Theorem elimasni 5195

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 3735	. . . . 5 |- -. C e. (/)
2		snprc 4035	. . . . . . . . 9 |- ( -. B e. _V <-> { B } = (/) )
3	2	biimpi 198	. . . . . . . 8 |- ( -. B e. _V -> { B } = (/) )
4	3	imaeq2d 5168	. . . . . . 7 |- ( -. B e. _V -> ( A " { B } ) = ( A " (/) ) )
5		ima0 5183	. . . . . . 7 |- ( A " (/) ) = (/)
6	4, 5	syl6eq 2501	. . . . . 6 |- ( -. B e. _V -> ( A " { B } ) = (/) )
7	6	eleq2d 2514	. . . . 5 |- ( -. B e. _V -> ( C e. ( A " { B } ) <-> C e. (/) ) )
8	1, 7	mtbiri 305	. . . 4 |- ( -. B e. _V -> -. C e. ( A " { B } ) )
9	8	con4i 134	. . 3 |- ( C e. ( A " { B } ) -> B e. _V )
10		elex 3054	. . 3 |- ( C e. ( A " { B } ) -> C e. _V )
11	9, 10	jca 535	. 2 |- ( C e. ( A " { B } ) -> ( B e. _V /\ C e. _V ) )
12		elimasng 5194	. . . 4 |- ( ( B e. _V /\ C e. _V ) -> ( C e. ( A " { B } ) <-> <. B , C >. e. A ) )
13		df-br 4403	. . . 4 |- ( B A C <-> <. B , C >. e. A )
14	12, 13	syl6bbr 267	. . 3 |- ( ( B e. _V /\ C e. _V ) -> ( C e. ( A " { B } ) <-> B A C ) )
15	14	biimpd 211	. 2 |- ( ( B e. _V /\ C e. _V ) -> ( C e. ( A " { B } ) -> B A C ) )
16	11, 15	mpcom 37	1 |- ( C e. ( A " { B } ) -> B A C )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 371   = wceq 1444   e. wcel 1887   _Vcvv 3045   (/)c0 3731   {csn 3968   <.cop 3974   class class class wbr 4402   "cima 4837

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pr 4639

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-sbc 3268  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-sn 3969  df-pr 3971  df-op 3975  df-br 4403  df-opab 4462  df-xp 4840  df-cnv 4842  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847

This theorem is referenced by:  dffv2  5938  poimirlem2  31942  poimirlem23  31963