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Theorem elimasni 5056
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 3472 . . . . 5  |-  -.  C  e.  (/)
2 snprc 3708 . . . . . . . . 9  |-  ( -.  B  e.  _V  <->  { B }  =  (/) )
32biimpi 186 . . . . . . . 8  |-  ( -.  B  e.  _V  ->  { B }  =  (/) )
43imaeq2d 5028 . . . . . . 7  |-  ( -.  B  e.  _V  ->  ( A " { B } )  =  ( A " (/) ) )
5 ima0 5046 . . . . . . 7  |-  ( A
" (/) )  =  (/)
64, 5syl6eq 2344 . . . . . 6  |-  ( -.  B  e.  _V  ->  ( A " { B } )  =  (/) )
76eleq2d 2363 . . . . 5  |-  ( -.  B  e.  _V  ->  ( C  e.  ( A
" { B }
)  <->  C  e.  (/) ) )
81, 7mtbiri 294 . . . 4  |-  ( -.  B  e.  _V  ->  -.  C  e.  ( A
" { B }
) )
98con4i 122 . . 3  |-  ( C  e.  ( A " { B } )  ->  B  e.  _V )
10 elex 2809 . . 3  |-  ( C  e.  ( A " { B } )  ->  C  e.  _V )
119, 10jca 518 . 2  |-  ( C  e.  ( A " { B } )  -> 
( B  e.  _V  /\  C  e.  _V )
)
12 elimasng 5055 . . . 4  |-  ( ( B  e.  _V  /\  C  e.  _V )  ->  ( C  e.  ( A " { B } )  <->  <. B ,  C >.  e.  A ) )
13 df-br 4040 . . . 4  |-  ( B A C  <->  <. B ,  C >.  e.  A )
1412, 13syl6bbr 254 . . 3  |-  ( ( B  e.  _V  /\  C  e.  _V )  ->  ( C  e.  ( A " { B } )  <->  B A C ) )
1514biimpd 198 . 2  |-  ( ( B  e.  _V  /\  C  e.  _V )  ->  ( C  e.  ( A " { B } )  ->  B A C ) )
1611, 15mpcom 32 1  |-  ( C  e.  ( A " { B } )  ->  B A C )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   _Vcvv 2801   (/)c0 3468   {csn 3653   <.cop 3656   class class class wbr 4039   "cima 4708
This theorem is referenced by:  dffv2  5608
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-xp 4711  df-cnv 4713  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718
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