Theorem meetdm2 50415

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 18615	. . 3 ⊢ (𝜑 → dom ∧ ⊆ (𝐵 × 𝐵))
5		eqss 3928	. . . 4 ⊢ (dom ∧ = (𝐵 × 𝐵) ↔ (dom ∧ ⊆ (𝐵 × 𝐵) ∧ (𝐵 × 𝐵) ⊆ dom ∧ ))
6	5	baib 545	. . 3 ⊢ (dom ∧ ⊆ (𝐵 × 𝐵) → (dom ∧ = (𝐵 × 𝐵) ↔ (𝐵 × 𝐵) ⊆ dom ∧ ))
7	4, 6	syl 18	. 2 ⊢ (𝜑 → (dom ∧ = (𝐵 × 𝐵) ↔ (𝐵 × 𝐵) ⊆ dom ∧ ))
8		relxp 5676	. . 3 ⊢ Rel (𝐵 × 𝐵)
9		ssrel 5767	. . 3 ⊢ (Rel (𝐵 × 𝐵) → ((𝐵 × 𝐵) ⊆ dom ∧ ↔ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐵) → ⟨𝑥, 𝑦⟩ ∈ dom ∧ )))
10	8, 9	mp1i 14	. 2 ⊢ (𝜑 → ((𝐵 × 𝐵) ⊆ dom ∧ ↔ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐵) → ⟨𝑥, 𝑦⟩ ∈ dom ∧ )))
11		opelxp 5694	. . . . . 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 3458	. . . . . . 7 ⊢ 𝑥 ∈ V
15	14	a1i 11	. . . . . 6 ⊢ (𝜑 → 𝑥 ∈ V)
16		vex 3458	. . . . . . 7 ⊢ 𝑦 ∈ V
17	16	a1i 11	. . . . . 6 ⊢ (𝜑 → 𝑦 ∈ V)
18	13, 2, 3, 15, 17	meetdef 18612	. . . . 5 ⊢ (𝜑 → (⟨𝑥, 𝑦⟩ ∈ dom ∧ ↔ {𝑥, 𝑦} ∈ dom 𝐺))
19	12, 18	imbi12d 347	. . . 4 ⊢ (𝜑 → ((⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐵) → ⟨𝑥, 𝑦⟩ ∈ dom ∧ ) ↔ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → {𝑥, 𝑦} ∈ dom 𝐺)))
20	19	2albidv 1958	. . 3 ⊢ (𝜑 → (∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐵) → ⟨𝑥, 𝑦⟩ ∈ dom ∧ ) ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → {𝑥, 𝑦} ∈ dom 𝐺)))
21		r2al 3202	. . 3 ⊢ (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 {𝑥, 𝑦} ∈ dom 𝐺 ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → {𝑥, 𝑦} ∈ dom 𝐺))
22	20, 21	bitr4di 292	. 2 ⊢ (𝜑 → (∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐵) → ⟨𝑥, 𝑦⟩ ∈ dom ∧ ) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 {𝑥, 𝑦} ∈ dom 𝐺))
23	7, 10, 22	3bitrd 308	1 ⊢ (𝜑 → (dom ∧ = (𝐵 × 𝐵) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 {𝑥, 𝑦} ∈ dom 𝐺))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1568   = wceq 1570   ∈ wcel 2147  ∀wral 3080  Vcvv 3454   ⊆ wss 3900  {cpr 4590  ⟨cop 4840   × cxp 5656  dom cdm 5658  Rel wrel 5663  ‘cfv 6566  Basecbs 17431  glbcglb 18534  meetcmee 18536

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-riota 7408  df-oprab 7455  df-glb 18569  df-meet 18571

This theorem is used by:  meetdm3  50416  toslat  50427