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Theorem elimasng 5039
Description: Membership in an image of a singleton. (Contributed by Raph Levien, 21-Oct-2006.)
Assertion
Ref Expression
elimasng  |-  ( ( B  e.  V  /\  C  e.  W )  ->  ( C  e.  ( A " { B } )  <->  <. B ,  C >.  e.  A ) )

Proof of Theorem elimasng
Dummy variables  y 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sneq 3651 . . . . 5  |-  ( y  =  B  ->  { y }  =  { B } )
21imaeq2d 5012 . . . 4  |-  ( y  =  B  ->  ( A " { y } )  =  ( A
" { B }
) )
32eleq2d 2350 . . 3  |-  ( y  =  B  ->  (
z  e.  ( A
" { y } )  <->  z  e.  ( A " { B } ) ) )
4 opeq1 3796 . . . 4  |-  ( y  =  B  ->  <. y ,  z >.  =  <. B ,  z >. )
54eleq1d 2349 . . 3  |-  ( y  =  B  ->  ( <. y ,  z >.  e.  A  <->  <. B ,  z
>.  e.  A ) )
63, 5bibi12d 312 . 2  |-  ( y  =  B  ->  (
( z  e.  ( A " { y } )  <->  <. y ,  z >.  e.  A
)  <->  ( z  e.  ( A " { B } )  <->  <. B , 
z >.  e.  A ) ) )
7 eleq1 2343 . . 3  |-  ( z  =  C  ->  (
z  e.  ( A
" { B }
)  <->  C  e.  ( A " { B }
) ) )
8 opeq2 3797 . . . 4  |-  ( z  =  C  ->  <. B , 
z >.  =  <. B ,  C >. )
98eleq1d 2349 . . 3  |-  ( z  =  C  ->  ( <. B ,  z >.  e.  A  <->  <. B ,  C >.  e.  A ) )
107, 9bibi12d 312 . 2  |-  ( z  =  C  ->  (
( z  e.  ( A " { B } )  <->  <. B , 
z >.  e.  A )  <-> 
( C  e.  ( A " { B } )  <->  <. B ,  C >.  e.  A ) ) )
11 vex 2791 . . 3  |-  y  e. 
_V
12 vex 2791 . . 3  |-  z  e. 
_V
1311, 12elimasn 5038 . 2  |-  ( z  e.  ( A " { y } )  <->  <. y ,  z >.  e.  A )
146, 10, 13vtocl2g 2847 1  |-  ( ( B  e.  V  /\  C  e.  W )  ->  ( C  e.  ( A " { B } )  <->  <. B ,  C >.  e.  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   {csn 3640   <.cop 3643   "cima 4692
This theorem is referenced by:  elimasni  5040  eliniseg  5042  dffv3  5521  fvimacnv  5640  elecg  6698  elpredim  24176  elpredg  24178  eltail  26323
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-xp 4695  df-cnv 4697  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702
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