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Theorem resoprab 5597
 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 4652 . . 3
2 19.42vv 1788 . . . . 5
3 an12 495 . . . . . . 7
4 eleq1 2100 . . . . . . . . . 10
5 opelxp 4374 . . . . . . . . . 10
64, 5syl6bb 185 . . . . . . . . 9
76anbi1d 438 . . . . . . . 8
87pm5.32i 427 . . . . . . 7
93, 8bitri 173 . . . . . 6
1092exbii 1497 . . . . 5
112, 10bitr3i 175 . . . 4
1211opabbii 3824 . . 3
131, 12eqtri 2060 . 2
14 dfoprab2 5552 . . 3
1514reseq1i 4608 . 2
16 dfoprab2 5552 . 2
1713, 15, 163eqtr4i 2070 1
 Colors of variables: wff set class Syntax hints:   wa 97   wceq 1243  wex 1381   wcel 1393  cop 3378  copab 3817   cxp 4343   cres 4347  coprab 5513 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 3875  ax-pow 3927  ax-pr 3944 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-opab 3819  df-xp 4351  df-rel 4352  df-res 4357  df-oprab 5516 This theorem is referenced by:  resoprab2  5598
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