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Theorem ressn 4888
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 4667 . 2  |-  Rel  ( A  |`  { B }
)
2 relxp 4475 . 2  |-  Rel  ( { B }  X.  ( A " { B }
) )
3 ancom 262 . . . 4  |-  ( (
<. x ,  y >.  e.  A  /\  x  e.  { B } )  <-> 
( x  e.  { B }  /\  <. x ,  y >.  e.  A
) )
4 vex 2605 . . . . . . 7  |-  x  e. 
_V
5 vex 2605 . . . . . . 7  |-  y  e. 
_V
64, 5elimasn 4722 . . . . . 6  |-  ( y  e.  ( A " { x } )  <->  <. x ,  y >.  e.  A )
7 elsni 3424 . . . . . . . . 9  |-  ( x  e.  { B }  ->  x  =  B )
87sneqd 3419 . . . . . . . 8  |-  ( x  e.  { B }  ->  { x }  =  { B } )
98imaeq2d 4698 . . . . . . 7  |-  ( x  e.  { B }  ->  ( A " {
x } )  =  ( A " { B } ) )
109eleq2d 2149 . . . . . 6  |-  ( x  e.  { B }  ->  ( y  e.  ( A " { x } )  <->  y  e.  ( A " { B } ) ) )
116, 10syl5bbr 192 . . . . 5  |-  ( x  e.  { B }  ->  ( <. x ,  y
>.  e.  A  <->  y  e.  ( A " { B } ) ) )
1211pm5.32i 442 . . . 4  |-  ( ( x  e.  { B }  /\  <. x ,  y
>.  e.  A )  <->  ( x  e.  { B }  /\  y  e.  ( A " { B } ) ) )
133, 12bitri 182 . . 3  |-  ( (
<. x ,  y >.  e.  A  /\  x  e.  { B } )  <-> 
( x  e.  { B }  /\  y  e.  ( A " { B } ) ) )
145opelres 4645 . . 3  |-  ( <.
x ,  y >.  e.  ( A  |`  { B } )  <->  ( <. x ,  y >.  e.  A  /\  x  e.  { B } ) )
15 opelxp 4400 . . 3  |-  ( <.
x ,  y >.  e.  ( { B }  X.  ( A " { B } ) )  <->  ( x  e.  { B }  /\  y  e.  ( A " { B } ) ) )
1613, 14, 153bitr4i 210 . 2  |-  ( <.
x ,  y >.  e.  ( A  |`  { B } )  <->  <. x ,  y >.  e.  ( { B }  X.  ( A " { B }
) ) )
171, 2, 16eqrelriiv 4460 1  |-  ( A  |`  { B } )  =  ( { B }  X.  ( A " { B } ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 102    = wceq 1285    e. wcel 1434   {csn 3406   <.cop 3409    X. cxp 4369    |` cres 4373   "cima 4374
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 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-sbc 2817  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-br 3794  df-opab 3848  df-xp 4377  df-rel 4378  df-cnv 4379  df-dm 4381  df-rn 4382  df-res 4383  df-ima 4384
This theorem is referenced by: (None)
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