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

Theorem cores 5493
Description: Restricted first member of a class composition. (Contributed by NM, 12-Oct-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
Assertion
Ref Expression
cores  |-  ( ran 
B  C_  C  ->  ( ( A  |`  C )  o.  B )  =  ( A  o.  B
) )

Proof of Theorem cores
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 3109 . . . . . . 7  |-  z  e. 
_V
2 vex 3109 . . . . . . 7  |-  y  e. 
_V
31, 2brelrn 5222 . . . . . 6  |-  ( z B y  ->  y  e.  ran  B )
4 ssel 3483 . . . . . 6  |-  ( ran 
B  C_  C  ->  ( y  e.  ran  B  ->  y  e.  C ) )
5 vex 3109 . . . . . . . 8  |-  x  e. 
_V
65brres 5268 . . . . . . 7  |-  ( y ( A  |`  C ) x  <->  ( y A x  /\  y  e.  C ) )
76rbaib 904 . . . . . 6  |-  ( y  e.  C  ->  (
y ( A  |`  C ) x  <->  y A x ) )
83, 4, 7syl56 34 . . . . 5  |-  ( ran 
B  C_  C  ->  ( z B y  -> 
( y ( A  |`  C ) x  <->  y A x ) ) )
98pm5.32d 637 . . . 4  |-  ( ran 
B  C_  C  ->  ( ( z B y  /\  y ( A  |`  C ) x )  <-> 
( z B y  /\  y A x ) ) )
109exbidv 1719 . . 3  |-  ( ran 
B  C_  C  ->  ( E. y ( z B y  /\  y
( A  |`  C ) x )  <->  E. y
( z B y  /\  y A x ) ) )
1110opabbidv 4502 . 2  |-  ( ran 
B  C_  C  ->  {
<. z ,  x >.  |  E. y ( z B y  /\  y
( A  |`  C ) x ) }  =  { <. z ,  x >.  |  E. y ( z B y  /\  y A x ) } )
12 df-co 4997 . 2  |-  ( ( A  |`  C )  o.  B )  =  { <. z ,  x >.  |  E. y ( z B y  /\  y
( A  |`  C ) x ) }
13 df-co 4997 . 2  |-  ( A  o.  B )  =  { <. z ,  x >.  |  E. y ( z B y  /\  y A x ) }
1411, 12, 133eqtr4g 2520 1  |-  ( ran 
B  C_  C  ->  ( ( A  |`  C )  o.  B )  =  ( A  o.  B
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1398   E.wex 1617    e. wcel 1823    C_ wss 3461   class class class wbr 4439   {copab 4496   ran crn 4989    |` cres 4990    o. ccom 4992
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-br 4440  df-opab 4498  df-xp 4994  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000
This theorem is referenced by:  cocnvcnv1  5501  cores2  5503  relcoi2  5518  fco2  5724  fcoi2  5742  domss2  7669  canthp1lem2  9020  imasdsval2  15005  frmdss2  16230  gsumval3OLD  17107  gsumval3lem1  17108  gsumzres  17113  gsumzresOLD  17117  gsumzaddlem  17133  gsumzaddlemOLD  17135  dprdf1  17275  kgencn2  20224  tsmsf1o  20813  hhssims  26389  eulerpartgbij  28575  lgamcvg2  28861  cvmlift2lem9a  29012  fourierdlem53  32181  funresfunco  32449
  Copyright terms: Public domain W3C validator