Theorem elecg 6809

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 5132	. . 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 6771	. . 3 |- [ B ] R = ( R " { B } )
4	3	eleq2i 2301	. 2 |- ( A e. [ B ] R <-> A e. ( R " { B } ) )
5		df-br 4112	. 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 2205   {csn 3691   <.cop 3694   class class class wbr 4111   "cima 4754   [cec 6767

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-pow 4289  ax-pr 4324

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-sbc 3045  df-un 3217  df-in 3219  df-ss 3226  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-br 4112  df-opab 4174  df-xp 4757  df-cnv 4759  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-ec 6771

This theorem is referenced by:  elec  6810  relelec  6811  ecdmn0m  6813  erth  6815  ecidg  6835  qsel  6848  xmetec  15319  blpnfctr  15321  xmetresbl  15322