Theorem dfec2 6700

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 6699	. 2 |- [ A ] R = ( R " { A } )
2		imasng 5099	. 2 |- ( A e. V -> ( R " { A } ) = { y | A R y } )
3	1, 2	eqtrid 2274	1 |- ( A e. V -> [ A ] R = { y | A R y } )

Syntax hints:   -> wi 4   = wceq 1395   e. wcel 2200   {cab 2215   {csn 3667   class class class wbr 4086   "cima 4726   [cec 6695

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2802  df-sbc 3030  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-br 4087  df-opab 4149  df-xp 4729  df-cnv 4731  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-ec 6699

This theorem is referenced by:  eqglact  13802