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Theorem opprval 15406
 Description: Value of the opposite ring. (Contributed by Mario Carneiro, 1-Dec-2014.)
Hypotheses
Ref Expression
opprval.1
opprval.2
opprval.3 oppr
Assertion
Ref Expression
opprval sSet tpos

Proof of Theorem opprval
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 opprval.3 . 2 oppr
2 id 19 . . . . 5
3 fveq2 5525 . . . . . . . 8
4 opprval.2 . . . . . . . 8
53, 4syl6eqr 2333 . . . . . . 7
6 tposeq 6236 . . . . . . 7 tpos tpos
75, 6syl 15 . . . . . 6 tpos tpos
87opeq2d 3803 . . . . 5 tpos tpos
92, 8oveq12d 5876 . . . 4 sSet tpos sSet tpos
10 df-oppr 15405 . . . 4 oppr sSet tpos
11 ovex 5883 . . . 4 sSet tpos
129, 10, 11fvmpt 5602 . . 3 oppr sSet tpos
13 fvprc 5519 . . . 4 oppr
14 reldmsets 13170 . . . . 5 sSet
1514ovprc1 5886 . . . 4 sSet tpos
1613, 15eqtr4d 2318 . . 3 oppr sSet tpos
1712, 16pm2.61i 156 . 2 oppr sSet tpos
181, 17eqtri 2303 1 sSet tpos
 Colors of variables: wff set class Syntax hints:   wn 3   wceq 1623   wcel 1684  cvv 2788  c0 3455  cop 3643  cfv 5255  (class class class)co 5858  tpos ctpos 6233  cnx 13145   sSet csts 13146  cbs 13148  cmulr 13209  opprcoppr 15404 This theorem is referenced by:  opprmulfval  15407  opprlem  15410 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-res 4701  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-tpos 6234  df-sets 13154  df-oppr 15405
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