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Theorem ecxrn 36849
Description: The (𝑅𝑆)-coset of 𝐴. (Contributed by Peter Mazsa, 18-Apr-2020.) (Revised by Peter Mazsa, 21-Sep-2021.)
Assertion
Ref Expression
ecxrn (𝐴𝑉 → [𝐴](𝑅𝑆) = {⟨𝑦, 𝑧⟩ ∣ (𝐴𝑅𝑦𝐴𝑆𝑧)})
Distinct variable groups:   𝑦,𝐴,𝑧   𝑦,𝑅,𝑧   𝑦,𝑆,𝑧   𝑦,𝑉,𝑧

Proof of Theorem ecxrn
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 elecxrn 36848 . . . 4 (𝐴𝑉 → (𝑥 ∈ [𝐴](𝑅𝑆) ↔ ∃𝑦𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝐴𝑅𝑦𝐴𝑆𝑧)))
2 3anass 1095 . . . . 5 ((𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝐴𝑅𝑦𝐴𝑆𝑧) ↔ (𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝐴𝑅𝑦𝐴𝑆𝑧)))
322exbii 1851 . . . 4 (∃𝑦𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝐴𝑅𝑦𝐴𝑆𝑧) ↔ ∃𝑦𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝐴𝑅𝑦𝐴𝑆𝑧)))
41, 3bitrdi 286 . . 3 (𝐴𝑉 → (𝑥 ∈ [𝐴](𝑅𝑆) ↔ ∃𝑦𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝐴𝑅𝑦𝐴𝑆𝑧))))
5 elopab 5484 . . 3 (𝑥 ∈ {⟨𝑦, 𝑧⟩ ∣ (𝐴𝑅𝑦𝐴𝑆𝑧)} ↔ ∃𝑦𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝐴𝑅𝑦𝐴𝑆𝑧)))
64, 5bitr4di 288 . 2 (𝐴𝑉 → (𝑥 ∈ [𝐴](𝑅𝑆) ↔ 𝑥 ∈ {⟨𝑦, 𝑧⟩ ∣ (𝐴𝑅𝑦𝐴𝑆𝑧)}))
76eqrdv 2734 1 (𝐴𝑉 → [𝐴](𝑅𝑆) = {⟨𝑦, 𝑧⟩ ∣ (𝐴𝑅𝑦𝐴𝑆𝑧)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1087   = wceq 1541  wex 1781  wcel 2106  cop 4592   class class class wbr 5105  {copab 5167  [cec 8646  cxrn 36633
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 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384  ax-un 7672
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 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-fo 6502  df-fv 6504  df-1st 7921  df-2nd 7922  df-ec 8650  df-xrn 36833
This theorem is referenced by:  disjecxrn  36851  br1cosscnvxrn  36936
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