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Theorem dfres2 5331
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 5306 . 2
2 relopab 5134 . 2
3 vex 3112 . . . . 5
43brres 5285 . . . 4
5 df-br 4453 . . . 4
6 ancom 450 . . . 4
74, 5, 63bitr3i 275 . . 3
8 vex 3112 . . . 4
9 eleq1 2529 . . . . 5
10 breq1 4455 . . . . 5
119, 10anbi12d 710 . . . 4
12 breq2 4456 . . . . 5
1312anbi2d 703 . . . 4
148, 3, 11, 13opelopab 4774 . . 3
157, 14bitr4i 252 . 2
161, 2, 15eqrelriiv 5102 1
Colors of variables: wff setvar class
Syntax hints:  /\wa 369  =wceq 1395  e.wcel 1818  <.cop 4035   class class class wbr 4452  {copab 4509  |`cres 5006
This theorem is referenced by:  shftidt2  12914
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pr 4691
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036  df-br 4453  df-opab 4511  df-xp 5010  df-rel 5011  df-res 5016
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