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Theorem elimasng 4769
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 3442 . . . . 5  |-  ( y  =  B  ->  { y }  =  { B } )
21imaeq2d 4743 . . . 4  |-  ( y  =  B  ->  ( A " { y } )  =  ( A
" { B }
) )
32eleq2d 2154 . . 3  |-  ( y  =  B  ->  (
z  e.  ( A
" { y } )  <->  z  e.  ( A " { B } ) ) )
4 opeq1 3607 . . . 4  |-  ( y  =  B  ->  <. y ,  z >.  =  <. B ,  z >. )
54eleq1d 2153 . . 3  |-  ( y  =  B  ->  ( <. y ,  z >.  e.  A  <->  <. B ,  z
>.  e.  A ) )
63, 5bibi12d 233 . 2  |-  ( y  =  B  ->  (
( z  e.  ( A " { y } )  <->  <. y ,  z >.  e.  A
)  <->  ( z  e.  ( A " { B } )  <->  <. B , 
z >.  e.  A ) ) )
7 eleq1 2147 . . 3  |-  ( z  =  C  ->  (
z  e.  ( A
" { B }
)  <->  C  e.  ( A " { B }
) ) )
8 opeq2 3608 . . . 4  |-  ( z  =  C  ->  <. B , 
z >.  =  <. B ,  C >. )
98eleq1d 2153 . . 3  |-  ( z  =  C  ->  ( <. B ,  z >.  e.  A  <->  <. B ,  C >.  e.  A ) )
107, 9bibi12d 233 . 2  |-  ( z  =  C  ->  (
( z  e.  ( A " { B } )  <->  <. B , 
z >.  e.  A )  <-> 
( C  e.  ( A " { B } )  <->  <. B ,  C >.  e.  A ) ) )
11 vex 2618 . . 3  |-  y  e. 
_V
12 vex 2618 . . 3  |-  z  e. 
_V
1311, 12elimasn 4768 . 2  |-  ( z  e.  ( A " { y } )  <->  <. y ,  z >.  e.  A )
146, 10, 13vtocl2g 2676 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    /\ wa 102    <-> wb 103    = wceq 1287    e. wcel 1436   {csn 3431   <.cop 3434   "cima 4416
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3934  ax-pow 3986  ax-pr 4012
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361  df-v 2617  df-sbc 2830  df-un 2992  df-in 2994  df-ss 3001  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-br 3823  df-opab 3877  df-xp 4419  df-cnv 4421  df-dm 4423  df-rn 4424  df-res 4425  df-ima 4426
This theorem is referenced by:  eliniseg  4771  inimasn  4817  dffv3g  5266  fvimacnv  5379  funfvima3  5491  elecg  6284
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