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Theorem oppcval 13715
 Description: Value of the opposite category. (Contributed by Mario Carneiro, 2-Jan-2017.)
Hypotheses
Ref Expression
oppcval.b
oppcval.h
oppcval.x comp
oppcval.o oppCat
Assertion
Ref Expression
oppcval sSet tpos sSet comp tpos
Distinct variable group:   ,,
Allowed substitution hints:   (,)   (,)   (,)   (,)   (,)

Proof of Theorem oppcval
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 oppcval.o . 2 oppCat
2 elex 2872 . . 3
3 id 19 . . . . . 6
4 fveq2 5608 . . . . . . . . 9
5 oppcval.h . . . . . . . . 9
64, 5syl6eqr 2408 . . . . . . . 8
7 tposeq 6323 . . . . . . . 8 tpos tpos
86, 7syl 15 . . . . . . 7 tpos tpos
98opeq2d 3884 . . . . . 6 tpos tpos
103, 9oveq12d 5963 . . . . 5 sSet tpos sSet tpos
11 fveq2 5608 . . . . . . . . 9
12 oppcval.b . . . . . . . . 9
1311, 12syl6eqr 2408 . . . . . . . 8
1413, 13xpeq12d 4796 . . . . . . 7
15 fveq2 5608 . . . . . . . . . 10 comp comp
16 oppcval.x . . . . . . . . . 10 comp
1715, 16syl6eqr 2408 . . . . . . . . 9 comp
1817oveqd 5962 . . . . . . . 8 comp
19 tposeq 6323 . . . . . . . 8 comp tpos comp tpos
2018, 19syl 15 . . . . . . 7 tpos comp tpos
2114, 13, 20mpt2eq123dv 5997 . . . . . 6 tpos comp tpos
2221opeq2d 3884 . . . . 5 comp tpos comp comp tpos
2310, 22oveq12d 5963 . . . 4 sSet tpos sSet comp tpos comp sSet tpos sSet comp tpos
24 df-oppc 13714 . . . 4 oppCat sSet tpos sSet comp tpos comp
25 ovex 5970 . . . 4 sSet tpos sSet comp tpos
2623, 24, 25fvmpt 5685 . . 3 oppCat sSet tpos sSet comp tpos
272, 26syl 15 . 2 oppCat sSet tpos sSet comp tpos
281, 27syl5eq 2402 1 sSet tpos sSet comp tpos
 Colors of variables: wff set class Syntax hints:   wi 4   wceq 1642   wcel 1710  cvv 2864  cop 3719   cxp 4769  cfv 5337  (class class class)co 5945   cmpt2 5947  c1st 6207  c2nd 6208  tpos ctpos 6320  cnx 13242   sSet csts 13243  cbs 13245   chom 13316  compcco 13317  oppCatcoppc 13713 This theorem is referenced by:  oppchomfval  13716  oppccofval  13718  oppcbas  13720  catcoppccl  14039 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-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pr 4295 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 3909  df-br 4105  df-opab 4159  df-mpt 4160  df-id 4391  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-res 4783  df-iota 5301  df-fun 5339  df-fv 5345  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-tpos 6321  df-oppc 13714
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