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Theorem resoprab 6166
 Description: Restriction of an operation class abstraction. (Contributed by NM, 10-Feb-2007.)
Assertion
Ref Expression
resoprab
Distinct variable groups:   ,,,   ,,,
Allowed substitution hints:   (,,)

Proof of Theorem resoprab
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 resopab 5187 . . 3
2 19.42vv 1930 . . . . 5
3 an12 773 . . . . . . 7
4 eleq1 2496 . . . . . . . . . 10
5 opelxp 4908 . . . . . . . . . 10
64, 5syl6bb 253 . . . . . . . . 9
76anbi1d 686 . . . . . . . 8
87pm5.32i 619 . . . . . . 7
93, 8bitri 241 . . . . . 6
1092exbii 1593 . . . . 5
112, 10bitr3i 243 . . . 4
1211opabbii 4272 . . 3
131, 12eqtri 2456 . 2
14 dfoprab2 6121 . . 3
1514reseq1i 5142 . 2
16 dfoprab2 6121 . 2
1713, 15, 163eqtr4i 2466 1
 Colors of variables: wff set class Syntax hints:   wa 359  wex 1550   wceq 1652   wcel 1725  cop 3817  copab 4265   cxp 4876   cres 4880  coprab 6082 This theorem is referenced by:  resoprab2  6167  df1stres  24091  df2ndres  24092 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-opab 4267  df-xp 4884  df-rel 4885  df-res 4890  df-oprab 6085
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