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Theorem joindef 17602
Description: Two ways to say that a join is defined. (Contributed by NM, 9-Sep-2018.)
Hypotheses
Ref Expression
joindef.u 𝑈 = (lub‘𝐾)
joindef.j = (join‘𝐾)
joindef.k (𝜑𝐾𝑉)
joindef.x (𝜑𝑋𝑊)
joindef.y (𝜑𝑌𝑍)
Assertion
Ref Expression
joindef (𝜑 → (⟨𝑋, 𝑌⟩ ∈ dom ↔ {𝑋, 𝑌} ∈ dom 𝑈))

Proof of Theorem joindef
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 joindef.k . . 3 (𝜑𝐾𝑉)
2 joindef.u . . . . 5 𝑈 = (lub‘𝐾)
3 joindef.j . . . . 5 = (join‘𝐾)
42, 3joindm 17601 . . . 4 (𝐾𝑉 → dom = {⟨𝑥, 𝑦⟩ ∣ {𝑥, 𝑦} ∈ dom 𝑈})
54eleq2d 2895 . . 3 (𝐾𝑉 → (⟨𝑋, 𝑌⟩ ∈ dom ↔ ⟨𝑋, 𝑌⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ {𝑥, 𝑦} ∈ dom 𝑈}))
61, 5syl 17 . 2 (𝜑 → (⟨𝑋, 𝑌⟩ ∈ dom ↔ ⟨𝑋, 𝑌⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ {𝑥, 𝑦} ∈ dom 𝑈}))
7 joindef.x . . 3 (𝜑𝑋𝑊)
8 joindef.y . . 3 (𝜑𝑌𝑍)
9 preq1 4661 . . . . 5 (𝑥 = 𝑋 → {𝑥, 𝑦} = {𝑋, 𝑦})
109eleq1d 2894 . . . 4 (𝑥 = 𝑋 → ({𝑥, 𝑦} ∈ dom 𝑈 ↔ {𝑋, 𝑦} ∈ dom 𝑈))
11 preq2 4662 . . . . 5 (𝑦 = 𝑌 → {𝑋, 𝑦} = {𝑋, 𝑌})
1211eleq1d 2894 . . . 4 (𝑦 = 𝑌 → ({𝑋, 𝑦} ∈ dom 𝑈 ↔ {𝑋, 𝑌} ∈ dom 𝑈))
1310, 12opelopabg 5416 . . 3 ((𝑋𝑊𝑌𝑍) → (⟨𝑋, 𝑌⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ {𝑥, 𝑦} ∈ dom 𝑈} ↔ {𝑋, 𝑌} ∈ dom 𝑈))
147, 8, 13syl2anc 584 . 2 (𝜑 → (⟨𝑋, 𝑌⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ {𝑥, 𝑦} ∈ dom 𝑈} ↔ {𝑋, 𝑌} ∈ dom 𝑈))
156, 14bitrd 280 1 (𝜑 → (⟨𝑋, 𝑌⟩ ∈ dom ↔ {𝑋, 𝑌} ∈ dom 𝑈))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207   = wceq 1528  wcel 2105  {cpr 4559  cop 4563  {copab 5119  dom cdm 5548  cfv 6348  lubclub 17540  joincjn 17542
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-oprab 7149  df-lub 17572  df-join 17574
This theorem is referenced by:  joinval  17603  joincl  17604  joindmss  17605  joineu  17608  clatl  17714
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