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Theorem prjsprel 39528
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 488 . . . . 5 (((𝑥 = 𝑋𝑦 = 𝑌) ∧ 𝑙 = 𝑚) → 𝑙 = 𝑚)
3 simplr 768 . . . . 5 (((𝑥 = 𝑋𝑦 = 𝑌) ∧ 𝑙 = 𝑚) → 𝑦 = 𝑌)
42, 3oveq12d 7158 . . . 4 (((𝑥 = 𝑋𝑦 = 𝑌) ∧ 𝑙 = 𝑚) → (𝑙 · 𝑦) = (𝑚 · 𝑌))
51, 4eqeq12d 2838 . . 3 (((𝑥 = 𝑋𝑦 = 𝑌) ∧ 𝑙 = 𝑚) → (𝑥 = (𝑙 · 𝑦) ↔ 𝑋 = (𝑚 · 𝑌)))
65cbvrexdva 3435 . 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 5621 1 (𝑋 𝑌 ↔ ((𝑋𝐵𝑌𝐵) ∧ ∃𝑚𝐾 𝑋 = (𝑚 · 𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399   = wceq 1538  wcel 2114  wrex 3131   class class class wbr 5042  {copab 5104  (class class class)co 7140
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pr 5307
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ral 3135  df-rex 3136  df-v 3471  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-br 5043  df-opab 5105  df-xp 5538  df-iota 6293  df-fv 6342  df-ov 7143
This theorem is referenced by:  prjspertr  39529  prjsperref  39530  prjspersym  39531  prjspreln0  39533  prjspvs  39534  0prjspnrel  39543
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