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Theorem joindmss 17596
Description: Subset property of domain of join. (Contributed by NM, 12-Sep-2018.)
Hypotheses
Ref Expression
joindmss.b 𝐵 = (Base‘𝐾)
joindmss.j = (join‘𝐾)
joindmss.k (𝜑𝐾𝑉)
Assertion
Ref Expression
joindmss (𝜑 → dom ⊆ (𝐵 × 𝐵))

Proof of Theorem joindmss
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relopab 5672 . . 3 Rel {⟨𝑥, 𝑦⟩ ∣ {𝑥, 𝑦} ∈ dom (lub‘𝐾)}
2 joindmss.k . . . . 5 (𝜑𝐾𝑉)
3 eqid 2820 . . . . . 6 (lub‘𝐾) = (lub‘𝐾)
4 joindmss.j . . . . . 6 = (join‘𝐾)
53, 4joindm 17592 . . . . 5 (𝐾𝑉 → dom = {⟨𝑥, 𝑦⟩ ∣ {𝑥, 𝑦} ∈ dom (lub‘𝐾)})
62, 5syl 17 . . . 4 (𝜑 → dom = {⟨𝑥, 𝑦⟩ ∣ {𝑥, 𝑦} ∈ dom (lub‘𝐾)})
76releqd 5629 . . 3 (𝜑 → (Rel dom ↔ Rel {⟨𝑥, 𝑦⟩ ∣ {𝑥, 𝑦} ∈ dom (lub‘𝐾)}))
81, 7mpbiri 260 . 2 (𝜑 → Rel dom )
9 vex 3476 . . . . 5 𝑥 ∈ V
109a1i 11 . . . 4 (𝜑𝑥 ∈ V)
11 vex 3476 . . . . 5 𝑦 ∈ V
1211a1i 11 . . . 4 (𝜑𝑦 ∈ V)
133, 4, 2, 10, 12joindef 17593 . . 3 (𝜑 → (⟨𝑥, 𝑦⟩ ∈ dom ↔ {𝑥, 𝑦} ∈ dom (lub‘𝐾)))
14 joindmss.b . . . . . 6 𝐵 = (Base‘𝐾)
15 eqid 2820 . . . . . 6 (le‘𝐾) = (le‘𝐾)
162adantr 483 . . . . . 6 ((𝜑 ∧ {𝑥, 𝑦} ∈ dom (lub‘𝐾)) → 𝐾𝑉)
17 simpr 487 . . . . . 6 ((𝜑 ∧ {𝑥, 𝑦} ∈ dom (lub‘𝐾)) → {𝑥, 𝑦} ∈ dom (lub‘𝐾))
1814, 15, 3, 16, 17lubelss 17571 . . . . 5 ((𝜑 ∧ {𝑥, 𝑦} ∈ dom (lub‘𝐾)) → {𝑥, 𝑦} ⊆ 𝐵)
1918ex 415 . . . 4 (𝜑 → ({𝑥, 𝑦} ∈ dom (lub‘𝐾) → {𝑥, 𝑦} ⊆ 𝐵))
209, 11prss 4729 . . . . 5 ((𝑥𝐵𝑦𝐵) ↔ {𝑥, 𝑦} ⊆ 𝐵)
21 opelxpi 5568 . . . . 5 ((𝑥𝐵𝑦𝐵) → ⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐵))
2220, 21sylbir 237 . . . 4 ({𝑥, 𝑦} ⊆ 𝐵 → ⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐵))
2319, 22syl6 35 . . 3 (𝜑 → ({𝑥, 𝑦} ∈ dom (lub‘𝐾) → ⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐵)))
2413, 23sylbid 242 . 2 (𝜑 → (⟨𝑥, 𝑦⟩ ∈ dom → ⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐵)))
258, 24relssdv 5637 1 (𝜑 → dom ⊆ (𝐵 × 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398   = wceq 1537  wcel 2114  Vcvv 3473  wss 3913  {cpr 4545  cop 4549  {copab 5104   × cxp 5529  dom cdm 5531  Rel wrel 5536  cfv 6331  Basecbs 16462  lecple 16551  lubclub 17531  joincjn 17533
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5242  ax-pr 5306  ax-un 7439
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-ral 3130  df-rex 3131  df-reu 3132  df-rab 3134  df-v 3475  df-sbc 3753  df-csb 3861  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4270  df-if 4444  df-pw 4517  df-sn 4544  df-pr 4546  df-op 4550  df-uni 4815  df-iun 4897  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5436  df-xp 5537  df-rel 5538  df-cnv 5539  df-co 5540  df-dm 5541  df-rn 5542  df-res 5543  df-ima 5544  df-iota 6290  df-fun 6333  df-fn 6334  df-f 6335  df-f1 6336  df-fo 6337  df-f1o 6338  df-fv 6339  df-riota 7091  df-oprab 7137  df-lub 17563  df-join 17565
This theorem is referenced by:  clatl  17705
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