Theorem meetdmss 18358

Description: Subset property of domain of meet. (Contributed by NM, 12-Sep-2018.)

Hypotheses

Ref	Expression
meetdmss.b	⊢ 𝐵 = (Base‘𝐾)
meetdmss.j	⊢ ∧ = (meet‘𝐾)
meetdmss.k	⊢ (𝜑 → 𝐾 ∈ 𝑉)

Assertion

Ref	Expression
meetdmss	⊢ (𝜑 → dom ∧ ⊆ (𝐵 × 𝐵))

Proof of Theorem meetdmss

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		relopabv 5814	. . 3 ⊢ Rel {⟨𝑥, 𝑦⟩ ∣ {𝑥, 𝑦} ∈ dom (glb‘𝐾)}
2		meetdmss.k	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ 𝑉)
3		eqid 2726	. . . . . 6 ⊢ (glb‘𝐾) = (glb‘𝐾)
4		meetdmss.j	. . . . . 6 ⊢ ∧ = (meet‘𝐾)
5	3, 4	meetdm 18354	. . . . 5 ⊢ (𝐾 ∈ 𝑉 → dom ∧ = {⟨𝑥, 𝑦⟩ ∣ {𝑥, 𝑦} ∈ dom (glb‘𝐾)})
6	2, 5	syl 17	. . . 4 ⊢ (𝜑 → dom ∧ = {⟨𝑥, 𝑦⟩ ∣ {𝑥, 𝑦} ∈ dom (glb‘𝐾)})
7	6	releqd 5771	. . 3 ⊢ (𝜑 → (Rel dom ∧ ↔ Rel {⟨𝑥, 𝑦⟩ ∣ {𝑥, 𝑦} ∈ dom (glb‘𝐾)}))
8	1, 7	mpbiri 258	. 2 ⊢ (𝜑 → Rel dom ∧ )
9		vex 3472	. . . . 5 ⊢ 𝑥 ∈ V
10	9	a1i 11	. . . 4 ⊢ (𝜑 → 𝑥 ∈ V)
11		vex 3472	. . . . 5 ⊢ 𝑦 ∈ V
12	11	a1i 11	. . . 4 ⊢ (𝜑 → 𝑦 ∈ V)
13	3, 4, 2, 10, 12	meetdef 18355	. . 3 ⊢ (𝜑 → (⟨𝑥, 𝑦⟩ ∈ dom ∧ ↔ {𝑥, 𝑦} ∈ dom (glb‘𝐾)))
14		meetdmss.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐾)
15		eqid 2726	. . . . . 6 ⊢ (le‘𝐾) = (le‘𝐾)
16	2	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ {𝑥, 𝑦} ∈ dom (glb‘𝐾)) → 𝐾 ∈ 𝑉)
17		simpr 484	. . . . . 6 ⊢ ((𝜑 ∧ {𝑥, 𝑦} ∈ dom (glb‘𝐾)) → {𝑥, 𝑦} ∈ dom (glb‘𝐾))
18	14, 15, 3, 16, 17	glbelss 18332	. . . . 5 ⊢ ((𝜑 ∧ {𝑥, 𝑦} ∈ dom (glb‘𝐾)) → {𝑥, 𝑦} ⊆ 𝐵)
19	18	ex 412	. . . 4 ⊢ (𝜑 → ({𝑥, 𝑦} ∈ dom (glb‘𝐾) → {𝑥, 𝑦} ⊆ 𝐵))
20	9, 11	prss 4818	. . . . 5 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ↔ {𝑥, 𝑦} ⊆ 𝐵)
21		opelxpi 5706	. . . . 5 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → ⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐵))
22	20, 21	sylbir 234	. . . 4 ⊢ ({𝑥, 𝑦} ⊆ 𝐵 → ⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐵))
23	19, 22	syl6 35	. . 3 ⊢ (𝜑 → ({𝑥, 𝑦} ∈ dom (glb‘𝐾) → ⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐵)))
24	13, 23	sylbid 239	. 2 ⊢ (𝜑 → (⟨𝑥, 𝑦⟩ ∈ dom ∧ → ⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐵)))
25	8, 24	relssdv 5781	1 ⊢ (𝜑 → dom ∧ ⊆ (𝐵 × 𝐵))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1533   ∈ wcel 2098  Vcvv 3468   ⊆ wss 3943  {cpr 4625  ⟨cop 4629  {copab 5203   × cxp 5667  dom cdm 5669  Rel wrel 5674  ‘cfv 6537  Basecbs 17153  lecple 17213  glbcglb 18275  meetcmee 18277

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7361  df-oprab 7409  df-glb 18312  df-meet 18314

This theorem is referenced by:  clatl  18473  meetdm2  47874