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Theorem ecexr 6111
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 4672 . . . . 5  |-  ( A  e.  ( R " { B } )  -> 
( A  e.  ( R " { B } )  <->  E. x  e.  { B } x R A ) )
21ibi 165 . . . 4  |-  ( A  e.  ( R " { B } )  ->  E. x  e.  { B } x R A )
3 df-ec 6108 . . . 4  |-  [ B ] R  =  ( R " { B }
)
42, 3eleq2s 2132 . . 3  |-  ( A  e.  [ B ] R  ->  E. x  e.  { B } x R A )
5 df-rex 2312 . . . 4  |-  ( E. x  e.  { B } x R A  <->  E. x ( x  e. 
{ B }  /\  x R A ) )
6 simpl 102 . . . . . 6  |-  ( ( x  e.  { B }  /\  x R A )  ->  x  e.  { B } )
7 velsn 3392 . . . . . 6  |-  ( x  e.  { B }  <->  x  =  B )
86, 7sylib 127 . . . . 5  |-  ( ( x  e.  { B }  /\  x R A )  ->  x  =  B )
98eximi 1491 . . . 4  |-  ( E. x ( x  e. 
{ B }  /\  x R A )  ->  E. x  x  =  B )
105, 9sylbi 114 . . 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 2561 . 2  |-  ( B  e.  _V  <->  E. x  x  =  B )
1311, 12sylibr 137 1  |-  ( A  e.  [ B ] R  ->  B  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 97    = wceq 1243   E.wex 1381    e. wcel 1393   E.wrex 2307   _Vcvv 2557   {csn 3375   class class class wbr 3764   "cima 4348   [cec 6104
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-xp 4351  df-cnv 4353  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-ec 6108
This theorem is referenced by:  relelec  6146  ecdmn0m  6148
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