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Theorem cores 4770
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 2557 . . . . . . 7 
_V
2 vex 2557 . . . . . . 7 
_V
31, 2brelrn 4513 . . . . . 6  ran
4 ssel 2936 . . . . . 6  ran  C_  C  ran  C
5 vex 2557 . . . . . . . 8 
_V
65brres 4564 . . . . . . 7  |`  C  C
76rbaib 830 . . . . . 6  C  |`  C
83, 4, 7syl56 30 . . . . 5  ran  C_  C  |`  C
98pm5.32d 423 . . . 4  ran  C_  C  |`  C
109exbidv 1706 . . 3  ran  C_  C  |`  C
1110opabbidv 3817 . 2  ran  C_  C  { <. ,  >.  |  |`  C }  { <. ,  >.  |  }
12 df-co 4300 . 2  |`  C  o.  { <. ,  >.  |  |`  C }
13 df-co 4300 . 2  o.  { <. ,  >.  |  }
1411, 12, 133eqtr4g 2097 1  ran  C_  C  |`  C  o.  o.
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wb 98   wceq 1243  wex 1381   wcel 1393    C_ wss 2914   class class class wbr 3758   {copab 3811   ran crn 4292    |` cres 4293    o. ccom 4295
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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3869  ax-pow 3921  ax-pr 3938
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2308  df-rex 2309  df-v 2556  df-un 2919  df-in 2921  df-ss 2928  df-pw 3356  df-sn 3376  df-pr 3377  df-op 3379  df-br 3759  df-opab 3813  df-xp 4297  df-cnv 4299  df-co 4300  df-dm 4301  df-rn 4302  df-res 4303
This theorem is referenced by:  cocnvcnv1  4777  cores2  4779  relcoi2  4794  fco2  5003  fcoi2  5017
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