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Theorem elimasn 4757
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 3818 . . 3  |-  ( x  =  C  ->  ( B A x  <->  B A C ) )
3 elimasn.1 . . . 4  |-  B  e. 
_V
4 imasng 4755 . . . 4  |-  ( B  e.  _V  ->  ( A " { B }
)  =  { x  |  B A x }
)
53, 4ax-mp 7 . . 3  |-  ( A
" { B }
)  =  { x  |  B A x }
61, 2, 5elab2 2753 . 2  |-  ( C  e.  ( A " { B } )  <->  B A C )
7 df-br 3815 . 2  |-  ( B A C  <->  <. B ,  C >.  e.  A )
86, 7bitri 182 1  |-  ( C  e.  ( A " { B } )  <->  <. B ,  C >.  e.  A )
Colors of variables: wff set class
Syntax hints:    <-> wb 103    = wceq 1287    e. wcel 1436   {cab 2071   _Vcvv 2614   {csn 3425   <.cop 3428   class class class wbr 3814   "cima 4407
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3925  ax-pow 3977  ax-pr 4003
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361  df-v 2616  df-sbc 2829  df-un 2990  df-in 2992  df-ss 2999  df-pw 3411  df-sn 3431  df-pr 3432  df-op 3434  df-br 3815  df-opab 3869  df-xp 4410  df-cnv 4412  df-dm 4414  df-rn 4415  df-res 4416  df-ima 4417
This theorem is referenced by:  elimasng  4758  dfco2  4887  dfco2a  4888  ressn  4928
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