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Theorem prjsprel 42561
Description: Utility theorem regarding the relation used in ℙ𝕣𝕠𝕛. (Contributed by Steven Nguyen, 29-Apr-2023.)
Hypothesis
Ref Expression
prjsprel.1 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ ∃𝑙𝐾 𝑥 = (𝑙 · 𝑦))}
Assertion
Ref Expression
prjsprel (𝑋 𝑌 ↔ ((𝑋𝐵𝑌𝐵) ∧ ∃𝑚𝐾 𝑋 = (𝑚 · 𝑌)))
Distinct variable groups:   𝑥,𝐵,𝑦   𝑥,𝑋,𝑦,𝑙,𝑚   𝑥,𝑌,𝑦,𝑙,𝑚   𝑥,𝐾,𝑦,𝑙,𝑚   𝑥, · ,𝑦,𝑙,𝑚
Allowed substitution hints:   𝐵(𝑚,𝑙)   (𝑥,𝑦,𝑚,𝑙)

Proof of Theorem prjsprel
StepHypRef Expression
1 simpll 766 . . . 4 (((𝑥 = 𝑋𝑦 = 𝑌) ∧ 𝑙 = 𝑚) → 𝑥 = 𝑋)
2 simpr 484 . . . . 5 (((𝑥 = 𝑋𝑦 = 𝑌) ∧ 𝑙 = 𝑚) → 𝑙 = 𝑚)
3 simplr 768 . . . . 5 (((𝑥 = 𝑋𝑦 = 𝑌) ∧ 𝑙 = 𝑚) → 𝑦 = 𝑌)
42, 3oveq12d 7468 . . . 4 (((𝑥 = 𝑋𝑦 = 𝑌) ∧ 𝑙 = 𝑚) → (𝑙 · 𝑦) = (𝑚 · 𝑌))
51, 4eqeq12d 2756 . . 3 (((𝑥 = 𝑋𝑦 = 𝑌) ∧ 𝑙 = 𝑚) → (𝑥 = (𝑙 · 𝑦) ↔ 𝑋 = (𝑚 · 𝑌)))
65cbvrexdva 3246 . 2 ((𝑥 = 𝑋𝑦 = 𝑌) → (∃𝑙𝐾 𝑥 = (𝑙 · 𝑦) ↔ ∃𝑚𝐾 𝑋 = (𝑚 · 𝑌)))
7 prjsprel.1 . 2 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐵𝑦𝐵) ∧ ∃𝑙𝐾 𝑥 = (𝑙 · 𝑦))}
86, 7brab2a 5793 1 (𝑋 𝑌 ↔ ((𝑋𝐵𝑌𝐵) ∧ ∃𝑚𝐾 𝑋 = (𝑚 · 𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1537  wcel 2108  wrex 3076   class class class wbr 5166  {copab 5228  (class class class)co 7450
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-xp 5706  df-iota 6527  df-fv 6583  df-ov 7453
This theorem is referenced by:  prjspertr  42562  prjsperref  42563  prjspersym  42564  prjspreln0  42566  prjspvs  42567  prjspner1  42583  0prjspnrel  42584
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