MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  joindmss Structured version   Visualization version   GIF version

Theorem joindmss 18376
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 relopabv 5825 . . 3 Rel {⟨𝑥, 𝑦⟩ ∣ {𝑥, 𝑦} ∈ dom (lub‘𝐾)}
2 joindmss.k . . . . 5 (𝜑𝐾𝑉)
3 eqid 2727 . . . . . 6 (lub‘𝐾) = (lub‘𝐾)
4 joindmss.j . . . . . 6 = (join‘𝐾)
53, 4joindm 18372 . . . . 5 (𝐾𝑉 → dom = {⟨𝑥, 𝑦⟩ ∣ {𝑥, 𝑦} ∈ dom (lub‘𝐾)})
62, 5syl 17 . . . 4 (𝜑 → dom = {⟨𝑥, 𝑦⟩ ∣ {𝑥, 𝑦} ∈ dom (lub‘𝐾)})
76releqd 5782 . . 3 (𝜑 → (Rel dom ↔ Rel {⟨𝑥, 𝑦⟩ ∣ {𝑥, 𝑦} ∈ dom (lub‘𝐾)}))
81, 7mpbiri 257 . 2 (𝜑 → Rel dom )
9 vex 3475 . . . . 5 𝑥 ∈ V
109a1i 11 . . . 4 (𝜑𝑥 ∈ V)
11 vex 3475 . . . . 5 𝑦 ∈ V
1211a1i 11 . . . 4 (𝜑𝑦 ∈ V)
133, 4, 2, 10, 12joindef 18373 . . 3 (𝜑 → (⟨𝑥, 𝑦⟩ ∈ dom ↔ {𝑥, 𝑦} ∈ dom (lub‘𝐾)))
14 joindmss.b . . . . . 6 𝐵 = (Base‘𝐾)
15 eqid 2727 . . . . . 6 (le‘𝐾) = (le‘𝐾)
162adantr 479 . . . . . 6 ((𝜑 ∧ {𝑥, 𝑦} ∈ dom (lub‘𝐾)) → 𝐾𝑉)
17 simpr 483 . . . . . 6 ((𝜑 ∧ {𝑥, 𝑦} ∈ dom (lub‘𝐾)) → {𝑥, 𝑦} ∈ dom (lub‘𝐾))
1814, 15, 3, 16, 17lubelss 18351 . . . . 5 ((𝜑 ∧ {𝑥, 𝑦} ∈ dom (lub‘𝐾)) → {𝑥, 𝑦} ⊆ 𝐵)
1918ex 411 . . . 4 (𝜑 → ({𝑥, 𝑦} ∈ dom (lub‘𝐾) → {𝑥, 𝑦} ⊆ 𝐵))
209, 11prss 4826 . . . . 5 ((𝑥𝐵𝑦𝐵) ↔ {𝑥, 𝑦} ⊆ 𝐵)
21 opelxpi 5717 . . . . 5 ((𝑥𝐵𝑦𝐵) → ⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐵))
2220, 21sylbir 234 . . . 4 ({𝑥, 𝑦} ⊆ 𝐵 → ⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐵))
2319, 22syl6 35 . . 3 (𝜑 → ({𝑥, 𝑦} ∈ dom (lub‘𝐾) → ⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐵)))
2413, 23sylbid 239 . 2 (𝜑 → (⟨𝑥, 𝑦⟩ ∈ dom → ⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐵)))
258, 24relssdv 5792 1 (𝜑 → dom ⊆ (𝐵 × 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1533  wcel 2098  Vcvv 3471  wss 3947  {cpr 4632  cop 4636  {copab 5212   × cxp 5678  dom cdm 5680  Rel wrel 5685  cfv 6551  Basecbs 17185  lecple 17245  lubclub 18306  joincjn 18308
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 2698  ax-rep 5287  ax-sep 5301  ax-nul 5308  ax-pow 5367  ax-pr 5431  ax-un 7744
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 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2937  df-ral 3058  df-rex 3067  df-rmo 3372  df-reu 3373  df-rab 3429  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4325  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4911  df-iun 5000  df-br 5151  df-opab 5213  df-mpt 5234  df-id 5578  df-xp 5686  df-rel 5687  df-cnv 5688  df-co 5689  df-dm 5690  df-rn 5691  df-res 5692  df-ima 5693  df-iota 6503  df-fun 6553  df-fn 6554  df-f 6555  df-f1 6556  df-fo 6557  df-f1o 6558  df-fv 6559  df-riota 7380  df-oprab 7428  df-lub 18343  df-join 18345
This theorem is referenced by:  clatl  18505  joindm2  48038
  Copyright terms: Public domain W3C validator