Theorem elimasn 5110

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 4097	. . 3 |- ( x = C -> ( B A x <-> B A C ) )
3		elimasn.1	. . . 4 |- B e. _V
4		imasng 5108	. . . 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 2955	. 2 |- ( C e. ( A " { B } ) <-> B A C )
7		df-br 4094	. 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 1398   e. wcel 2202   {cab 2217   _Vcvv 2803   {csn 3673   <.cop 3676   class class class wbr 4093   "cima 4734

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-sbc 3033  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-br 4094  df-opab 4156  df-xp 4737  df-cnv 4739  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744

This theorem is referenced by:  elimasng  5111  dfco2  5243  dfco2a  5244  ressn  5284