Theorem elimasn 5068

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 4063	. . 3 |- ( x = C -> ( B A x <-> B A C ) )
3		elimasn.1	. . . 4 |- B e. _V
4		imasng 5066	. . . 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 2928	. 2 |- ( C e. ( A " { B } ) <-> B A C )
7		df-br 4060	. 2 |- ( B A C <-> <. B , C >. e. A )
8	6, 7	bitri 184	1 |- ( C e. ( A " { B } ) <-> <. B , C >. e. A )

Syntax hints:   <-> wb 105   = wceq 1373   e. wcel 2178   {cab 2193   _Vcvv 2776   {csn 3643   <.cop 3646   class class class wbr 4059   "cima 4696

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2778  df-sbc 3006  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-br 4060  df-opab 4122  df-xp 4699  df-cnv 4701  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706

This theorem is referenced by:  elimasng  5069  dfco2  5201  dfco2a  5202  ressn  5242