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Theorem elimasn 4997
Description: Membership in an image of a singleton. (Contributed by NM, 15-Mar-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
Hypotheses
Ref Expression
elimasn.1  |-  B  e. 
_V
elimasn.2  |-  C  e. 
_V
Assertion
Ref Expression
elimasn  |-  ( C  e.  ( A " { B } )  <->  <. B ,  C >.  e.  A )

Proof of Theorem elimasn
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 elimasn.2 . . 3  |-  C  e. 
_V
2 breq2 4009 . . 3  |-  ( x  =  C  ->  ( B A x  <->  B A C ) )
3 elimasn.1 . . . 4  |-  B  e. 
_V
4 imasng 4995 . . . 4  |-  ( B  e.  _V  ->  ( A " { B }
)  =  { x  |  B A x }
)
53, 4ax-mp 5 . . 3  |-  ( A
" { B }
)  =  { x  |  B A x }
61, 2, 5elab2 2887 . 2  |-  ( C  e.  ( A " { B } )  <->  B A C )
7 df-br 4006 . 2  |-  ( B A C  <->  <. B ,  C >.  e.  A )
86, 7bitri 184 1  |-  ( C  e.  ( A " { B } )  <->  <. B ,  C >.  e.  A )
Colors of variables: wff set class
Syntax hints:    <-> wb 105    = wceq 1353    e. wcel 2148   {cab 2163   _Vcvv 2739   {csn 3594   <.cop 3597   class class class wbr 4005   "cima 4631
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-sbc 2965  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
This theorem is referenced by:  elimasng  4998  dfco2  5130  dfco2a  5131  ressn  5171
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