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Theorem dfres2 4658
 Description: Alternate definition of the restriction operation. (Contributed by Mario Carneiro, 5-Nov-2013.)
Assertion
Ref Expression
dfres2
Distinct variable groups:   ,,   ,,

Proof of Theorem dfres2
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relres 4639 . 2
2 relopab 4464 . 2
3 vex 2560 . . . . 5
43brres 4618 . . . 4
5 df-br 3765 . . . 4
6 ancom 253 . . . 4
74, 5, 63bitr3i 199 . . 3
8 vex 2560 . . . 4
9 eleq1 2100 . . . . 5
10 breq1 3767 . . . . 5
119, 10anbi12d 442 . . . 4
12 breq2 3768 . . . . 5
1312anbi2d 437 . . . 4
148, 3, 11, 13opelopab 4008 . . 3
157, 14bitr4i 176 . 2
161, 2, 15eqrelriiv 4434 1
 Colors of variables: wff set class Syntax hints:   wa 97   wceq 1243   wcel 1393  cop 3378   class class class wbr 3764  copab 3817   cres 4347 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-eu 1903  df-mo 1904  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-br 3765  df-opab 3819  df-xp 4351  df-rel 4352  df-res 4357 This theorem is referenced by:  shftidt2  9433
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