Theorem dfec2 6504

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 e. V -> [ A ] R = { y | A R y } )

Distinct variable groups:   y, A   y, R

Allowed substitution hint:   V( y)

Proof of Theorem dfec2

Step	Hyp	Ref	Expression
1		df-ec 6503	. 2 |- [ A ] R = ( R " { A } )
2		imasng 4969	. 2 |- ( A e. V -> ( R " { A } ) = { y | A R y } )
3	1, 2	syl5eq 2211	1 |- ( A e. V -> [ A ] R = { y | A R y } )

Syntax hints:   -> wi 4   = wceq 1343   e. wcel 2136   {cab 2151   {csn 3576   class class class wbr 3982   "cima 4607   [cec 6499

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-sbc 2952  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-opab 4044  df-xp 4610  df-cnv 4612  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-ec 6503

This theorem is referenced by: (None)