Theorem elimasn 5103

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 4092	. . 3 |- ( x = C -> ( B A x <-> B A C ) )
3		elimasn.1	. . . 4 |- B e. _V
4		imasng 5101	. . . 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 2954	. 2 |- ( C e. ( A " { B } ) <-> B A C )
7		df-br 4089	. 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 1397   e. wcel 2202   {cab 2217   _Vcvv 2802   {csn 3669   <.cop 3672   class class class wbr 4088   "cima 4728

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-sbc 3032  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-br 4089  df-opab 4151  df-xp 4731  df-cnv 4733  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738

This theorem is referenced by:  elimasng  5104  dfco2  5236  dfco2a  5237  ressn  5277