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Theorem ressn 5549
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 5307 . 2  |-  Rel  ( A  |`  { B }
)
2 relxp 5116 . 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 3121 . . . . . . 7  |-  x  e. 
_V
5 vex 3121 . . . . . . 7  |-  y  e. 
_V
64, 5elimasn 5368 . . . . . 6  |-  ( y  e.  ( A " { x } )  <->  <. x ,  y >.  e.  A )
7 elsni 4058 . . . . . . . . 9  |-  ( x  e.  { B }  ->  x  =  B )
87sneqd 4045 . . . . . . . 8  |-  ( x  e.  { B }  ->  { x }  =  { B } )
98imaeq2d 5343 . . . . . . 7  |-  ( x  e.  { B }  ->  ( A " {
x } )  =  ( A " { B } ) )
109eleq2d 2537 . . . . . 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 5285 . . 3  |-  ( <.
x ,  y >.  e.  ( A  |`  { B } )  <->  ( <. x ,  y >.  e.  A  /\  x  e.  { B } ) )
15 opelxp 5035 . . 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 5103 1  |-  ( A  |`  { B } )  =  ( { B }  X.  ( A " { B } ) )
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    = wceq 1379    e. wcel 1767   {csn 4033   <.cop 4039    X. cxp 5003    |` cres 5007   "cima 5008
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pr 4692
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-br 4454  df-opab 4512  df-xp 5011  df-rel 5012  df-cnv 5013  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018
This theorem is referenced by:  gsum2dlem2  16871  gsum2dOLD  16873  dprd2da  16963  ustneism  20594
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