Theorem dfec2 6516

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 6515	. 2 |- [ A ] R = ( R " { A } )
2		imasng 4976	. 2 |- ( A e. V -> ( R " { A } ) = { y | A R y } )
3	1, 2	eqtrid 2215	1 |- ( A e. V -> [ A ] R = { y | A R y } )

Syntax hints:   -> wi 4   = wceq 1348   e. wcel 2141   {cab 2156   {csn 3583   class class class wbr 3989   "cima 4614   [cec 6511

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051  df-xp 4617  df-cnv 4619  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-ec 6515

This theorem is referenced by: (None)