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Theorem elecg 6210
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
StepHypRef Expression
1 elimasng 4723 . . 3  |-  ( ( B  e.  W  /\  A  e.  V )  ->  ( A  e.  ( R " { B } )  <->  <. B ,  A >.  e.  R ) )
21ancoms 264 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  e.  ( R " { B } )  <->  <. B ,  A >.  e.  R ) )
3 df-ec 6174 . . 3  |-  [ B ] R  =  ( R " { B }
)
43eleq2i 2146 . 2  |-  ( A  e.  [ B ] R 
<->  A  e.  ( R
" { B }
) )
5 df-br 3794 . 2  |-  ( B R A  <->  <. B ,  A >.  e.  R )
62, 4, 53bitr4g 221 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  e.  [ B ] R  <->  B R A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    e. wcel 1434   {csn 3406   <.cop 3409   class class class wbr 3793   "cima 4374   [cec 6170
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-sbc 2817  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-br 3794  df-opab 3848  df-xp 4377  df-cnv 4379  df-dm 4381  df-rn 4382  df-res 4383  df-ima 4384  df-ec 6174
This theorem is referenced by:  elec  6211  relelec  6212  ecdmn0m  6214  erth  6216  ecidg  6236  qsel  6249
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