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Theorem oppgval 14913
 Description: Value of the opposite group. (Contributed by Stefan O'Rear, 25-Aug-2015.) (Revised by Mario Carneiro, 16-Sep-2015.) (Revised by Fan Zheng, 26-Jun-2016.)
Hypotheses
Ref Expression
oppgval.2
oppgval.3 oppg
Assertion
Ref Expression
oppgval sSet tpos

Proof of Theorem oppgval
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 oppgval.3 . 2 oppg
2 id 19 . . . . 5
3 fveq2 5605 . . . . . . . 8
4 oppgval.2 . . . . . . . 8
53, 4syl6eqr 2408 . . . . . . 7
6 tposeq 6320 . . . . . . 7 tpos tpos
75, 6syl 15 . . . . . 6 tpos tpos
87opeq2d 3882 . . . . 5 tpos tpos
92, 8oveq12d 5960 . . . 4 sSet tpos sSet tpos
10 df-oppg 14912 . . . 4 oppg sSet tpos
11 ovex 5967 . . . 4 sSet tpos
129, 10, 11fvmpt 5682 . . 3 oppg sSet tpos
13 fvprc 5599 . . . 4 oppg
14 reldmsets 13261 . . . . 5 sSet
1514ovprc1 5970 . . . 4 sSet tpos
1613, 15eqtr4d 2393 . . 3 oppg sSet tpos
1712, 16pm2.61i 156 . 2 oppg sSet tpos
181, 17eqtri 2378 1 sSet tpos
 Colors of variables: wff set class Syntax hints:   wn 3   wceq 1642   wcel 1710  cvv 2864  c0 3531  cop 3719  cfv 5334  (class class class)co 5942  tpos ctpos 6317  cnx 13236   sSet csts 13237   cplusg 13299  oppgcoppg 14911 This theorem is referenced by:  oppgplusfval  14914  oppglem  14916 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3907  df-br 4103  df-opab 4157  df-mpt 4158  df-id 4388  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-res 4780  df-iota 5298  df-fun 5336  df-fv 5342  df-ov 5945  df-oprab 5946  df-mpt2 5947  df-tpos 6318  df-sets 13245  df-oppg 14912
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