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Theorem elimasn 5353
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 4444 . . 3  |-  ( x  =  C  ->  ( B A x  <->  B A C ) )
3 elimasn.1 . . . 4  |-  B  e. 
_V
4 imasng 5350 . . . 4  |-  ( B  e.  _V  ->  ( A " { B }
)  =  { x  |  B A x }
)
53, 4ax-mp 5 . . 3  |-  ( A
" { B }
)  =  { x  |  B A x }
61, 2, 5elab2 3246 . 2  |-  ( C  e.  ( A " { B } )  <->  B A C )
7 df-br 4441 . 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 1374    e. wcel 1762   {cab 2445   _Vcvv 3106   {csn 4020   <.cop 4026   class class class wbr 4440   "cima 4995
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pr 4679
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-sbc 3325  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-op 4027  df-br 4441  df-opab 4499  df-xp 4998  df-cnv 5000  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005
This theorem is referenced by:  elimasng  5354  dfco2  5497  dfco2a  5498  ressn  5534  funfvima3  6128  frxp  6883  marypha1lem  7882  gsum2dlem1  16781  gsum2dlem2  16782  gsum2d  16783  gsum2dOLD  16784  gsum2d2  16786  ovoliunlem1  21641  dfcnv2  27175  gsummpt2co  27420  funpartfun  29156  areaquad  30778
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