Theorem elimasng 5096

Description: Membership in an image of a singleton. (Contributed by Raph Levien, 21-Oct-2006.)

Assertion

Ref	Expression
elimasng	|- ( ( B e. V /\ C e. W ) -> ( C e. ( A " { B } ) <-> <. B , C >. e. A ) )

Proof of Theorem elimasng

Dummy variables y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		sneq 3677	. . . . 5 |- ( y = B -> { y } = { B } )
2	1	imaeq2d 5068	. . . 4 |- ( y = B -> ( A " { y } ) = ( A " { B } ) )
3	2	eleq2d 2299	. . 3 |- ( y = B -> ( z e. ( A " { y } ) <-> z e. ( A " { B } ) ) )
4		opeq1 3857	. . . 4 |- ( y = B -> <. y , z >. = <. B , z >. )
5	4	eleq1d 2298	. . 3 |- ( y = B -> ( <. y , z >. e. A <-> <. B , z >. e. A ) )
6	3, 5	bibi12d 235	. 2 |- ( y = B -> ( ( z e. ( A " { y } ) <-> <. y , z >. e. A ) <-> ( z e. ( A " { B } ) <-> <. B , z >. e. A ) ) )
7		eleq1 2292	. . 3 |- ( z = C -> ( z e. ( A " { B } ) <-> C e. ( A " { B } ) ) )
8		opeq2 3858	. . . 4 |- ( z = C -> <. B , z >. = <. B , C >. )
9	8	eleq1d 2298	. . 3 |- ( z = C -> ( <. B , z >. e. A <-> <. B , C >. e. A ) )
10	7, 9	bibi12d 235	. 2 |- ( z = C -> ( ( z e. ( A " { B } ) <-> <. B , z >. e. A ) <-> ( C e. ( A " { B } ) <-> <. B , C >. e. A ) ) )
11		vex 2802	. . 3 |- y e. _V
12		vex 2802	. . 3 |- z e. _V
13	11, 12	elimasn 5095	. 2 |- ( z e. ( A " { y } ) <-> <. y , z >. e. A )
14	6, 10, 13	vtocl2g 2865	1 |- ( ( B e. V /\ C e. W ) -> ( C e. ( A " { B } ) <-> <. B , C >. e. A ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1395   e. wcel 2200   {csn 3666   <.cop 3669   "cima 4722

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-sbc 3029  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-br 4084  df-opab 4146  df-xp 4725  df-cnv 4727  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732

This theorem is referenced by:  eliniseg  5098  inimasn  5146  dffv3g  5623  fvimacnv  5750  funfvima3  5873  elecg  6720  imasnopn  14973