MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  oppcval Unicode version

Theorem oppcval 13715
Description: Value of the opposite category. (Contributed by Mario Carneiro, 2-Jan-2017.)
Hypotheses
Ref Expression
oppcval.b  |-  B  =  ( Base `  C
)
oppcval.h  |-  H  =  (  Hom  `  C
)
oppcval.x  |-  .x.  =  (comp `  C )
oppcval.o  |-  O  =  (oppCat `  C )
Assertion
Ref Expression
oppcval  |-  ( C  e.  V  ->  O  =  ( ( C sSet  <. (  Hom  `  ndx ) , tpos  H >. ) sSet  <.
(comp `  ndx ) ,  ( u  e.  ( B  X.  B ) ,  z  e.  B  |-> tpos  ( <. z ,  ( 2nd `  u )
>.  .x.  ( 1st `  u
) ) ) >.
) )
Distinct variable group:    z, u, C
Allowed substitution hints:    B( z, u)    .x. ( z, u)    H( z, u)    O( z, u)    V( z, u)

Proof of Theorem oppcval
Dummy variable  c is distinct from all other variables.
StepHypRef Expression
1 oppcval.o . 2  |-  O  =  (oppCat `  C )
2 elex 2872 . . 3  |-  ( C  e.  V  ->  C  e.  _V )
3 id 19 . . . . . 6  |-  ( c  =  C  ->  c  =  C )
4 fveq2 5608 . . . . . . . . 9  |-  ( c  =  C  ->  (  Hom  `  c )  =  (  Hom  `  C
) )
5 oppcval.h . . . . . . . . 9  |-  H  =  (  Hom  `  C
)
64, 5syl6eqr 2408 . . . . . . . 8  |-  ( c  =  C  ->  (  Hom  `  c )  =  H )
7 tposeq 6323 . . . . . . . 8  |-  ( (  Hom  `  c )  =  H  -> tpos  (  Hom  `  c )  = tpos  H
)
86, 7syl 15 . . . . . . 7  |-  ( c  =  C  -> tpos  (  Hom  `  c )  = tpos  H
)
98opeq2d 3884 . . . . . 6  |-  ( c  =  C  ->  <. (  Hom  `  ndx ) , tpos  (  Hom  `  c
) >.  =  <. (  Hom  `  ndx ) , tpos 
H >. )
103, 9oveq12d 5963 . . . . 5  |-  ( c  =  C  ->  (
c sSet  <. (  Hom  `  ndx ) , tpos  (  Hom  `  c
) >. )  =  ( C sSet  <. (  Hom  `  ndx ) , tpos  H >. )
)
11 fveq2 5608 . . . . . . . . 9  |-  ( c  =  C  ->  ( Base `  c )  =  ( Base `  C
) )
12 oppcval.b . . . . . . . . 9  |-  B  =  ( Base `  C
)
1311, 12syl6eqr 2408 . . . . . . . 8  |-  ( c  =  C  ->  ( Base `  c )  =  B )
1413, 13xpeq12d 4796 . . . . . . 7  |-  ( c  =  C  ->  (
( Base `  c )  X.  ( Base `  c
) )  =  ( B  X.  B ) )
15 fveq2 5608 . . . . . . . . . 10  |-  ( c  =  C  ->  (comp `  c )  =  (comp `  C ) )
16 oppcval.x . . . . . . . . . 10  |-  .x.  =  (comp `  C )
1715, 16syl6eqr 2408 . . . . . . . . 9  |-  ( c  =  C  ->  (comp `  c )  =  .x.  )
1817oveqd 5962 . . . . . . . 8  |-  ( c  =  C  ->  ( <. z ,  ( 2nd `  u ) >. (comp `  c ) ( 1st `  u ) )  =  ( <. z ,  ( 2nd `  u )
>.  .x.  ( 1st `  u
) ) )
19 tposeq 6323 . . . . . . . 8  |-  ( (
<. z ,  ( 2nd `  u ) >. (comp `  c ) ( 1st `  u ) )  =  ( <. z ,  ( 2nd `  u )
>.  .x.  ( 1st `  u
) )  -> tpos  ( <.
z ,  ( 2nd `  u ) >. (comp `  c ) ( 1st `  u ) )  = tpos  ( <. z ,  ( 2nd `  u )
>.  .x.  ( 1st `  u
) ) )
2018, 19syl 15 . . . . . . 7  |-  ( c  =  C  -> tpos  ( <.
