MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ressn Structured version   Unicode version

Theorem ressn 5408
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 5174 . 2  |-  Rel  ( A  |`  { B }
)
2 relxp 4983 . 2  |-  Rel  ( { B }  X.  ( A " { B }
) )
3 ancom 438 . . . 4  |-  ( (
<. x ,  y >.  e.  A  /\  x  e.  { B } )  <-> 
( x  e.  { B }  /\  <. x ,  y >.  e.  A
) )
4 vex 2959 . . . . . . 7  |-  x  e. 
_V
5 vex 2959 . . . . . . 7  |-  y  e. 
_V
64, 5elimasn 5229 . . . . . 6  |-  ( y  e.  ( A " { x } )  <->  <. x ,  y >.  e.  A )
7 elsni 3838 . . . . . . . . 9  |-  ( x  e.  { B }  ->  x  =  B )
87sneqd 3827 . . . . . . . 8  |-  ( x  e.  { B }  ->  { x }  =  { B } )
98imaeq2d 5203 . . . . . . 7  |-  ( x  e.  { B }  ->  ( A " {
x } )  =  ( A " { B } ) )
109eleq2d 2503 . . . . . 6  |-  ( x  e.  { B }  ->  ( y  e.  ( A " { x } )  <->  y  e.  ( A " { B } ) ) )
116, 10syl5bbr 251 . . . . 5  |-  ( x  e.  { B }  ->  ( <. x ,  y
>.  e.  A  <->  y  e.  ( A " { B } ) ) )
1211pm5.32i 619 . . . 4  |-  ( ( x  e.  { B }  /\  <. x ,  y
>.  e.  A )  <->  ( x  e.  { B }  /\  y  e.  ( A " { B } ) ) )
133, 12bitri 241 . . 3  |-  ( (
<. x ,  y >.  e.  A  /\  x  e.  { B } )  <-> 
( x  e.  { B }  /\  y  e.  ( A " { B } ) ) )
145opelres 5151 . . 3  |-  ( <.
x ,  y >.  e.  ( A  |`  { B } )  <->  ( <. x ,  y >.  e.  A  /\  x  e.  { B } ) )
15 opelxp 4908 . . 3  |-  ( <.
x ,  y >.  e.  ( { B }  X.  ( A " { B } ) )  <->  ( x  e.  { B }  /\  y  e.  ( A " { B } ) ) )
1613, 14, 153bitr4i 269 . 2  |-  ( <.
x ,  y >.  e.  ( A  |`  { B } )  <->  <. x ,  y >.  e.  ( { B }  X.  ( A " { B }
) ) )
171, 2, 16eqrelriiv 4970 1  |-  ( A  |`  { B } )  =  ( { B }  X.  ( A " { B } ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 359    = wceq 1652    e. wcel 1725   {csn 3814   <.cop 3817    X. cxp 4876    |` cres 4880   "cima 4881
This theorem is referenced by:  gsum2d  15546  dprd2da  15600  ustneism  18253
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-br 4213  df-opab 4267  df-xp 4884  df-rel 4885  df-cnv 4886  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891
  Copyright terms: Public domain W3C validator