Theorem joindm2 48998

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 18280	. . 3 ⊢ (𝜑 → dom ∨ ⊆ (𝐵 × 𝐵))
5		eqss 3950	. . . 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 5634	. . 3 ⊢ Rel (𝐵 × 𝐵)
9		ssrel 5723	. . 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 5652	. . . . . 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 3440	. . . . . . 7 ⊢ 𝑥 ∈ V
15	14	a1i 11	. . . . . 6 ⊢ (𝜑 → 𝑥 ∈ V)
16		vex 3440	. . . . . . 7 ⊢ 𝑦 ∈ V
17	16	a1i 11	. . . . . 6 ⊢ (𝜑 → 𝑦 ∈ V)
18	13, 2, 3, 15, 17	joindef 18277	. . . . 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 1924	. . 3 ⊢ (𝜑 → (∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐵) → ⟨𝑥, 𝑦⟩ ∈ dom ∨ ) ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → {𝑥, 𝑦} ∈ dom 𝑈)))
21		r2al 3168	. . 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 1539   = wceq 1541   ∈ wcel 2111  ∀wral 3047  Vcvv 3436   ⊆ wss 3902  {cpr 4578  ⟨cop 4582   × cxp 5614  dom cdm 5616  Rel wrel 5621  ‘cfv 6481  Basecbs 17117  lubclub 18212  joincjn 18214

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5217  ax-sep 5234  ax-nul 5244  ax-pow 5303  ax-pr 5370  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-iun 4943  df-br 5092  df-opab 5154  df-mpt 5173  df-id 5511  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-riota 7303  df-oprab 7350  df-lub 18247  df-join 18249

This theorem is referenced by:  joindm3  48999  toslat  49012