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Theorem prjsprel 39132
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 763 . . . 4 (((𝑥 = 𝑋𝑦 = 𝑌) ∧ 𝑙 = 𝑚) → 𝑥 = 𝑋)
2 simpr 485 . . . . 5 (((𝑥 = 𝑋𝑦 = 𝑌) ∧ 𝑙 = 𝑚) → 𝑙 = 𝑚)
3 simplr 765 . . . . 5 (((𝑥 = 𝑋𝑦 = 𝑌) ∧ 𝑙 = 𝑚) → 𝑦 = 𝑌)
42, 3oveq12d 7163 . . . 4 (((𝑥 = 𝑋𝑦 = 𝑌) ∧ 𝑙 = 𝑚) → (𝑙 · 𝑦) = (𝑚 · 𝑌))
51, 4eqeq12d 2834 . . 3 (((𝑥 = 𝑋𝑦 = 𝑌) ∧ 𝑙 = 𝑚) → (𝑥 = (𝑙 · 𝑦) ↔ 𝑋 = (𝑚 · 𝑌)))
65cbvrexdva 3458 . 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 5637 1 (𝑋 𝑌 ↔ ((𝑋𝐵𝑌𝐵) ∧ ∃𝑚𝐾 𝑋 = (𝑚 · 𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wb 207  wa 396   = wceq 1528  wcel 2105  wrex 3136   class class class wbr 5057  {copab 5119  (class class class)co 7145
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-sep 5194  ax-nul 5201  ax-pr 5320
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-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-xp 5554  df-iota 6307  df-fv 6356  df-ov 7148
This theorem is referenced by:  prjspertr  39133  prjsperref  39134  prjspersym  39135  prjspreln0  39137  prjspvs  39138  0prjspnrel  39147
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