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Theorem cores 5192
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 2804 . . . . . . 7  |-  z  e. 
_V
2 vex 2804 . . . . . . 7  |-  y  e. 
_V
31, 2brelrn 4925 . . . . . 6  |-  ( z B y  ->  y  e.  ran  B )
4 ssel 3187 . . . . . 6  |-  ( ran 
B  C_  C  ->  ( y  e.  ran  B  ->  y  e.  C ) )
5 vex 2804 . . . . . . . 8  |-  x  e. 
_V
65brres 4977 . . . . . . 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 1616 . . 3  |-  ( ran 
B  C_  C  ->  ( E. y ( z B y  /\  y
( A  |`  C ) x )  <->  E. y
( z B y  /\  y A x ) ) )
1110opabbidv 4098 . 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 4714 . 2  |-  ( ( A  |`  C )  o.  B )  =  { <. z ,  x >.  |  E. y ( z B y  /\  y
( A  |`  C ) x ) }
13 df-co 4714 . 2  |-  ( A  o.  B )  =  { <. z ,  x >.  |  E. y ( z B y  /\  y A x ) }
1411, 12, 133eqtr4g 2353 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 1531    = wceq 1632    e. wcel 1696    C_ wss 3165   class class class wbr 4039   {copab 4092   ran crn 4706    |` cres 4707    o. ccom 4709
This theorem is referenced by:  cocnvcnv1  5199  cores2  5201  relcoi2  5216  fco2  5415  fcoi2  5432  domss2  7036  canthp1lem2  8291  imasdsval2  13435  frmdss2  14501  gsumval3  15207  gsumzres  15210  gsumzaddlem  15219  dprdf1  15284  kgencn2  17268  tsmsf1o  17843  hhssims  21868  cvmlift2lem9a  23849  dfps2  25392  funresfunco  28093
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-xp 4711  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717
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