Theorem dfec2 6704

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 6703	. 2 |- [ A ] R = ( R " { A } )
2		imasng 5101	. 2 |- ( A e. V -> ( R " { A } ) = { y | A R y } )
3	1, 2	eqtrid 2276	1 |- ( A e. V -> [ A ] R = { y | A R y } )

Syntax hints:   -> wi 4   = wceq 1397   e. wcel 2202   {cab 2217   {csn 3669   class class class wbr 4088   "cima 4728   [cec 6699

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-sbc 3032  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-br 4089  df-opab 4151  df-xp 4731  df-cnv 4733  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-ec 6703

This theorem is referenced by:  eqglact  13811