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Theorem elimasn 5182
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 4399 . . 3  |-  ( x  =  C  ->  ( B A x  <->  B A C ) )
3 elimasn.1 . . . 4  |-  B  e. 
_V
4 imasng 5179 . . . 4  |-  ( B  e.  _V  ->  ( A " { B }
)  =  { x  |  B A x }
)
53, 4ax-mp 5 . . 3  |-  ( A
" { B }
)  =  { x  |  B A x }
61, 2, 5elab2 3199 . 2  |-  ( C  e.  ( A " { B } )  <->  B A C )
7 df-br 4396 . 2  |-  ( B A C  <->  <. B ,  C >.  e.  A )
86, 7bitri 249 1  |-  ( C  e.  ( A " { B } )  <->  <. B ,  C >.  e.  A )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    = wceq 1405    e. wcel 1842   {cab 2387   _Vcvv 3059   {csn 3972   <.cop 3978   class class class wbr 4395   "cima 4826
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pr 4630
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-rab 2763  df-v 3061  df-sbc 3278  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-sn 3973  df-pr 3975  df-op 3979  df-br 4396  df-opab 4454  df-xp 4829  df-cnv 4831  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836
This theorem is referenced by:  elimasng  5183  dfco2  5322  dfco2a  5323  ressn  5360  funfvima3  6130  frxp  6894  marypha1lem  7927  gsum2dlem1  17318  gsum2dlem2  17319  gsum2d  17320  gsum2dOLD  17321  gsum2d2  17323  ovoliunlem1  22205  iunsnima  27907  dfcnv2  27961  gsummpt2co  28222  gsummpt2d  28223  funpartfun  30281  areaquad  35548  dffrege76  35920  frege97  35941  frege98  35942  frege109  35953  frege110  35954  frege131  35975  frege133  35977
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