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Theorem elimasng 5200
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 3969 . . . . 5  |-  ( y  =  B  ->  { y }  =  { B } )
21imaeq2d 5174 . . . 4  |-  ( y  =  B  ->  ( A " { y } )  =  ( A
" { B }
) )
32eleq2d 2534 . . 3  |-  ( y  =  B  ->  (
z  e.  ( A
" { y } )  <->  z  e.  ( A " { B } ) ) )
4 opeq1 4158 . . . 4  |-  ( y  =  B  ->  <. y ,  z >.  =  <. B ,  z >. )
54eleq1d 2533 . . 3  |-  ( y  =  B  ->  ( <. y ,  z >.  e.  A  <->  <. B ,  z
>.  e.  A ) )
63, 5bibi12d 328 . 2  |-  ( y  =  B  ->  (
( z  e.  ( A " { y } )  <->  <. y ,  z >.  e.  A
)  <->  ( z  e.  ( A " { B } )  <->  <. B , 
z >.  e.  A ) ) )
7 eleq1 2537 . . 3  |-  ( z  =  C  ->  (
z  e.  ( A
" { B }
)  <->  C  e.  ( A " { B }
) ) )
8 opeq2 4159 . . . 4  |-  ( z  =  C  ->  <. B , 
z >.  =  <. B ,  C >. )
98eleq1d 2533 . . 3  |-  ( z  =  C  ->  ( <. B ,  z >.  e.  A  <->  <. B ,  C >.  e.  A ) )
107, 9bibi12d 328 . 2  |-  ( z  =  C  ->  (
( z  e.  ( A " { B } )  <->  <. B , 
z >.  e.  A )  <-> 
( C  e.  ( A " { B } )  <->  <. B ,  C >.  e.  A ) ) )
11 vex 3034 . . 3  |-  y  e. 
_V
12 vex 3034 . . 3  |-  z  e. 
_V
1311, 12elimasn 5199 . 2  |-  ( z  e.  ( A " { y } )  <->  <. y ,  z >.  e.  A )
146, 10, 13vtocl2g 3097 1  |-  ( ( B  e.  V  /\  C  e.  W )  ->  ( C  e.  ( A " { B } )  <->  <. B ,  C >.  e.  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 376    = wceq 1452    e. wcel 1904   {csn 3959   <.cop 3965   "cima 4842
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-br 4396  df-opab 4455  df-xp 4845  df-cnv 4847  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852
This theorem is referenced by:  elimasni  5201  eliniseg  5203  inimasn  5259  elpredim  5399  elpredg  5401  dffv3  5875  fvimacnv  6012  fvrnressn  6095  elecg  7420  imasnopn  20782  imasncld  20783  imasncls  20784  ustelimasn  21315  blval2  21655  elbl4  21656  1stpreimas  28361  opelco3  30491  funpartfv  30783  eltail  31101  brtrclfv2  36390  frege77d  36409
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