Theorem dfec2 6810

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 6809	. 2 |- [ A ] R = ( R " { A } )
2		imasng 5152	. 2 |- ( A e. V -> ( R " { A } ) = { y | A R y } )
3	1, 2	eqtrid 2283	1 |- ( A e. V -> [ A ] R = { y | A R y } )

This proof depends on syntax axioms:   -> wi 4   = wceq 1402   e. wcel 2209   {cab 2224   {csn 3709   class class class wbr 4130   "cima 4777   [cec 6805

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-br 4131  df-opab 4193  df-xp 4780  df-cnv 4782  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-ec 6809

This theorem is used by:  eqglact  14028