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Theorem elecg 6470
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 4910 . . 3  |-  ( ( B  e.  W  /\  A  e.  V )  ->  ( A  e.  ( R " { B } )  <->  <. B ,  A >.  e.  R ) )
21ancoms 266 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  e.  ( R " { B } )  <->  <. B ,  A >.  e.  R ) )
3 df-ec 6434 . . 3  |-  [ B ] R  =  ( R " { B }
)
43eleq2i 2206 . 2  |-  ( A  e.  [ B ] R 
<->  A  e.  ( R
" { B }
) )
5 df-br 3933 . 2  |-  ( B R A  <->  <. B ,  A >.  e.  R )
62, 4, 53bitr4g 222 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 103    <-> wb 104    e. wcel 1480   {csn 3527   <.cop 3530   class class class wbr 3932   "cima 4545   [cec 6430
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4049  ax-pow 4101  ax-pr 4134
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-sbc 2910  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-br 3933  df-opab 3993  df-xp 4548  df-cnv 4550  df-dm 4552  df-rn 4553  df-res 4554  df-ima 4555  df-ec 6434
This theorem is referenced by:  elec  6471  relelec  6472  ecdmn0m  6474  erth  6476  ecidg  6496  qsel  6509  xmetec  12632  blpnfctr  12634  xmetresbl  12635
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