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Theorem ecexr 6142
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 4700 . . . . 5  |-  ( A  e.  ( R " { B } )  -> 
( A  e.  ( R " { B } )  <->  E. x  e.  { B } x R A ) )
21ibi 169 . . . 4  |-  ( A  e.  ( R " { B } )  ->  E. x  e.  { B } x R A )
3 df-ec 6139 . . . 4  |-  [ B ] R  =  ( R " { B }
)
42, 3eleq2s 2148 . . 3  |-  ( A  e.  [ B ] R  ->  E. x  e.  { B } x R A )
5 df-rex 2329 . . . 4  |-  ( E. x  e.  { B } x R A  <->  E. x ( x  e. 
{ B }  /\  x R A ) )
6 simpl 106 . . . . . 6  |-  ( ( x  e.  { B }  /\  x R A )  ->  x  e.  { B } )
7 velsn 3420 . . . . . 6  |-  ( x  e.  { B }  <->  x  =  B )
86, 7sylib 131 . . . . 5  |-  ( ( x  e.  { B }  /\  x R A )  ->  x  =  B )
98eximi 1507 . . . 4  |-  ( E. x ( x  e. 
{ B }  /\  x R A )  ->  E. x  x  =  B )
105, 9sylbi 118 . . 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 2578 . 2  |-  ( B  e.  _V  <->  E. x  x  =  B )
1311, 12sylibr 141 1  |-  ( A  e.  [ B ] R  ->  B  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101    = wceq 1259   E.wex 1397    e. wcel 1409   E.wrex 2324   _Vcvv 2574   {csn 3403   class class class wbr 3792   "cima 4376   [cec 6135
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-br 3793  df-opab 3847  df-xp 4379  df-cnv 4381  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-ec 6139
This theorem is referenced by:  relelec  6177  ecdmn0m  6179
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