑥,𝑦)

Proof of Theorem joindm2

Step	Hyp	Ref	Expression
1		joindm2.b	. . . 4 ⊢ 𝐵 = (Base‘𝐾)
2		joindm2.j	. . . 4 ⊢ ∨ = (join‘𝐾)
3		joindm2.k	. . . 4 ⊢ (𝜑 → 𝐾 ∈ 𝑉)
4	1, 2, 3	joindmss 18394	. . 3 ⊢ (𝜑 → dom ∨ ⊆ (𝐵 × 𝐵))
5		eqss 3979	. . . 4 ⊢ (dom ∨ = (𝐵 × 𝐵) ↔ (dom ∨ ⊆ (𝐵 × 𝐵) ∧ (𝐵 × 𝐵) ⊆ dom ∨ ))
6	5	baib 535	. . 3 ⊢ (dom ∨ ⊆ (𝐵 × 𝐵) → (dom ∨ = (𝐵 × 𝐵) ↔ (𝐵 × 𝐵) ⊆ dom ∨ ))
7	4, 6	syl 17	. 2 ⊢ (𝜑 → (dom ∨ = (𝐵 × 𝐵) ↔ (𝐵 × 𝐵) ⊆ dom ∨ ))
8		relxp 5677	. . 3 ⊢ Rel (𝐵 × 𝐵)
9		ssrel 5766	. . 3 ⊢ (Rel (𝐵 × 𝐵) → ((𝐵 × 𝐵) ⊆ dom ∨ ↔ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐵) → ⟨𝑥, 𝑦⟩ ∈ dom ∨ )))
10	8, 9	mp1i 13	. 2 ⊢ (𝜑 → ((𝐵 × 𝐵) ⊆ dom ∨ ↔ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐵) → ⟨𝑥, 𝑦⟩ ∈ dom ∨ )))
11		opelxp 5695	. . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐵) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵))
12	11	a1i 11	. . . . 5 ⊢ (𝜑 → (⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐵) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)))
13		joindm2.u	. . . . . 6 ⊢ 𝑈 = (lub‘𝐾)
14		vex 3468	. . . . . . 7 ⊢ 𝑥 ∈ V
15	14	a1i 11	. . . . . 6 ⊢ (𝜑 → 𝑥 ∈ V)
16		vex 3468	. . . . . . 7 ⊢ 𝑦 ∈ V
17	16	a1i 11	. . . . . 6 ⊢ (𝜑 → 𝑦 ∈ V)
18	13, 2, 3, 15, 17	joindef 18391	. . . . 5 ⊢ (𝜑 → (⟨𝑥, 𝑦⟩ ∈ dom ∨ ↔ {𝑥, 𝑦} ∈ dom 𝑈))
19	12, 18	imbi12d 344	. . . 4 ⊢ (𝜑 → ((⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐵) → ⟨𝑥, 𝑦⟩ ∈ dom ∨ ) ↔ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → {𝑥, 𝑦} ∈ dom 𝑈)))
20	19	2albidv 1923	. . 3 ⊢ (𝜑 → (∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐵) → ⟨𝑥, 𝑦⟩ ∈ dom ∨ ) ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → {𝑥, 𝑦} ∈ dom 𝑈)))
21		r2al 3181	. . 3 ⊢ (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 {𝑥, 𝑦} ∈ dom 𝑈 ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → {𝑥, 𝑦} ∈ dom 𝑈))
22	20, 21	bitr4di 289	. 2 ⊢ (𝜑 → (∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐵) → ⟨𝑥, 𝑦⟩ ∈ dom ∨ ) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 {𝑥, 𝑦} ∈ dom 𝑈))
23	7, 10, 22	3bitrd 305	1 ⊢ (𝜑 → (dom ∨ = (𝐵 × 𝐵) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 {𝑥, 𝑦} ∈ dom 𝑈))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1538   = wceq 1540   ∈ wcel 2109  ∀wral 3052  Vcvv 3464   ⊆ wss 3931  {cpr 4608  ⟨cop 4612   × cxp 5657  dom cdm 5659  Rel wrel 5664  ‘cfv 6536  Basecbs 17233  lubclub 18326  joincjn 18328

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-rep 5254  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rmo 3364  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-id 5553  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-oprab 7414  df-lub 18361  df-join 18363

This theorem is referenced by:  joindm3  48923  toslat  48936