Theorem meetdmss 18446

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 5809	. . 3 ⊢ Rel {⟨𝑥, 𝑦⟩ ∣ {𝑥, 𝑦} ∈ dom (glb‘𝐾)}
2		meetdmss.k	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ 𝑉)
3		eqid 2769	. . . . . 6 ⊢ (glb‘𝐾) = (glb‘𝐾)
4		meetdmss.j	. . . . . 6 ⊢ ∧ = (meet‘𝐾)
5	3, 4	meetdm 18442	. . . . 5 ⊢ (𝐾 ∈ 𝑉 → dom ∧ = {⟨𝑥, 𝑦⟩ ∣ {𝑥, 𝑦} ∈ dom (glb‘𝐾)})
6	2, 5	syl 18	. . . 4 ⊢ (𝜑 → dom ∧ = {⟨𝑥, 𝑦⟩ ∣ {𝑥, 𝑦} ∈ dom (glb‘𝐾)})
7	6	releqd 5766	. . 3 ⊢ (𝜑 → (Rel dom ∧ ↔ Rel {⟨𝑥, 𝑦⟩ ∣ {𝑥, 𝑦} ∈ dom (glb‘𝐾)}))
8	1, 7	mpbiri 261	. 2 ⊢ (𝜑 → Rel dom ∧ )
9		vex 3467	. . . . 5 ⊢ 𝑥 ∈ V
10	9	a1i 11	. . . 4 ⊢ (𝜑 → 𝑥 ∈ V)
11		vex 3467	. . . . 5 ⊢ 𝑦 ∈ V
12	11	a1i 11	. . . 4 ⊢ (𝜑 → 𝑦 ∈ V)
13	3, 4, 2, 10, 12	meetdef 18443	. . 3 ⊢ (𝜑 → (⟨𝑥, 𝑦⟩ ∈ dom ∧ ↔ {𝑥, 𝑦} ∈ dom (glb‘𝐾)))
14		meetdmss.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐾)
15		eqid 2769	. . . . . 6 ⊢ (le‘𝐾) = (le‘𝐾)
16	2	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ {𝑥, 𝑦} ∈ dom (glb‘𝐾)) → 𝐾 ∈ 𝑉)
17		simpr 489	. . . . . 6 ⊢ ((𝜑 ∧ {𝑥, 𝑦} ∈ dom (glb‘𝐾)) → {𝑥, 𝑦} ∈ dom (glb‘𝐾))
18	14, 15, 3, 16, 17	glbelss 18420	. . . . 5 ⊢ ((𝜑 ∧ {𝑥, 𝑦} ∈ dom (glb‘𝐾)) → {𝑥, 𝑦} ⊆ 𝐵)
19	18	ex 417	. . . 4 ⊢ (𝜑 → ({𝑥, 𝑦} ∈ dom (glb‘𝐾) → {𝑥, 𝑦} ⊆ 𝐵))
20	9, 11	prss 4790	. . . . 5 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ↔ {𝑥, 𝑦} ⊆ 𝐵)
21		opelxpi 5699	. . . . 5 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → ⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐵))
22	20, 21	sylbir 238	. . . 4 ⊢ ({𝑥, 𝑦} ⊆ 𝐵 → ⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐵))
23	19, 22	syl6 36	. . 3 ⊢ (𝜑 → ({𝑥, 𝑦} ∈ dom (glb‘𝐾) → ⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐵)))
24	13, 23	sylbid 243	. 2 ⊢ (𝜑 → (⟨𝑥, 𝑦⟩ ∈ dom ∧ → ⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐵)))
25	8, 24	relssdv 5775	1 ⊢ (𝜑 → dom ∧ ⊆ (𝐵 × 𝐵))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1567   ∈ wcel 2149  Vcvv 3463   ⊆ wss 3913  {cpr 4596  ⟨cop 4600  {copab 5177   × cxp 5660  dom cdm 5662  Rel wrel 5667  ‘cfv 6537  Basecbs 17268  lecple 17316  glbcglb 18365  meetcmee 18367

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7368  df-oprab 7415  df-glb 18400  df-meet 18402

This theorem is referenced by:  clatl  18563  meetdm2  49632