Theorem meetdmss 18392

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 5827	. . 3 ⊢ Rel {⟨𝑥, 𝑦⟩ ∣ {𝑥, 𝑦} ∈ dom (glb‘𝐾)}
2		meetdmss.k	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ 𝑉)
3		eqid 2728	. . . . . 6 ⊢ (glb‘𝐾) = (glb‘𝐾)
4		meetdmss.j	. . . . . 6 ⊢ ∧ = (meet‘𝐾)
5	3, 4	meetdm 18388	. . . . 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 5784	. . 3 ⊢ (𝜑 → (Rel dom ∧ ↔ Rel {⟨𝑥, 𝑦⟩ ∣ {𝑥, 𝑦} ∈ dom (glb‘𝐾)}))
8	1, 7	mpbiri 257	. 2 ⊢ (𝜑 → Rel dom ∧ )
9		vex 3477	. . . . 5 ⊢ 𝑥 ∈ V
10	9	a1i 11	. . . 4 ⊢ (𝜑 → 𝑥 ∈ V)
11		vex 3477	. . . . 5 ⊢ 𝑦 ∈ V
12	11	a1i 11	. . . 4 ⊢ (𝜑 → 𝑦 ∈ V)
13	3, 4, 2, 10, 12	meetdef 18389	. . 3 ⊢ (𝜑 → (⟨𝑥, 𝑦⟩ ∈ dom ∧ ↔ {𝑥, 𝑦} ∈ dom (glb‘𝐾)))
14		meetdmss.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐾)
15		eqid 2728	. . . . . 6 ⊢ (le‘𝐾) = (le‘𝐾)
16	2	adantr 479	. . . . . 6 ⊢ ((𝜑 ∧ {𝑥, 𝑦} ∈ dom (glb‘𝐾)) → 𝐾 ∈ 𝑉)
17		simpr 483	. . . . . 6 ⊢ ((𝜑 ∧ {𝑥, 𝑦} ∈ dom (glb‘𝐾)) → {𝑥, 𝑦} ∈ dom (glb‘𝐾))
18	14, 15, 3, 16, 17	glbelss 18366	. . . . 5 ⊢ ((𝜑 ∧ {𝑥, 𝑦} ∈ dom (glb‘𝐾)) → {𝑥, 𝑦} ⊆ 𝐵)
19	18	ex 411	. . . 4 ⊢ (𝜑 → ({𝑥, 𝑦} ∈ dom (glb‘𝐾) → {𝑥, 𝑦} ⊆ 𝐵))
20	9, 11	prss 4828	. . . . 5 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ↔ {𝑥, 𝑦} ⊆ 𝐵)
21		opelxpi 5719	. . . . 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 5794	1 ⊢ (𝜑 → dom ∧ ⊆ (𝐵 × 𝐵))

Syntax hints:   → wi 4   ∧ wa 394   = wceq 1533   ∈ wcel 2098  Vcvv 3473   ⊆ wss 3949  {cpr 4634  ⟨cop 4638  {copab 5214   × cxp 5680  dom cdm 5682  Rel wrel 5687  ‘cfv 6553  Basecbs 17187  lecple 17247  glbcglb 18309  meetcmee 18311

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 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-riota 7382  df-oprab 7430  df-glb 18346  df-meet 18348

This theorem is referenced by:  clatl  18507  meetdm2  48067