Theorem elimasni 5354

Description: Membership in an image of a singleton. (Contributed by NM, 5-Aug-2010.)

Assertion

Ref	Expression
elimasni	|- ( C e. ( A " { B } ) -> B A C )

Proof of Theorem elimasni

Step	Hyp	Ref	Expression
1		noel 3774	. . . . 5 |- -. C e. (/)
2		snprc 4078	. . . . . . . . 9 |- ( -. B e. _V <-> { B } = (/) )
3	2	biimpi 194	. . . . . . . 8 |- ( -. B e. _V -> { B } = (/) )
4	3	imaeq2d 5327	. . . . . . 7 |- ( -. B e. _V -> ( A " { B } ) = ( A " (/) ) )
5		ima0 5342	. . . . . . 7 |- ( A " (/) ) = (/)
6	4, 5	syl6eq 2500	. . . . . 6 |- ( -. B e. _V -> ( A " { B } ) = (/) )
7	6	eleq2d 2513	. . . . 5 |- ( -. B e. _V -> ( C e. ( A " { B } ) <-> C e. (/) ) )
8	1, 7	mtbiri 303	. . . 4 |- ( -. B e. _V -> -. C e. ( A " { B } ) )
9	8	con4i 130	. . 3 |- ( C e. ( A " { B } ) -> B e. _V )
10		elex 3104	. . 3 |- ( C e. ( A " { B } ) -> C e. _V )
11	9, 10	jca 532	. 2 |- ( C e. ( A " { B } ) -> ( B e. _V /\ C e. _V ) )
12		elimasng 5353	. . . 4 |- ( ( B e. _V /\ C e. _V ) -> ( C e. ( A " { B } ) <-> <. B , C >. e. A ) )
13		df-br 4438	. . . 4 |- ( B A C <-> <. B , C >. e. A )
14	12, 13	syl6bbr 263	. . 3 |- ( ( B e. _V /\ C e. _V ) -> ( C e. ( A " { B } ) <-> B A C ) )
15	14	biimpd 207	. 2 |- ( ( B e. _V /\ C e. _V ) -> ( C e. ( A " { B } ) -> B A C ) )
16	11, 15	mpcom 36	1 |- ( C e. ( A " { B } ) -> B A C )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 369   = wceq 1383   e. wcel 1804   _Vcvv 3095   (/)c0 3770   {csn 4014   <.cop 4020   class class class wbr 4437   "cima 4992

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pr 4676

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-sbc 3314  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-br 4438  df-opab 4496  df-xp 4995  df-cnv 4997  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002

This theorem is referenced by:  dffv2  5931