Theorem ecxrn 38389

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 38388	. . . 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 1848	. . . 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 5531	. . 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 2734	1 ⊢ (𝐴 ∈ 𝑉 → [𝐴](𝑅 ⋉ 𝑆) = {⟨𝑦, 𝑧⟩ ∣ (𝐴𝑅𝑦 ∧ 𝐴𝑆𝑧)})

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1539  ∃wex 1778   ∈ wcel 2107  ⟨cop 4631   class class class wbr 5142  {copab 5204  [cec 8744   ⋉ cxrn 38182

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431  ax-un 7756

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-fo 6566  df-fv 6568  df-1st 8015  df-2nd 8016  df-ec 8748  df-xrn 38373

This theorem is referenced by:  disjecxrn  38391  br1cosscnvxrn  38476