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Theorem elimasni 5171
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 3575 . . . . 5  |-  -.  C  e.  (/)
2 snprc 3814 . . . . . . . . 9  |-  ( -.  B  e.  _V  <->  { B }  =  (/) )
32biimpi 187 . . . . . . . 8  |-  ( -.  B  e.  _V  ->  { B }  =  (/) )
43imaeq2d 5143 . . . . . . 7  |-  ( -.  B  e.  _V  ->  ( A " { B } )  =  ( A " (/) ) )
5 ima0 5161 . . . . . . 7  |-  ( A
" (/) )  =  (/)
64, 5syl6eq 2435 . . . . . 6  |-  ( -.  B  e.  _V  ->  ( A " { B } )  =  (/) )
76eleq2d 2454 . . . . 5  |-  ( -.  B  e.  _V  ->  ( C  e.  ( A
" { B }
)  <->  C  e.  (/) ) )
81, 7mtbiri 295 . . . 4  |-  ( -.  B  e.  _V  ->  -.  C  e.  ( A
" { B }
) )
98con4i 124 . . 3  |-  ( C  e.  ( A " { B } )  ->  B  e.  _V )
10 elex 2907 . . 3  |-  ( C  e.  ( A " { B } )  ->  C  e.  _V )
119, 10jca 519 . 2  |-  ( C  e.  ( A " { B } )  -> 
( B  e.  _V  /\  C  e.  _V )
)
12 elimasng 5170 . . . 4  |-  ( ( B  e.  _V  /\  C  e.  _V )  ->  ( C  e.  ( A " { B } )  <->  <. B ,  C >.  e.  A ) )
13 df-br 4154 . . . 4  |-  ( B A C  <->  <. B ,  C >.  e.  A )
1412, 13syl6bbr 255 . . 3  |-  ( ( B  e.  _V  /\  C  e.  _V )  ->  ( C  e.  ( A " { B } )  <->  B A C ) )
1514biimpd 199 . 2  |-  ( ( B  e.  _V  /\  C  e.  _V )  ->  ( C  e.  ( A " { B } )  ->  B A C ) )
1611, 15mpcom 34 1  |-  ( C  e.  ( A " { B } )  ->  B A C )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   _Vcvv 2899   (/)c0 3571   {csn 3757   <.cop 3760   class class class wbr 4153   "cima 4821
This theorem is referenced by:  dffv2  5735
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pr 4344
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-sbc 3105  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-br 4154  df-opab 4208  df-xp 4824  df-cnv 4826  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831
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