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Theorem ecexr 6543
Description: An inhabited equivalence class implies the representative is a set. (Contributed by Mario Carneiro, 9-Jul-2014.)
Assertion
Ref Expression
ecexr  |-  ( A  e.  [ B ] R  ->  B  e.  _V )

Proof of Theorem ecexr
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 elimag 4976 . . . . 5  |-  ( A  e.  ( R " { B } )  -> 
( A  e.  ( R " { B } )  <->  E. x  e.  { B } x R A ) )
21ibi 176 . . . 4  |-  ( A  e.  ( R " { B } )  ->  E. x  e.  { B } x R A )
3 df-ec 6540 . . . 4  |-  [ B ] R  =  ( R " { B }
)
42, 3eleq2s 2272 . . 3  |-  ( A  e.  [ B ] R  ->  E. x  e.  { B } x R A )
5 df-rex 2461 . . . 4  |-  ( E. x  e.  { B } x R A  <->  E. x ( x  e. 
{ B }  /\  x R A ) )
6 simpl 109 . . . . . 6  |-  ( ( x  e.  { B }  /\  x R A )  ->  x  e.  { B } )
7 velsn 3611 . . . . . 6  |-  ( x  e.  { B }  <->  x  =  B )
86, 7sylib 122 . . . . 5  |-  ( ( x  e.  { B }  /\  x R A )  ->  x  =  B )
98eximi 1600 . . . 4  |-  ( E. x ( x  e. 
{ B }  /\  x R A )  ->  E. x  x  =  B )
105, 9sylbi 121 . . 3  |-  ( E. x  e.  { B } x R A  ->  E. x  x  =  B )
114, 10syl 14 . 2  |-  ( A  e.  [ B ] R  ->  E. x  x  =  B )
12 isset 2745 . 2  |-  ( B  e.  _V  <->  E. x  x  =  B )
1311, 12sylibr 134 1  |-  ( A  e.  [ B ] R  ->  B  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1353   E.wex 1492    e. wcel 2148   E.wrex 2456   _Vcvv 2739   {csn 3594   class class class wbr 4005   "cima 4631   [cec 6536
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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-br 4006  df-opab 4067  df-xp 4634  df-cnv 4636  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-ec 6540
This theorem is referenced by:  relelec  6578  ecdmn0m  6580
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