Theorem elecg 6720

Description: Membership in an equivalence class. Theorem 72 of [Suppes] p. 82. (Contributed by Mario Carneiro, 9-Jul-2014.)

Assertion

Ref	Expression
elecg	|- ( ( A e. V /\ B e. W ) -> ( A e. [ B ] R <-> B R A ) )

Proof of Theorem elecg

Step	Hyp	Ref	Expression
1		elimasng 5096	. . 3 |- ( ( B e. W /\ A e. V ) -> ( A e. ( R " { B } ) <-> <. B , A >. e. R ) )
2	1	ancoms 268	. 2 |- ( ( A e. V /\ B e. W ) -> ( A e. ( R " { B } ) <-> <. B , A >. e. R ) )
3		df-ec 6682	. . 3 |- [ B ] R = ( R " { B } )
4	3	eleq2i 2296	. 2 |- ( A e. [ B ] R <-> A e. ( R " { B } ) )
5		df-br 4084	. 2 |- ( B R A <-> <. B , A >. e. R )
6	2, 4, 5	3bitr4g 223	1 |- ( ( A e. V /\ B e. W ) -> ( A e. [ B ] R <-> B R A ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   e. wcel 2200   {csn 3666   <.cop 3669   class class class wbr 4083   "cima 4722   [cec 6678

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 4202  ax-pow 4258  ax-pr 4293

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 2801  df-sbc 3029  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-br 4084  df-opab 4146  df-xp 4725  df-cnv 4727  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-ec 6682

This theorem is referenced by:  elec  6721  relelec  6722  ecdmn0m  6724  erth  6726  ecidg  6746  qsel  6759  xmetec  15111  blpnfctr  15113  xmetresbl  15114