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Theorem joindmss 17360
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 5480 . . 3 Rel {⟨𝑥, 𝑦⟩ ∣ {𝑥, 𝑦} ∈ dom (lub‘𝐾)}
2 joindmss.k . . . . 5 (𝜑𝐾𝑉)
3 eqid 2825 . . . . . 6 (lub‘𝐾) = (lub‘𝐾)
4 joindmss.j . . . . . 6 = (join‘𝐾)
53, 4joindm 17356 . . . . 5 (𝐾𝑉 → dom = {⟨𝑥, 𝑦⟩ ∣ {𝑥, 𝑦} ∈ dom (lub‘𝐾)})
62, 5syl 17 . . . 4 (𝜑 → dom = {⟨𝑥, 𝑦⟩ ∣ {𝑥, 𝑦} ∈ dom (lub‘𝐾)})
76releqd 5438 . . 3 (𝜑 → (Rel dom ↔ Rel {⟨𝑥, 𝑦⟩ ∣ {𝑥, 𝑦} ∈ dom (lub‘𝐾)}))
81, 7mpbiri 250 . 2 (𝜑 → Rel dom )
9 vex 3417 . . . . 5 𝑥 ∈ V
109a1i 11 . . . 4 (𝜑𝑥 ∈ V)
11 vex 3417 . . . . 5 𝑦 ∈ V
1211a1i 11 . . . 4 (𝜑𝑦 ∈ V)
133, 4, 2, 10, 12joindef 17357 . . 3 (𝜑 → (⟨𝑥, 𝑦⟩ ∈ dom ↔ {𝑥, 𝑦} ∈ dom (lub‘𝐾)))
14 joindmss.b . . . . . 6 𝐵 = (Base‘𝐾)
15 eqid 2825 . . . . . 6 (le‘𝐾) = (le‘𝐾)
162adantr 474 . . . . . 6 ((𝜑 ∧ {𝑥, 𝑦} ∈ dom (lub‘𝐾)) → 𝐾𝑉)
17 simpr 479 . . . . . 6 ((𝜑 ∧ {𝑥, 𝑦} ∈ dom (lub‘𝐾)) → {𝑥, 𝑦} ∈ dom (lub‘𝐾))
1814, 15, 3, 16, 17lubelss 17335 . . . . 5 ((𝜑 ∧ {𝑥, 𝑦} ∈ dom (lub‘𝐾)) → {𝑥, 𝑦} ⊆ 𝐵)
1918ex 403 . . . 4 (𝜑 → ({𝑥, 𝑦} ∈ dom (lub‘𝐾) → {𝑥, 𝑦} ⊆ 𝐵))
209, 11prss 4569 . . . . 5 ((𝑥𝐵𝑦𝐵) ↔ {𝑥, 𝑦} ⊆ 𝐵)
21 opelxpi 5379 . . . . 5 ((𝑥𝐵𝑦𝐵) → ⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐵))
2220, 21sylbir 227 . . . 4 ({𝑥, 𝑦} ⊆ 𝐵 → ⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐵))
2319, 22syl6 35 . . 3 (𝜑 → ({𝑥, 𝑦} ∈ dom (lub‘𝐾) → ⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐵)))
2413, 23sylbid 232 . 2 (𝜑 → (⟨𝑥, 𝑦⟩ ∈ dom → ⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐵)))
258, 24relssdv 5446 1 (𝜑 → dom ⊆ (𝐵 × 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386   = wceq 1656  wcel 2164  Vcvv 3414  wss 3798  {cpr 4399  cop 4403  {copab 4935   × cxp 5340  dom cdm 5342  Rel wrel 5347  cfv 6123  Basecbs 16222  lecple 16312  lubclub 17295  joincjn 17297
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-rep 4994  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-iun 4742  df-br 4874  df-opab 4936  df-mpt 4953  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-fv 6131  df-riota 6866  df-oprab 6909  df-lub 17327  df-join 17329
This theorem is referenced by:  clatl  17469
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