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Theorem cores 5176
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 2791 . . . . . . 7  |-  z  e. 
_V
2 vex 2791 . . . . . . 7  |-  y  e. 
_V
31, 2brelrn 4909 . . . . . 6  |-  ( z B y  ->  y  e.  ran  B )
4 ssel 3174 . . . . . 6  |-  ( ran 
B  C_  C  ->  ( y  e.  ran  B  ->  y  e.  C ) )
5 vex 2791 . . . . . . . 8  |-  x  e. 
_V
65brres 4961 . . . . . . 7  |-  ( y ( A  |`  C ) x  <->  ( y A x  /\  y  e.  C ) )
76rbaib 873 . . . . . 6  |-  ( y  e.  C  ->  (
y ( A  |`  C ) x  <->  y A x ) )
83, 4, 7syl56 30 . . . . 5  |-  ( ran 
B  C_  C  ->  ( z B y  -> 
( y ( A  |`  C ) x  <->  y A x ) ) )
98pm5.32d 620 . . . 4  |-  ( ran 
B  C_  C  ->  ( ( z B y  /\  y ( A  |`  C ) x )  <-> 
( z B y  /\  y A x ) ) )
109exbidv 1612 . . 3  |-  ( ran 
B  C_  C  ->  ( E. y ( z B y  /\  y
( A  |`  C ) x )  <->  E. y
( z B y  /\  y A x ) ) )
1110opabbidv 4082 . 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 4698 . 2  |-  ( ( A  |`  C )  o.  B )  =  { <. z ,  x >.  |  E. y ( z B y  /\  y
( A  |`  C ) x ) }
13 df-co 4698 . 2  |-  ( A  o.  B )  =  { <. z ,  x >.  |  E. y ( z B y  /\  y A x ) }
1411, 12, 133eqtr4g 2340 1  |-  ( ran 
B  C_  C  ->  ( ( A  |`  C )  o.  B )  =  ( A  o.  B
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1684    C_ wss 3152   class class class wbr 4023   {copab 4076   ran crn 4690    |` cres 4691    o. ccom 4693
This theorem is referenced by:  cocnvcnv1  5183  cores2  5185  relcoi2  5200  fco2  5399  fcoi2  5416  domss2  7020  canthp1lem2  8275  imasdsval2  13419  frmdss2  14485  gsumval3  15191  gsumzres  15194  gsumzaddlem  15203  dprdf1  15268  kgencn2  17252  tsmsf1o  17827  hhssims  21852  cvmlift2lem9a  23834  dfps2  25289  funresfunco  27988
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-xp 4695  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701
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