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Theorem dfres2 5185
 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 5166 . 2
2 relopab 4993 . 2
3 vex 2951 . . . . 5
43brres 5144 . . . 4
5 df-br 4205 . . . 4
6 ancom 438 . . . 4
74, 5, 63bitr3i 267 . . 3
8 vex 2951 . . . 4
9 eleq1 2495 . . . . 5
10 breq1 4207 . . . . 5
119, 10anbi12d 692 . . . 4
12 breq2 4208 . . . . 5
1312anbi2d 685 . . . 4
148, 3, 11, 13opelopab 4468 . . 3
157, 14bitr4i 244 . 2
161, 2, 15eqrelriiv 4962 1
 Colors of variables: wff set class Syntax hints:   wa 359   wceq 1652   wcel 1725  cop 3809   class class class wbr 4204  copab 4257   cres 4872 This theorem is referenced by:  shftidt2  11888 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 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395 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-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-br 4205  df-opab 4259  df-xp 4876  df-rel 4877  df-res 4882
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