Theorem ecqs 8786

Description: Equivalence class in terms of quotient set. (Contributed by NM, 29-Jan-1999.)

Hypothesis

Ref	Expression
ecqs.1	⊢ 𝑅 ∈ V

Assertion

Ref	Expression
ecqs	⊢ [𝐴]𝑅 = ∪ ({𝐴} / 𝑅)

Proof of Theorem ecqs

Step	Hyp	Ref	Expression
1		df-ec 8705	. 2 ⊢ [𝐴]𝑅 = (𝑅 “ {𝐴})
2		ecqs.1	. . 3 ⊢ 𝑅 ∈ V
3		uniqsw 8781	. . 3 ⊢ (𝑅 ∈ V → ∪ ({𝐴} / 𝑅) = (𝑅 “ {𝐴}))
4	2, 3	ax-mp 5	. 2 ⊢ ∪ ({𝐴} / 𝑅) = (𝑅 “ {𝐴})
5	1, 4	eqtr4i 2792	1 ⊢ [𝐴]𝑅 = ∪ ({𝐴} / 𝑅)

This proof depends on syntax axioms:   = wceq 1570   ∈ wcel 2146  Vcvv 3458  {csn 4594  ∪ cuni 4877   “ cima 5669  [cec 8701   / cqs 8702

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-11 2195  ax-ext 2738  ax-sep 5262  ax-pr 5409  ax-un 7745

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-iun 4963  df-br 5115  df-opab 5179  df-xp 5672  df-rel 5673  df-cnv 5674  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-ec 8705  df-qs 8709

This theorem is used by: (None)