Theorem elimasn 5149

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 4129	. . 3 |- ( x = C -> ( B A x <-> B A C ) )
3		elimasn.1	. . . 4 |- B e. _V
4		imasng 5147	. . . 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 2974	. 2 |- ( C e. ( A " { B } ) <-> B A C )
7		df-br 4126	. 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 1402   e. wcel 2209   {cab 2224   _Vcvv 2821   {csn 3705   <.cop 3708   class class class wbr 4125   "cima 4772

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-br 4126  df-opab 4188  df-xp 4775  df-cnv 4777  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782

This theorem is referenced by:  elimasng  5150  dfco2  5282  dfco2a  5283  ressn  5323