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Theorem elimasni 5364
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 3789 . . . . 5  |-  -.  C  e.  (/)
2 snprc 4091 . . . . . . . . 9  |-  ( -.  B  e.  _V  <->  { B }  =  (/) )
32biimpi 194 . . . . . . . 8  |-  ( -.  B  e.  _V  ->  { B }  =  (/) )
43imaeq2d 5337 . . . . . . 7  |-  ( -.  B  e.  _V  ->  ( A " { B } )  =  ( A " (/) ) )
5 ima0 5352 . . . . . . 7  |-  ( A
" (/) )  =  (/)
64, 5syl6eq 2524 . . . . . 6  |-  ( -.  B  e.  _V  ->  ( A " { B } )  =  (/) )
76eleq2d 2537 . . . . 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 3122 . . 3  |-  ( C  e.  ( A " { B } )  ->  C  e.  _V )
119, 10jca 532 . 2  |-  ( C  e.  ( A " { B } )  -> 
( B  e.  _V  /\  C  e.  _V )
)
12 elimasng 5363 . . . 4  |-  ( ( B  e.  _V  /\  C  e.  _V )  ->  ( C  e.  ( A " { B } )  <->  <. B ,  C >.  e.  A ) )
13 df-br 4448 . . . 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 1379    e. wcel 1767   _Vcvv 3113   (/)c0 3785   {csn 4027   <.cop 4033   class class class wbr 4447   "cima 5002
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-br 4448  df-opab 4506  df-xp 5005  df-cnv 5007  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012
This theorem is referenced by:  dffv2  5940
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