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Theorem dfres3 25382
 Description: Alternate definition of restriction. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Assertion
Ref Expression
dfres3

Proof of Theorem dfres3
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-res 4890 . 2
2 eleq1 2496 . . . . . . . . . 10
3 vex 2959 . . . . . . . . . . . 12
43biantru 492 . . . . . . . . . . 11
5 vex 2959 . . . . . . . . . . . . 13
65, 3opelrn 5101 . . . . . . . . . . . 12
76biantrud 494 . . . . . . . . . . 11
84, 7syl5bbr 251 . . . . . . . . . 10
92, 8syl6bi 220 . . . . . . . . 9
109com12 29 . . . . . . . 8
1110pm5.32d 621 . . . . . . 7
12112exbidv 1638 . . . . . 6
13 elxp 4895 . . . . . 6
14 elxp 4895 . . . . . 6
1512, 13, 143bitr4g 280 . . . . 5
1615pm5.32i 619 . . . 4
17 elin 3530 . . . 4
1816, 17bitr4i 244 . . 3
1918ineqri 3534 . 2
201, 19eqtri 2456 1
 Colors of variables: wff set class Syntax hints:   wb 177   wa 359  wex 1550   wceq 1652   wcel 1725  cvv 2956   cin 3319  cop 3817   cxp 4876   crn 4879   cres 4880 This theorem is referenced by:  brrestrict  25794 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-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  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-br 4213  df-opab 4267  df-xp 4884  df-cnv 4886  df-dm 4888  df-rn 4889  df-res 4890
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