Theorem ecqs 8794

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 8721	. 2 ⊢ [𝐴]𝑅 = (𝑅 “ {𝐴})
2		ecqs.1	. . 3 ⊢ 𝑅 ∈ V
3		uniqs 8790	. . 3 ⊢ (𝑅 ∈ V → ∪ ({𝐴} / 𝑅) = (𝑅 “ {𝐴}))
4	2, 3	ax-mp 5	. 2 ⊢ ∪ ({𝐴} / 𝑅) = (𝑅 “ {𝐴})
5	1, 4	eqtr4i 2759	1 ⊢ [𝐴]𝑅 = ∪ ({𝐴} / 𝑅)

Syntax hints:   = wceq 1534   ∈ wcel 2099  Vcvv 3470  {csn 4625  ∪ cuni 4904   “ cima 5676  [cec 8717   / cqs 8718

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-11 2147  ax-ext 2699  ax-sep 5294  ax-nul 5301  ax-pr 5424  ax-un 7735

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3472  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4320  df-if 4526  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4905  df-iun 4994  df-br 5144  df-opab 5206  df-xp 5679  df-cnv 5681  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  df-ec 8721  df-qs 8725

This theorem is referenced by: (None)