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Theorem joindmss 18368
Description: Subset property of domain of join. (Contributed by NM, 12-Sep-2018.)
Hypotheses
Ref Expression
joindmss.b 𝐡 = (Baseβ€˜πΎ)
joindmss.j ∨ = (joinβ€˜πΎ)
joindmss.k (πœ‘ β†’ 𝐾 ∈ 𝑉)
Assertion
Ref Expression
joindmss (πœ‘ β†’ dom ∨ βŠ† (𝐡 Γ— 𝐡))

Proof of Theorem joindmss
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relopabv 5815 . . 3 Rel {⟨π‘₯, π‘¦βŸ© ∣ {π‘₯, 𝑦} ∈ dom (lubβ€˜πΎ)}
2 joindmss.k . . . . 5 (πœ‘ β†’ 𝐾 ∈ 𝑉)
3 eqid 2725 . . . . . 6 (lubβ€˜πΎ) = (lubβ€˜πΎ)
4 joindmss.j . . . . . 6 ∨ = (joinβ€˜πΎ)
53, 4joindm 18364 . . . . 5 (𝐾 ∈ 𝑉 β†’ dom ∨ = {⟨π‘₯, π‘¦βŸ© ∣ {π‘₯, 𝑦} ∈ dom (lubβ€˜πΎ)})
62, 5syl 17 . . . 4 (πœ‘ β†’ dom ∨ = {⟨π‘₯, π‘¦βŸ© ∣ {π‘₯, 𝑦} ∈ dom (lubβ€˜πΎ)})
76releqd 5772 . . 3 (πœ‘ β†’ (Rel dom ∨ ↔ Rel {⟨π‘₯, π‘¦βŸ© ∣ {π‘₯, 𝑦} ∈ dom (lubβ€˜πΎ)}))
81, 7mpbiri 257 . 2 (πœ‘ β†’ Rel dom ∨ )
9 vex 3467 . . . . 5 π‘₯ ∈ V
109a1i 11 . . . 4 (πœ‘ β†’ π‘₯ ∈ V)
11 vex 3467 . . . . 5 𝑦 ∈ V
1211a1i 11 . . . 4 (πœ‘ β†’ 𝑦 ∈ V)
133, 4, 2, 10, 12joindef 18365 . . 3 (πœ‘ β†’ (⟨π‘₯, π‘¦βŸ© ∈ dom ∨ ↔ {π‘₯, 𝑦} ∈ dom (lubβ€˜πΎ)))
14 joindmss.b . . . . . 6 𝐡 = (Baseβ€˜πΎ)
15 eqid 2725 . . . . . 6 (leβ€˜πΎ) = (leβ€˜πΎ)
162adantr 479 . . . . . 6 ((πœ‘ ∧ {π‘₯, 𝑦} ∈ dom (lubβ€˜πΎ)) β†’ 𝐾 ∈ 𝑉)
17 simpr 483 . . . . . 6 ((πœ‘ ∧ {π‘₯, 𝑦} ∈ dom (lubβ€˜πΎ)) β†’ {π‘₯, 𝑦} ∈ dom (lubβ€˜πΎ))
1814, 15, 3, 16, 17lubelss 18343 . . . . 5 ((πœ‘ ∧ {π‘₯, 𝑦} ∈ dom (lubβ€˜πΎ)) β†’ {π‘₯, 𝑦} βŠ† 𝐡)
1918ex 411 . . . 4 (πœ‘ β†’ ({π‘₯, 𝑦} ∈ dom (lubβ€˜πΎ) β†’ {π‘₯, 𝑦} βŠ† 𝐡))
209, 11prss 4817 . . . . 5 ((π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡) ↔ {π‘₯, 𝑦} βŠ† 𝐡)
21 opelxpi 5707 . . . . 5 ((π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡) β†’ ⟨π‘₯, π‘¦βŸ© ∈ (𝐡 Γ— 𝐡))
2220, 21sylbir 234 . . . 4 ({π‘₯, 𝑦} βŠ† 𝐡 β†’ ⟨π‘₯, π‘¦βŸ© ∈ (𝐡 Γ— 𝐡))
2319, 22syl6 35 . . 3 (πœ‘ β†’ ({π‘₯, 𝑦} ∈ dom (lubβ€˜πΎ) β†’ ⟨π‘₯, π‘¦βŸ© ∈ (𝐡 Γ— 𝐡)))
2413, 23sylbid 239 . 2 (πœ‘ β†’ (⟨π‘₯, π‘¦βŸ© ∈ dom ∨ β†’ ⟨π‘₯, π‘¦βŸ© ∈ (𝐡 Γ— 𝐡)))
258, 24relssdv 5782 1 (πœ‘ β†’ dom ∨ βŠ† (𝐡 Γ— 𝐡))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394   = wceq 1533   ∈ wcel 2098  Vcvv 3463   βŠ† wss 3939  {cpr 4624  βŸ¨cop 4628  {copab 5203   Γ— cxp 5668  dom cdm 5670  Rel wrel 5675  β€˜cfv 6541  Basecbs 17177  lecple 17237  lubclub 18298  joincjn 18300
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 2696  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5357  ax-pr 5421  ax-un 7736
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 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3769  df-csb 3885  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4317  df-if 4523  df-pw 4598  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-iun 4991  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5568  df-xp 5676  df-rel 5677  df-cnv 5678  df-co 5679  df-dm 5680  df-rn 5681  df-res 5682  df-ima 5683  df-iota 6493  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-riota 7370  df-oprab 7418  df-lub 18335  df-join 18337
This theorem is referenced by:  clatl  18497  joindm2  48071
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