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Theorem ecxrn 34643
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 34642 . . . 4 (𝐴𝑉 → (𝑥 ∈ [𝐴](𝑅𝑆) ↔ ∃𝑦𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝐴𝑅𝑦𝐴𝑆𝑧)))
2 3anass 1117 . . . . 5 ((𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝐴𝑅𝑦𝐴𝑆𝑧) ↔ (𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝐴𝑅𝑦𝐴𝑆𝑧)))
322exbii 1945 . . . 4 (∃𝑦𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝐴𝑅𝑦𝐴𝑆𝑧) ↔ ∃𝑦𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝐴𝑅𝑦𝐴𝑆𝑧)))
41, 3syl6bb 279 . . 3 (𝐴𝑉 → (𝑥 ∈ [𝐴](𝑅𝑆) ↔ ∃𝑦𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝐴𝑅𝑦𝐴𝑆𝑧))))
5 elopab 5179 . . 3 (𝑥 ∈ {⟨𝑦, 𝑧⟩ ∣ (𝐴𝑅𝑦𝐴𝑆𝑧)} ↔ ∃𝑦𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝐴𝑅𝑦𝐴𝑆𝑧)))
64, 5syl6bbr 281 . 2 (𝐴𝑉 → (𝑥 ∈ [𝐴](𝑅𝑆) ↔ 𝑥 ∈ {⟨𝑦, 𝑧⟩ ∣ (𝐴𝑅𝑦𝐴𝑆𝑧)}))
76eqrdv 2797 1 (𝐴𝑉 → [𝐴](𝑅𝑆) = {⟨𝑦, 𝑧⟩ ∣ (𝐴𝑅𝑦𝐴𝑆𝑧)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385  w3a 1108   = wceq 1653  wex 1875  wcel 2157  cop 4374   class class class wbr 4843  {copab 4905  [cec 7980  cxrn 34468
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-fo 6107  df-fv 6109  df-1st 7401  df-2nd 7402  df-ec 7984  df-xrn 34627
This theorem is referenced by:  br1cosscnvxrn  34718
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