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Theorem cores 5037
 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

Proof of Theorem cores
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 2684 . . . . . . 7
2 vex 2684 . . . . . . 7
31, 2brelrn 4767 . . . . . 6
4 ssel 3086 . . . . . 6
5 vex 2684 . . . . . . . 8
65brres 4820 . . . . . . 7
76rbaib 906 . . . . . 6
83, 4, 7syl56 34 . . . . 5
98pm5.32d 445 . . . 4
109exbidv 1797 . . 3
1110opabbidv 3989 . 2
12 df-co 4543 . 2
13 df-co 4543 . 2
1411, 12, 133eqtr4g 2195 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 103   wb 104   wceq 1331  wex 1468   wcel 1480   wss 3066   class class class wbr 3924  copab 3983   crn 4535   cres 4536   ccom 4538 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126 This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-br 3925  df-opab 3985  df-xp 4540  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546 This theorem is referenced by:  cocnvcnv1  5044  cores2  5046  cocnvres  5058  relcoi2  5064  fco2  5284  fcoi2  5299
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