Theorem ecelqsw 8719

Description: Membership of an equivalence class in a quotient set. More restrictive antecedent; kept for backward compatibility; for new work, prefer ecelqs 8718. (Contributed by Jeff Madsen, 10-Jun-2010.) (Revised by Mario Carneiro, 9-Jul-2014.) (Proof shortened by AV, 25-Nov-2025.)

Assertion

Ref	Expression
ecelqsw	⊢ ((𝑅 ∈ 𝑉 ∧ 𝐵 ∈ 𝐴) → [𝐵]𝑅 ∈ (𝐴 / 𝑅))

Proof of Theorem ecelqsw

Step	Hyp	Ref	Expression
1		resexg 5987	. 2 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ↾ 𝐴) ∈ V)
2		ecelqs 8718	. 2 ⊢ (((𝑅 ↾ 𝐴) ∈ V ∧ 𝐵 ∈ 𝐴) → [𝐵]𝑅 ∈ (𝐴 / 𝑅))
3	1, 2	sylan 580	1 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐵 ∈ 𝐴) → [𝐵]𝑅 ∈ (𝐴 / 𝑅))

Syntax hints:   → wi 4   ∧ wa 395   ∈ wcel 2109  Vcvv 3444   ↾ cres 5633  [cec 8646   / cqs 8647

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 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382  ax-un 7691

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-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5103  df-opab 5165  df-xp 5637  df-rel 5638  df-cnv 5639  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-ec 8650  df-qs 8654

This theorem is referenced by:  ecelqsi  8720  qliftlem  8748  erov  8764  eroprf  8765  sylow2a  19525  sylow2blem1  19526  sylow2blem2  19527  cldsubg  23974  tgjustr  28377