Theorem elimasn 4971

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.

Step	Hyp	Ref	Expression
1		elimasn.2	. . 3 |- C e. _V
2		breq2 3986	. . 3 |- ( x = C -> ( B A x <-> B A C ) )
3		elimasn.1	. . . 4 |- B e. _V
4		imasng 4969	. . . 4 |- ( B e. _V -> ( A " { B } ) = { x | B A x } )
5	3, 4	ax-mp 5	. . 3 |- ( A " { B } ) = { x | B A x }
6	1, 2, 5	elab2 2874	. 2 |- ( C e. ( A " { B } ) <-> B A C )
7		df-br 3983	. 2 |- ( B A C <-> <. B , C >. e. A )
8	6, 7	bitri 183	1 |- ( C e. ( A " { B } ) <-> <. B , C >. e. A )

Syntax hints:   <-> wb 104   = wceq 1343   e. wcel 2136   {cab 2151   _Vcvv 2726   {csn 3576   <.cop 3579   class class class wbr 3982   "cima 4607

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-sbc 2952  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-opab 4044  df-xp 4610  df-cnv 4612  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617

This theorem is referenced by:  elimasng  4972  dfco2  5103  dfco2a  5104  ressn  5144