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Theorem opprval 17930
 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 22 . . . . 5
3 fveq2 5879 . . . . . . . 8
4 opprval.2 . . . . . . . 8
53, 4syl6eqr 2523 . . . . . . 7
65tposeqd 6994 . . . . . 6 tpos tpos
76opeq2d 4165 . . . . 5 tpos tpos
82, 7oveq12d 6326 . . . 4 sSet tpos sSet tpos
9 df-oppr 17929 . . . 4 oppr sSet tpos
10 ovex 6336 . . . 4 sSet tpos
118, 9, 10fvmpt 5963 . . 3 oppr sSet tpos
12 fvprc 5873 . . . 4 oppr
13 reldmsets 15222 . . . . 5 sSet
1413ovprc1 6339 . . . 4 sSet tpos
1512, 14eqtr4d 2508 . . 3 oppr sSet tpos
1611, 15pm2.61i 169 . 2 oppr sSet tpos
171, 16eqtri 2493 1 sSet tpos
 Colors of variables: wff setvar class Syntax hints:   wn 3   wceq 1452   wcel 1904  cvv 3031  c0 3722  cop 3965  cfv 5589  (class class class)co 6308  tpos ctpos 6990  cnx 15196   sSet csts 15197  cbs 15199  cmulr 15269  opprcoppr 17928 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639 This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-res 4851  df-iota 5553  df-fun 5591  df-fv 5597  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-tpos 6991  df-sets 15205  df-oppr 17929 This theorem is referenced by:  opprmulfval  17931  opprlem  17934
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