Theorem elecxrn 35653

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

Assertion

Ref	Expression
elecxrn	⊢ (𝐴 ∈ 𝑉 → (𝐵 ∈ [𝐴](𝑅 ⋉ 𝑆) ↔ ∃𝑥∃𝑦(𝐵 = ⟨𝑥, 𝑦⟩ ∧ 𝐴𝑅𝑥 ∧ 𝐴𝑆𝑦)))

Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦   𝑥,𝑅,𝑦   𝑥,𝑆,𝑦   𝑥,𝑉,𝑦

Proof of Theorem elecxrn

Step	Hyp	Ref	Expression
1		xrnrel 35640	. . 3 ⊢ Rel (𝑅 ⋉ 𝑆)
2		relelec 8334	. . 3 ⊢ (Rel (𝑅 ⋉ 𝑆) → (𝐵 ∈ [𝐴](𝑅 ⋉ 𝑆) ↔ 𝐴(𝑅 ⋉ 𝑆)𝐵))
3	1, 2	ax-mp 5	. 2 ⊢ (𝐵 ∈ [𝐴](𝑅 ⋉ 𝑆) ↔ 𝐴(𝑅 ⋉ 𝑆)𝐵)
4		brxrn2 35642	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴(𝑅 ⋉ 𝑆)𝐵 ↔ ∃𝑥∃𝑦(𝐵 = ⟨𝑥, 𝑦⟩ ∧ 𝐴𝑅𝑥 ∧ 𝐴𝑆𝑦)))
5	3, 4	syl5bb 285	1 ⊢ (𝐴 ∈ 𝑉 → (𝐵 ∈ [𝐴](𝑅 ⋉ 𝑆) ↔ ∃𝑥∃𝑦(𝐵 = ⟨𝑥, 𝑦⟩ ∧ 𝐴𝑅𝑥 ∧ 𝐴𝑆𝑦)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ w3a 1083   = wceq 1537  ∃wex 1780   ∈ wcel 2114  ⟨cop 4573   class class class wbr 5066  Rel wrel 5560  [cec 8287   ⋉ cxrn 35467

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3773  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-br 5067  df-opab 5129  df-mpt 5147  df-id 5460  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-fo 6361  df-fv 6363  df-1st 7689  df-2nd 7690  df-ec 8291  df-xrn 35638

This theorem is referenced by:  ecxrn  35654