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Theorem ressn 5169
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 4935 . 2  |-  Rel  ( A  |`  { B }
)
2 relxp 4735 . 2  |-  Rel  ( { B }  X.  ( A " { B }
) )
3 ancom 266 . . . 4  |-  ( (
<. x ,  y >.  e.  A  /\  x  e.  { B } )  <-> 
( x  e.  { B }  /\  <. x ,  y >.  e.  A
) )
4 vex 2740 . . . . . . 7  |-  x  e. 
_V
5 vex 2740 . . . . . . 7  |-  y  e. 
_V
64, 5elimasn 4995 . . . . . 6  |-  ( y  e.  ( A " { x } )  <->  <. x ,  y >.  e.  A )
7 elsni 3610 . . . . . . . . 9  |-  ( x  e.  { B }  ->  x  =  B )
87sneqd 3605 . . . . . . . 8  |-  ( x  e.  { B }  ->  { x }  =  { B } )
98imaeq2d 4970 . . . . . . 7  |-  ( x  e.  { B }  ->  ( A " {
x } )  =  ( A " { B } ) )
109eleq2d 2247 . . . . . 6  |-  ( x  e.  { B }  ->  ( y  e.  ( A " { x } )  <->  y  e.  ( A " { B } ) ) )
116, 10bitr3id 194 . . . . 5  |-  ( x  e.  { B }  ->  ( <. x ,  y
>.  e.  A  <->  y  e.  ( A " { B } ) ) )
1211pm5.32i 454 . . . 4  |-  ( ( x  e.  { B }  /\  <. x ,  y
>.  e.  A )  <->  ( x  e.  { B }  /\  y  e.  ( A " { B } ) ) )
133, 12bitri 184 . . 3  |-  ( (
<. x ,  y >.  e.  A  /\  x  e.  { B } )  <-> 
( x  e.  { B }  /\  y  e.  ( A " { B } ) ) )
145opelres 4912 . . 3  |-  ( <.
x ,  y >.  e.  ( A  |`  { B } )  <->  ( <. x ,  y >.  e.  A  /\  x  e.  { B } ) )
15 opelxp 4656 . . 3  |-  ( <.
x ,  y >.  e.  ( { B }  X.  ( A " { B } ) )  <->  ( x  e.  { B }  /\  y  e.  ( A " { B } ) ) )
1613, 14, 153bitr4i 212 . 2  |-  ( <.
x ,  y >.  e.  ( A  |`  { B } )  <->  <. x ,  y >.  e.  ( { B }  X.  ( A " { B }
) ) )
171, 2, 16eqrelriiv 4720 1  |-  ( A  |`  { B } )  =  ( { B }  X.  ( A " { B } ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 104    = wceq 1353    e. wcel 2148   {csn 3592   <.cop 3595    X. cxp 4624    |` cres 4628   "cima 4629
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-sbc 2963  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-br 4004  df-opab 4065  df-xp 4632  df-rel 4633  df-cnv 4634  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639
This theorem is referenced by: (None)
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