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Theorem elimasni 5223
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 3624 . . . . 5  |-  -.  C  e.  (/)
2 snprc 3863 . . . . . . . . 9  |-  ( -.  B  e.  _V  <->  { B }  =  (/) )
32biimpi 187 . . . . . . . 8  |-  ( -.  B  e.  _V  ->  { B }  =  (/) )
43imaeq2d 5195 . . . . . . 7  |-  ( -.  B  e.  _V  ->  ( A " { B } )  =  ( A " (/) ) )
5 ima0 5213 . . . . . . 7  |-  ( A
" (/) )  =  (/)
64, 5syl6eq 2483 . . . . . 6  |-  ( -.  B  e.  _V  ->  ( A " { B } )  =  (/) )
76eleq2d 2502 . . . . 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 2956 . . 3  |-  ( C  e.  ( A " { B } )  ->  C  e.  _V )
119, 10jca 519 . 2  |-  ( C  e.  ( A " { B } )  -> 
( B  e.  _V  /\  C  e.  _V )
)
12 elimasng 5222 . . . 4  |-  ( ( B  e.  _V  /\  C  e.  _V )  ->  ( C  e.  ( A " { B } )  <->  <. B ,  C >.  e.  A ) )
13 df-br 4205 . . . 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 1652    e. wcel 1725   _Vcvv 2948   (/)c0 3620   {csn 3806   <.cop 3809   class class class wbr 4204   "cima 4873
This theorem is referenced by:  dffv2  5788
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-br 4205  df-opab 4259  df-xp 4876  df-cnv 4878  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883
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