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Theorem joindmss 17221
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 5385 . . 3 Rel {⟨𝑥, 𝑦⟩ ∣ {𝑥, 𝑦} ∈ dom (lub‘𝐾)}
2 joindmss.k . . . . 5 (𝜑𝐾𝑉)
3 eqid 2769 . . . . . 6 (lub‘𝐾) = (lub‘𝐾)
4 joindmss.j . . . . . 6 = (join‘𝐾)
53, 4joindm 17217 . . . . 5 (𝐾𝑉 → dom = {⟨𝑥, 𝑦⟩ ∣ {𝑥, 𝑦} ∈ dom (lub‘𝐾)})
62, 5syl 17 . . . 4 (𝜑 → dom = {⟨𝑥, 𝑦⟩ ∣ {𝑥, 𝑦} ∈ dom (lub‘𝐾)})
76releqd 5342 . . 3 (𝜑 → (Rel dom ↔ Rel {⟨𝑥, 𝑦⟩ ∣ {𝑥, 𝑦} ∈ dom (lub‘𝐾)}))
81, 7mpbiri 248 . 2 (𝜑 → Rel dom )
9 vex 3351 . . . . 5 𝑥 ∈ V
109a1i 11 . . . 4 (𝜑𝑥 ∈ V)
11 vex 3351 . . . . 5 𝑦 ∈ V
1211a1i 11 . . . 4 (𝜑𝑦 ∈ V)
133, 4, 2, 10, 12joindef 17218 . . 3 (𝜑 → (⟨𝑥, 𝑦⟩ ∈ dom ↔ {𝑥, 𝑦} ∈ dom (lub‘𝐾)))
14 joindmss.b . . . . . 6 𝐵 = (Base‘𝐾)
15 eqid 2769 . . . . . 6 (le‘𝐾) = (le‘𝐾)
162adantr 473 . . . . . 6 ((𝜑 ∧ {𝑥, 𝑦} ∈ dom (lub‘𝐾)) → 𝐾𝑉)
17 simpr 480 . . . . . 6 ((𝜑 ∧ {𝑥, 𝑦} ∈ dom (lub‘𝐾)) → {𝑥, 𝑦} ∈ dom (lub‘𝐾))
1814, 15, 3, 16, 17lubelss 17196 . . . . 5 ((𝜑 ∧ {𝑥, 𝑦} ∈ dom (lub‘𝐾)) → {𝑥, 𝑦} ⊆ 𝐵)
1918ex 448 . . . 4 (𝜑 → ({𝑥, 𝑦} ∈ dom (lub‘𝐾) → {𝑥, 𝑦} ⊆ 𝐵))
209, 11prss 4483 . . . . 5 ((𝑥𝐵𝑦𝐵) ↔ {𝑥, 𝑦} ⊆ 𝐵)
21 opelxpi 5287 . . . . 5 ((𝑥𝐵𝑦𝐵) → ⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐵))
2220, 21sylbir 225 . . . 4 ({𝑥, 𝑦} ⊆ 𝐵 → ⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐵))
2319, 22syl6 35 . . 3 (𝜑 → ({𝑥, 𝑦} ∈ dom (lub‘𝐾) → ⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐵)))
2413, 23sylbid 230 . 2 (𝜑 → (⟨𝑥, 𝑦⟩ ∈ dom → ⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐵)))
258, 24relssdv 5351 1 (𝜑 → dom ⊆ (𝐵 × 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1629  wcel 2143  Vcvv 3348  wss 3720  {cpr 4315  cop 4319  {copab 4843   × cxp 5246  dom cdm 5248  Rel wrel 5253  cfv 6030  Basecbs 16070  lecple 16162  lubclub 17156  joincjn 17158
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1868  ax-4 1883  ax-5 1989  ax-6 2055  ax-7 2091  ax-8 2145  ax-9 2152  ax-10 2172  ax-11 2188  ax-12 2201  ax-13 2406  ax-ext 2749  ax-rep 4901  ax-sep 4911  ax-nul 4919  ax-pow 4970  ax-pr 5033  ax-un 7094
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1071  df-tru 1632  df-ex 1851  df-nf 1856  df-sb 2048  df-eu 2620  df-mo 2621  df-clab 2756  df-cleq 2762  df-clel 2765  df-nfc 2900  df-ne 2942  df-ral 3064  df-rex 3065  df-reu 3066  df-rab 3068  df-v 3350  df-sbc 3585  df-csb 3680  df-dif 3723  df-un 3725  df-in 3727  df-ss 3734  df-nul 4061  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-op 4320  df-uni 4572  df-iun 4653  df-br 4784  df-opab 4844  df-mpt 4861  df-id 5156  df-xp 5254  df-rel 5255  df-cnv 5256  df-co 5257  df-dm 5258  df-rn 5259  df-res 5260  df-ima 5261  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-riota 6752  df-oprab 6795  df-lub 17188  df-join 17190
This theorem is referenced by:  clatl  17330
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