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Theorem elecxrn 37059
Description: Elementhood in the (𝑅𝑆)-coset of 𝐴. (Contributed by Peter Mazsa, 18-Apr-2020.) (Revised by Peter Mazsa, 21-Sep-2021.)
Assertion
Ref Expression
elecxrn (𝐴𝑉 → (𝐵 ∈ [𝐴](𝑅𝑆) ↔ ∃𝑥𝑦(𝐵 = ⟨𝑥, 𝑦⟩ ∧ 𝐴𝑅𝑥𝐴𝑆𝑦)))
Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦   𝑥,𝑅,𝑦   𝑥,𝑆,𝑦   𝑥,𝑉,𝑦

Proof of Theorem elecxrn
StepHypRef Expression
1 xrnrel 37046 . . 3 Rel (𝑅𝑆)
2 relelec 8731 . . 3 (Rel (𝑅𝑆) → (𝐵 ∈ [𝐴](𝑅𝑆) ↔ 𝐴(𝑅𝑆)𝐵))
31, 2ax-mp 5 . 2 (𝐵 ∈ [𝐴](𝑅𝑆) ↔ 𝐴(𝑅𝑆)𝐵)
4 brxrn2 37048 . 2 (𝐴𝑉 → (𝐴(𝑅𝑆)𝐵 ↔ ∃𝑥𝑦(𝐵 = ⟨𝑥, 𝑦⟩ ∧ 𝐴𝑅𝑥𝐴𝑆𝑦)))
53, 4bitrid 282 1 (𝐴𝑉 → (𝐵 ∈ [𝐴](𝑅𝑆) ↔ ∃𝑥𝑦(𝐵 = ⟨𝑥, 𝑦⟩ ∧ 𝐴𝑅𝑥𝐴𝑆𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  w3a 1087   = wceq 1541  wex 1781  wcel 2106  cop 4628   class class class wbr 5141  Rel wrel 5674  [cec 8684  cxrn 36845
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7708
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4523  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6484  df-fun 6534  df-fn 6535  df-f 6536  df-fo 6538  df-fv 6540  df-1st 7957  df-2nd 7958  df-ec 8688  df-xrn 37044
This theorem is referenced by:  ecxrn  37060
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