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Theorem elimasng 5230
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 3825 . . . . 5  |-  ( y  =  B  ->  { y }  =  { B } )
21imaeq2d 5203 . . . 4  |-  ( y  =  B  ->  ( A " { y } )  =  ( A
" { B }
) )
32eleq2d 2503 . . 3  |-  ( y  =  B  ->  (
z  e.  ( A
" { y } )  <->  z  e.  ( A " { B } ) ) )
4 opeq1 3984 . . . 4  |-  ( y  =  B  ->  <. y ,  z >.  =  <. B ,  z >. )
54eleq1d 2502 . . 3  |-  ( y  =  B  ->  ( <. y ,  z >.  e.  A  <->  <. B ,  z
>.  e.  A ) )
63, 5bibi12d 313 . 2  |-  ( y  =  B  ->  (
( z  e.  ( A " { y } )  <->  <. y ,  z >.  e.  A
)  <->  ( z  e.  ( A " { B } )  <->  <. B , 
z >.  e.  A ) ) )
7 eleq1 2496 . . 3  |-  ( z  =  C  ->  (
z  e.  ( A
" { B }
)  <->  C  e.  ( A " { B }
) ) )
8 opeq2 3985 . . . 4  |-  ( z  =  C  ->  <. B , 
z >.  =  <. B ,  C >. )
98eleq1d 2502 . . 3  |-  ( z  =  C  ->  ( <. B ,  z >.  e.  A  <->  <. B ,  C >.  e.  A ) )
107, 9bibi12d 313 . 2  |-  ( z  =  C  ->  (
( z  e.  ( A " { B } )  <->  <. B , 
z >.  e.  A )  <-> 
( C  e.  ( A " { B } )  <->  <. B ,  C >.  e.  A ) ) )
11 vex 2959 . . 3  |-  y  e. 
_V
12 vex 2959 . . 3  |-  z  e. 
_V
1311, 12elimasn 5229 . 2  |-  ( z  e.  ( A " { y } )  <->  <. y ,  z >.  e.  A )
146, 10, 13vtocl2g 3015 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 177    /\ wa 359    = wceq 1652    e. wcel 1725   {csn 3814   <.cop 3817   "cima 4881
This theorem is referenced by:  elimasni  5231  eliniseg  5233  inimasn  5289  dffv3  5724  fvimacnv  5845  elecg  6943  imasnopn  17722  imasncld  17723  imasncls  17724  ustelimasn  18252  blval2  18605  elbl4  18606  opelco3  25403  elpredim  25451  elpredg  25453  funpartfv  25790  eltail  26403
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 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-br 4213  df-opab 4267  df-xp 4884  df-cnv 4886  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891
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