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Theorem cores 5500
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 3098 . . . . . . 7  |-  z  e. 
_V
2 vex 3098 . . . . . . 7  |-  y  e. 
_V
31, 2brelrn 5223 . . . . . 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 3098 . . . . . . . 8  |-  x  e. 
_V
65brres 5270 . . . . . . 7  |-  ( y ( A  |`  C ) x  <->  ( y A x  /\  y  e.  C ) )
76rbaib 906 . . . . . 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 639 . . . 4  |-  ( ran 
B  C_  C  ->  ( ( z B y  /\  y ( A  |`  C ) x )  <-> 
( z B y  /\  y A x ) ) )
109exbidv 1701 . . 3  |-  ( ran 
B  C_  C  ->  ( E. y ( z B y  /\  y
( A  |`  C ) x )  <->  E. y
( z B y  /\  y A x ) ) )
1110opabbidv 4500 . 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 4998 . 2  |-  ( ( A  |`  C )  o.  B )  =  { <. z ,  x >.  |  E. y ( z B y  /\  y
( A  |`  C ) x ) }
13 df-co 4998 . 2  |-  ( A  o.  B )  =  { <. z ,  x >.  |  E. y ( z B y  /\  y A x ) }
1411, 12, 133eqtr4g 2509 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 369    = wceq 1383   E.wex 1599    e. wcel 1804    C_ wss 3461   class class class wbr 4437   {copab 4494   ran crn 4990    |` cres 4991    o. ccom 4993
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-br 4438  df-opab 4496  df-xp 4995  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001
This theorem is referenced by:  cocnvcnv1  5508  cores2  5510  relcoi2  5525  fco2  5732  fcoi2  5750  domss2  7678  canthp1lem2  9034  imasdsval2  14790  frmdss2  15905  gsumval3OLD  16782  gsumval3lem1  16783  gsumzres  16788  gsumzresOLD  16792  gsumzaddlem  16808  gsumzaddlemOLD  16810  dprdf1  16954  kgencn2  19931  tsmsf1o  20520  hhssims  26063  eulerpartgbij  28184  lgamcvg2  28470  cvmlift2lem9a  28621  fourierdlem53  31831  funresfunco  32048
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