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Theorem elimasn 5221
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 4208 . . 3  |-  ( x  =  C  ->  ( B A x  <->  B A C ) )
3 elimasn.1 . . . 4  |-  B  e. 
_V
4 imasng 5218 . . . 4  |-  ( B  e.  _V  ->  ( A " { B }
)  =  { x  |  B A x }
)
53, 4ax-mp 8 . . 3  |-  ( A
" { B }
)  =  { x  |  B A x }
61, 2, 5elab2 3077 . 2  |-  ( C  e.  ( A " { B } )  <->  B A C )
7 df-br 4205 . 2  |-  ( B A C  <->  <. B ,  C >.  e.  A )
86, 7bitri 241 1  |-  ( C  e.  ( A " { B } )  <->  <. B ,  C >.  e.  A )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    = wceq 1652    e. wcel 1725   {cab 2421   _Vcvv 2948   {csn 3806   <.cop 3809   class class class wbr 4204   "cima 4873
This theorem is referenced by:  elimasng  5222  dfco2  5361  dfco2a  5362  ressn  5400  funfvima3  5967  frxp  6448  marypha1lem  7430  gsum2d  15538  gsum2d2  15540  ovoliunlem1  19390  funpartfun  25780
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-br 4205  df-opab 4259  df-xp 4876  df-cnv 4878  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883
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