Theorem ecxrn 36517

Description: The (𝑅 ⋉ 𝑆)-coset of 𝐴. (Contributed by Peter Mazsa, 18-Apr-2020.) (Revised by Peter Mazsa, 21-Sep-2021.)

Assertion

Ref	Expression
ecxrn	⊢ (𝐴 ∈ 𝑉 → [𝐴](𝑅 ⋉ 𝑆) = {⟨𝑦, 𝑧⟩ ∣ (𝐴𝑅𝑦 ∧ 𝐴𝑆𝑧)})

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.

Step	Hyp	Ref	Expression
1		elecxrn 36516	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ [𝐴](𝑅 ⋉ 𝑆) ↔ ∃𝑦∃𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝐴𝑅𝑦 ∧ 𝐴𝑆𝑧)))
2		3anass 1094	. . . . 5 ⊢ ((𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝐴𝑅𝑦 ∧ 𝐴𝑆𝑧) ↔ (𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝐴𝑅𝑦 ∧ 𝐴𝑆𝑧)))
3	2	2exbii 1851	. . . 4 ⊢ (∃𝑦∃𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ 𝐴𝑅𝑦 ∧ 𝐴𝑆𝑧) ↔ ∃𝑦∃𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝐴𝑅𝑦 ∧ 𝐴𝑆𝑧)))
4	1, 3	bitrdi 287	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ [𝐴](𝑅 ⋉ 𝑆) ↔ ∃𝑦∃𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝐴𝑅𝑦 ∧ 𝐴𝑆𝑧))))
5		elopab 5440	. . 3 ⊢ (𝑥 ∈ {⟨𝑦, 𝑧⟩ ∣ (𝐴𝑅𝑦 ∧ 𝐴𝑆𝑧)} ↔ ∃𝑦∃𝑧(𝑥 = ⟨𝑦, 𝑧⟩ ∧ (𝐴𝑅𝑦 ∧ 𝐴𝑆𝑧)))
6	4, 5	bitr4di 289	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ [𝐴](𝑅 ⋉ 𝑆) ↔ 𝑥 ∈ {⟨𝑦, 𝑧⟩ ∣ (𝐴𝑅𝑦 ∧ 𝐴𝑆𝑧)}))
7	6	eqrdv 2736	1 ⊢ (𝐴 ∈ 𝑉 → [𝐴](𝑅 ⋉ 𝑆) = {⟨𝑦, 𝑧⟩ ∣ (𝐴𝑅𝑦 ∧ 𝐴𝑆𝑧)})

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1086   = wceq 1539  ∃wex 1782   ∈ wcel 2106  ⟨cop 4567   class class class wbr 5074  {copab 5136  [cec 8496   ⋉ cxrn 36332

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  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 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-fo 6439  df-fv 6441  df-1st 7831  df-2nd 7832  df-ec 8500  df-xrn 36501

This theorem is referenced by:  br1cosscnvxrn  36592