Theorem dfec2 6540

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 6539	. 2 ⊢ [𝐴]𝑅 = (𝑅 “ {𝐴})
2		imasng 4995	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝑅 “ {𝐴}) = {𝑦 ∣ 𝐴𝑅𝑦})
3	1, 2	eqtrid 2222	1 ⊢ (𝐴 ∈ 𝑉 → [𝐴]𝑅 = {𝑦 ∣ 𝐴𝑅𝑦})

Syntax hints:   → wi 4   = wceq 1353   ∈ wcel 2148  {cab 2163  {csn 3594   class class class wbr 4005   “ cima 4631  [cec 6535

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-sbc 2965  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-br 4006  df-opab 4067  df-xp 4634  df-cnv 4636  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-ec 6539

This theorem is referenced by:  eqglact  13089