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Theorem dfec2 6020
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 = {yA𝑅y})
Distinct variable groups:   y,A   y,𝑅
Allowed substitution hint:   𝑉(y)

Proof of Theorem dfec2
StepHypRef Expression
1 df-ec 6019 . 2 [A]𝑅 = (𝑅 “ {A})
2 imasng 4617 . 2 (A 𝑉 → (𝑅 “ {A}) = {yA𝑅y})
31, 2syl5eq 2066 1 (A 𝑉 → [A]𝑅 = {yA𝑅y})
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1228   wcel 1374  {cab 2008  {csn 3350   class class class wbr 3738  cima 4275  [cec 6015
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-sbc 2742  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-br 3739  df-opab 3793  df-xp 4278  df-cnv 4280  df-dm 4282  df-rn 4283  df-res 4284  df-ima 4285  df-ec 6019
This theorem is referenced by: (None)
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