Theorem dfec2 8702

Description: Alternate definition of 𝑅-coset of 𝐴. Definition 34 of [Suppes] p. 81. (Contributed by NM, 3-Jan-1997.) (Proof shortened by Mario Carneiro, 9-Jul-2014.)

Assertion

Ref	Expression
dfec2	⊢ (𝐴 ∈ 𝑉 → [𝐴]𝑅 = {𝑦 ∣ 𝐴𝑅𝑦})

Distinct variable groups:   𝑦,𝐴   𝑦,𝑅

Allowed substitution hint:   𝑉(𝑦)

Proof of Theorem dfec2

Step	Hyp	Ref	Expression
1		df-ec 8701	. 2 ⊢ [𝐴]𝑅 = (𝑅 “ {𝐴})
2		imasng 6072	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝑅 “ {𝐴}) = {𝑦 ∣ 𝐴𝑅𝑦})
3	1, 2	eqtrid 2776	1 ⊢ (𝐴 ∈ 𝑉 → [𝐴]𝑅 = {𝑦 ∣ 𝐴𝑅𝑦})

Syntax hints:   → wi 4   = wceq 1533   ∈ wcel 2098  {cab 2701  {csn 4620   class class class wbr 5138   “ cima 5669  [cec 8697

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pr 5417

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-sn 4621  df-pr 4623  df-op 4627  df-br 5139  df-opab 5201  df-xp 5672  df-cnv 5674  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-ec 8701

This theorem is referenced by:  elqsecl  8761  eqglact  19096  tgpconncompeqg  23938  fvline  35611  ellines  35619  ecres2  37637