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Theorem ressn 5385
Description: Restriction of a class to a singleton. (Contributed by Mario Carneiro, 28-Dec-2014.)
Assertion
Ref Expression
ressn  |-  ( A  |`  { B } )  =  ( { B }  X.  ( A " { B } ) )

Proof of Theorem ressn
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relres 5150 . 2  |-  Rel  ( A  |`  { B }
)
2 relxp 4959 . 2  |-  Rel  ( { B }  X.  ( A " { B }
) )
3 ancom 450 . . . 4  |-  ( (
<. x ,  y >.  e.  A  /\  x  e.  { B } )  <-> 
( x  e.  { B }  /\  <. x ,  y >.  e.  A
) )
4 vex 2987 . . . . . . 7  |-  x  e. 
_V
5 vex 2987 . . . . . . 7  |-  y  e. 
_V
64, 5elimasn 5206 . . . . . 6  |-  ( y  e.  ( A " { x } )  <->  <. x ,  y >.  e.  A )
7 elsni 3914 . . . . . . . . 9  |-  ( x  e.  { B }  ->  x  =  B )
87sneqd 3901 . . . . . . . 8  |-  ( x  e.  { B }  ->  { x }  =  { B } )
98imaeq2d 5181 . . . . . . 7  |-  ( x  e.  { B }  ->  ( A " {
x } )  =  ( A " { B } ) )
109eleq2d 2510 . . . . . 6  |-  ( x  e.  { B }  ->  ( y  e.  ( A " { x } )  <->  y  e.  ( A " { B } ) ) )
116, 10syl5bbr 259 . . . . 5  |-  ( x  e.  { B }  ->  ( <. x ,  y
>.  e.  A  <->  y  e.  ( A " { B } ) ) )
1211pm5.32i 637 . . . 4  |-  ( ( x  e.  { B }  /\  <. x ,  y
>.  e.  A )  <->  ( x  e.  { B }  /\  y  e.  ( A " { B } ) ) )
133, 12bitri 249 . . 3  |-  ( (
<. x ,  y >.  e.  A  /\  x  e.  { B } )  <-> 
( x  e.  { B }  /\  y  e.  ( A " { B } ) ) )
145opelres 5128 . . 3  |-  ( <.
x ,  y >.  e.  ( A  |`  { B } )  <->  ( <. x ,  y >.  e.  A  /\  x  e.  { B } ) )
15 opelxp 4881 . . 3  |-  ( <.
x ,  y >.  e.  ( { B }  X.  ( A " { B } ) )  <->  ( x  e.  { B }  /\  y  e.  ( A " { B } ) ) )
1613, 14, 153bitr4i 277 . 2  |-  ( <.
x ,  y >.  e.  ( A  |`  { B } )  <->  <. x ,  y >.  e.  ( { B }  X.  ( A " { B }
) ) )
171, 2, 16eqrelriiv 4946 1  |-  ( A  |`  { B } )  =  ( { B }  X.  ( A " { B } ) )
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    = wceq 1369    e. wcel 1756   {csn 3889   <.cop 3895    X. cxp 4850    |` cres 4854   "cima 4855
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4425  ax-nul 4433  ax-pr 4543
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-ral 2732  df-rex 2733  df-rab 2736  df-v 2986  df-sbc 3199  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-nul 3650  df-if 3804  df-sn 3890  df-pr 3892  df-op 3896  df-br 4305  df-opab 4363  df-xp 4858  df-rel 4859  df-cnv 4860  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865
This theorem is referenced by:  gsum2dlem2  16474  gsum2dOLD  16476  dprd2da  16553  ustneism  19810
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