Theorem elimasng 5150

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 3716	. . . . 5 |- ( y = B -> { y } = { B } )
2	1	imaeq2d 5121	. . . 4 |- ( y = B -> ( A " { y } ) = ( A " { B } ) )
3	2	eleq2d 2308	. . 3 |- ( y = B -> ( z e. ( A " { y } ) <-> z e. ( A " { B } ) ) )
4		opeq1 3899	. . . 4 |- ( y = B -> <. y , z >. = <. B , z >. )
5	4	eleq1d 2307	. . 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 2301	. . 3 |- ( z = C -> ( z e. ( A " { B } ) <-> C e. ( A " { B } ) ) )
8		opeq2 3900	. . . 4 |- ( z = C -> <. B , z >. = <. B , C >. )
9	8	eleq1d 2307	. . 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 2824	. . 3 |- y e. _V
12		vex 2824	. . 3 |- z e. _V
13	11, 12	elimasn 5149	. 2 |- ( z e. ( A " { y } ) <-> <. y , z >. e. A )
14	6, 10, 13	vtocl2g 2887	1 |- ( ( B e. V /\ C e. W ) -> ( C e. ( A " { B } ) <-> <. B , C >. e. A ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   {csn 3705   <.cop 3708   "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:  eliniseg  5152  inimasn  5200  dffv3g  5686  fvimacnv  5815  funfvima3  5942  elecg  6837  imasnopn  15323