Theorem ecqs 8716

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 8637	. 2 ⊢ [𝐴]𝑅 = (𝑅 “ {𝐴})
2		ecqs.1	. . 3 ⊢ 𝑅 ∈ V
3		uniqsw 8712	. . 3 ⊢ (𝑅 ∈ V → ∪ ({𝐴} / 𝑅) = (𝑅 “ {𝐴}))
4	2, 3	ax-mp 5	. 2 ⊢ ∪ ({𝐴} / 𝑅) = (𝑅 “ {𝐴})
5	1, 4	eqtr4i 2762	1 ⊢ [𝐴]𝑅 = ∪ ({𝐴} / 𝑅)

Syntax hints:   = wceq 1541   ∈ wcel 2113  Vcvv 3440  {csn 4580  ∪ cuni 4863   “ cima 5627  [cec 8633   / cqs 8634

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 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-11 2162  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-xp 5630  df-rel 5631  df-cnv 5632  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-ec 8637  df-qs 8641

This theorem is referenced by: (None)