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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
StepHypRef Expression
1 noel 3774 . . . . 5  |-  -.  C  e.  (/)
2 snprc 4078 . . . . . . . . 9  |-  ( -.  B  e.  _V  <->  { B }  =  (/) )
32biimpi 194 . . . . . . . 8  |-  ( -.  B  e.  _V  ->  { B }  =  (/) )
43imaeq2d 5327 . . . . . . 7  |-  ( -.  B  e.  _V  ->  ( A " { B } )  =  ( A " (/) ) )
5 ima0 5342 . . . . . . 7  |-  ( A
" (/) )  =  (/)
64, 5syl6eq 2500 . . . . . 6  |-  ( -.  B  e.  _V  ->  ( A " { B } )  =  (/) )
76eleq2d 2513 . . . . 5  |-  ( -.  B  e.  _V  ->  ( C  e.  ( A
" { B }
)  <->  C  e.  (/) ) )
81, 7mtbiri 303 . . . 4  |-  ( -.  B  e.  _V  ->  -.  C  e.  ( A
" { B }
) )
98con4i 130 . . 3  |-  ( C  e.  ( A " { B } )  ->  B  e.  _V )
10 elex 3104 . . 3  |-  ( C  e.  ( A " { B } )  ->  C  e.  _V )
119, 10jca 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 )
1412, 13syl6bbr 263 . . 3  |-  ( ( B  e.  _V  /\  C  e.  _V )  ->  ( C  e.  ( A " { B } )  <->  B A C ) )
1514biimpd 207 . 2  |-  ( ( B  e.  _V  /\  C  e.  _V )  ->  ( C  e.  ( A " { B } )  ->  B A C ) )
1611, 15mpcom 36 1  |-  ( C  e.  ( A " { B } )  ->  B A C )
Colors of variables: wff setvar class
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
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