Theorem ecxrn 38372

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 38371	. . . 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 1849	. . . 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 5495	. . 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 2728	1 ⊢ (𝐴 ∈ 𝑉 → [𝐴](𝑅 ⋉ 𝑆) = {⟨𝑦, 𝑧⟩ ∣ (𝐴𝑅𝑦 ∧ 𝐴𝑆𝑧)})

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540  ∃wex 1779   ∈ wcel 2109  ⟨cop 4603   class class class wbr 5115  {copab 5177  [cec 8680   ⋉ cxrn 38165

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5259  ax-nul 5269  ax-pr 5395  ax-un 7718

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2880  df-ne 2928  df-ral 3047  df-rex 3056  df-rab 3412  df-v 3457  df-dif 3925  df-un 3927  df-in 3929  df-ss 3939  df-nul 4305  df-if 4497  df-sn 4598  df-pr 4600  df-op 4604  df-uni 4880  df-br 5116  df-opab 5178  df-mpt 5197  df-id 5541  df-xp 5652  df-rel 5653  df-cnv 5654  df-co 5655  df-dm 5656  df-rn 5657  df-res 5658  df-ima 5659  df-iota 6472  df-fun 6521  df-fn 6522  df-f 6523  df-fo 6525  df-fv 6527  df-1st 7977  df-2nd 7978  df-ec 8684  df-xrn 38356

This theorem is referenced by:  disjecxrn  38374  br1cosscnvxrn  38459