Theorem relecxrn 39415

Description: The (𝑅 ⋉ 𝑆)-coset of a set is a relation. (Contributed by Peter Mazsa, 15-Oct-2020.)

Assertion

Ref	Expression
relecxrn	⊢ (𝐴 ∈ 𝑉 → Rel [𝐴](𝑅 ⋉ 𝑆))

Proof of Theorem relecxrn

Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		relopab 5801	. 2 ⊢ Rel {⟨𝑦, 𝑧⟩ ∣ (𝐴𝑅𝑦 ∧ 𝐴𝑆𝑧)}
2		ecxrn 39414	. . 3 ⊢ (𝐴 ∈ 𝑉 → [𝐴](𝑅 ⋉ 𝑆) = {⟨𝑦, 𝑧⟩ ∣ (𝐴𝑅𝑦 ∧ 𝐴𝑆𝑧)})
3	2	releqd 5754	. 2 ⊢ (𝐴 ∈ 𝑉 → (Rel [𝐴](𝑅 ⋉ 𝑆) ↔ Rel {⟨𝑦, 𝑧⟩ ∣ (𝐴𝑅𝑦 ∧ 𝐴𝑆𝑧)}))
4	1, 3	mpbiri 261	1 ⊢ (𝐴 ∈ 𝑉 → Rel [𝐴](𝑅 ⋉ 𝑆))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∈ wcel 2144   class class class wbr 5102  {copab 5166  Rel wrel 5655  [cec 8716   ⋉ cxrn 39182

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-fo 6544  df-fv 6546  df-1st 8001  df-2nd 8002  df-ec 8720  df-xrn 39388

This theorem is used by:  ecxrn2  39416