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Theorem cores 4767
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  C_  C  |`  C  o.  o.

Proof of Theorem cores
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 2554 . . . . . . 7 
_V
2 vex 2554 . . . . . . 7 
_V
31, 2brelrn 4510 . . . . . 6  ran
4 ssel 2933 . . . . . 6  ran  C_  C  ran  C
5 vex 2554 . . . . . . . 8 
_V
65brres 4561 . . . . . . 7  |`  C  C
76rbaib 829 . . . . . 6  C  |`  C
83, 4, 7syl56 30 . . . . 5  ran  C_  C  |`  C
98pm5.32d 423 . . . 4  ran  C_  C  |`  C
109exbidv 1703 . . 3  ran  C_  C  |`  C
1110opabbidv 3814 . 2  ran  C_  C  { <. ,  >.  |  |`  C }  { <. ,  >.  |  }
12 df-co 4297 . 2  |`  C  o.  { <. ,  >.  |  |`  C }
13 df-co 4297 . 2  o.  { <. ,  >.  |  }
1411, 12, 133eqtr4g 2094 1  ran  C_  C  |`  C  o.  o.
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wb 98   wceq 1242  wex 1378   wcel 1390    C_ wss 2911   class class class wbr 3755   {copab 3808   ran crn 4289    |` cres 4290    o. ccom 4292
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-xp 4294  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300
This theorem is referenced by:  cocnvcnv1  4774  cores2  4776  relcoi2  4791  fco2  5000  fcoi2  5014
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