Theorem meetdm2 48867

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

Hypotheses

Ref	Expression
joindm2.b	⊢ 𝐵 = (Base‘𝐾)
joindm2.k	⊢ (𝜑 → 𝐾 ∈ 𝑉)
meetdm2.g	⊢ 𝐺 = (glb‘𝐾)
meetdm2.m	⊢ ∧ = (meet‘𝐾)

Assertion

Ref	Expression
meetdm2	⊢ (𝜑 → (dom ∧ = (𝐵 × 𝐵) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 {𝑥, 𝑦} ∈ dom 𝐺))

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

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

Proof of Theorem meetdm2

Step	Hyp	Ref	Expression
1		joindm2.b	. . . 4 ⊢ 𝐵 = (Base‘𝐾)
2		meetdm2.m	. . . 4 ⊢ ∧ = (meet‘𝐾)
3		joindm2.k	. . . 4 ⊢ (𝜑 → 𝐾 ∈ 𝑉)
4	1, 2, 3	meetdmss 18438	. . 3 ⊢ (𝜑 → dom ∧ ⊆ (𝐵 × 𝐵))
5		eqss 3999	. . . 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 5703	. . 3 ⊢ Rel (𝐵 × 𝐵)
9		ssrel 5792	. . 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 5721	. . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐵) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵))
12	11	a1i 11	. . . . 5 ⊢ (𝜑 → (⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐵) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)))
13		meetdm2.g	. . . . . 6 ⊢ 𝐺 = (glb‘𝐾)
14		vex 3484	. . . . . . 7 ⊢ 𝑥 ∈ V
15	14	a1i 11	. . . . . 6 ⊢ (𝜑 → 𝑥 ∈ V)
16		vex 3484	. . . . . . 7 ⊢ 𝑦 ∈ V
17	16	a1i 11	. . . . . 6 ⊢ (𝜑 → 𝑦 ∈ V)
18	13, 2, 3, 15, 17	meetdef 18435	. . . . 5 ⊢ (𝜑 → (⟨𝑥, 𝑦⟩ ∈ dom ∧ ↔ {𝑥, 𝑦} ∈ dom 𝐺))
19	12, 18	imbi12d 344	. . . 4 ⊢ (𝜑 → ((⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐵) → ⟨𝑥, 𝑦⟩ ∈ dom ∧ ) ↔ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → {𝑥, 𝑦} ∈ dom 𝐺)))
20	19	2albidv 1923	. . 3 ⊢ (𝜑 → (∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐵) → ⟨𝑥, 𝑦⟩ ∈ dom ∧ ) ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → {𝑥, 𝑦} ∈ dom 𝐺)))
21		r2al 3195	. . 3 ⊢ (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 {𝑥, 𝑦} ∈ dom 𝐺 ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → {𝑥, 𝑦} ∈ dom 𝐺))
22	20, 21	bitr4di 289	. 2 ⊢ (𝜑 → (∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐵) → ⟨𝑥, 𝑦⟩ ∈ dom ∧ ) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 {𝑥, 𝑦} ∈ dom 𝐺))
23	7, 10, 22	3bitrd 305	1 ⊢ (𝜑 → (dom ∧ = (𝐵 × 𝐵) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 {𝑥, 𝑦} ∈ dom 𝐺))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1538   = wceq 1540   ∈ wcel 2108  ∀wral 3061  Vcvv 3480   ⊆ wss 3951  {cpr 4628  ⟨cop 4632   × cxp 5683  dom cdm 5685  Rel wrel 5690  ‘cfv 6561  Basecbs 17247  glbcglb 18356  meetcmee 18358

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-oprab 7435  df-glb 18392  df-meet 18394

This theorem is referenced by:  meetdm3  48868  toslat  48871