Theorem meetdm2 48954

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 18297	. . 3 ⊢ (𝜑 → dom ∧ ⊆ (𝐵 × 𝐵))
5		eqss 3951	. . . 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 5637	. . 3 ⊢ Rel (𝐵 × 𝐵)
9		ssrel 5726	. . 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 5655	. . . . . 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 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	meetdef 18294	. . . . 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 3165	. . 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 2109  ∀wral 3044  Vcvv 3436   ⊆ wss 3903  {cpr 4579  ⟨cop 4583   × cxp 5617  dom cdm 5619  Rel wrel 5624  ‘cfv 6482  Basecbs 17120  glbcglb 18216  meetcmee 18218

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 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671

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 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3343  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-riota 7306  df-oprab 7353  df-glb 18251  df-meet 18253

This theorem is referenced by:  meetdm3  48955  toslat  48966