z ,  ( 2nd `  u ) >. (comp `  c ) ( 1st `  u ) )  = tpos  ( <. z ,  ( 2nd `  u )
>.  .x.  ( 1st `  u
) ) )
2114, 13, 20mpt2eq123dv 5997 . . . . . 6  |-  ( c  =  C  ->  (
u  e.  ( (
Base `  c )  X.  ( Base `  c
) ) ,  z  e.  ( Base `  c
)  |-> tpos  ( <. z ,  ( 2nd `  u )
>. (comp `  c )
( 1st `  u
) ) )  =  ( u  e.  ( B  X.  B ) ,  z  e.  B  |-> tpos  ( <. z ,  ( 2nd `  u )
>.  .x.  ( 1st `  u
) ) ) )
2221opeq2d 3884 . . . . 5  |-  ( c  =  C  ->  <. (comp ` 
ndx ) ,  ( u  e.  ( (
Base `  c )  X.  ( Base `  c
) ) ,  z  e.  ( Base `  c
)  |-> tpos  ( <. z ,  ( 2nd `  u )
>. (comp `  c )
( 1st `  u
) ) ) >.  =  <. (comp `  ndx ) ,  ( u  e.  ( B  X.  B
) ,  z  e.  B  |-> tpos  ( <. z ,  ( 2nd `  u )
>.  .x.  ( 1st `  u
) ) ) >.
)
2310, 22oveq12d 5963 . . . 4  |-  ( c  =  C  ->  (
( c sSet  <. (  Hom  `  ndx ) , tpos  (  Hom  `  c
) >. ) sSet  <. (comp ` 
ndx ) ,  ( u  e.  ( (
Base `  c )  X.  ( Base `  c
) ) ,  z  e.  ( Base `  c
)  |-> tpos  ( <. z ,  ( 2nd `  u )
>. (comp `  c )
( 1st `  u
) ) ) >.
)  =  ( ( C sSet  <. (  Hom  `  ndx ) , tpos  H >. ) sSet  <.
(comp `  ndx ) ,  ( u  e.  ( B  X.  B ) ,  z  e.  B  |-> tpos  ( <. z ,  ( 2nd `  u )
>.  .x.  ( 1st `  u
) ) ) >.
) )
24 df-oppc 13714 . . . 4  |- oppCat  =  ( c  e.  _V  |->  ( ( c sSet  <. (  Hom  `  ndx ) , tpos  (  Hom  `  c
) >. ) sSet  <. (comp ` 
ndx ) ,  ( u  e.  ( (
Base `  c )  X.  ( Base `  c
) ) ,  z  e.  ( Base `  c
)  |-> tpos  ( <. z ,  ( 2nd `  u )
>. (comp `  c )
( 1st `  u
) ) ) >.
) )
25 ovex 5970 . . . 4  |-  ( ( C sSet  <. (  Hom  `  ndx ) , tpos  H >. ) sSet  <.
(comp `  ndx ) ,  ( u  e.  ( B  X.  B ) ,  z  e.  B  |-> tpos  ( <. z ,  ( 2nd `  u )
>.  .x.  ( 1st `  u
) ) ) >.
)  e.  _V
2623, 24, 25fvmpt 5685 . . 3  |-  ( C  e.  _V  ->  (oppCat `  C )  =  ( ( C sSet  <. (  Hom  `  ndx ) , tpos 
H >. ) sSet  <. (comp ` 
ndx ) ,  ( u  e.  ( B  X.  B ) ,  z  e.  B  |-> tpos  (
<. z ,  ( 2nd `  u ) >.  .x.  ( 1st `  u ) ) ) >. ) )
272, 26syl 15 . 2  |-  ( C  e.  V  ->  (oppCat `  C )  =  ( ( C sSet  <. (  Hom  `  ndx ) , tpos 
H >. ) sSet  <. (comp ` 
ndx ) ,  ( u  e.  ( B  X.  B ) ,  z  e.  B  |-> tpos  (
<. z ,  ( 2nd `  u ) >.  .x.  ( 1st `  u ) ) ) >. ) )
281, 27syl5eq 2402 1  |-  ( C  e.  V  ->  O  =  ( ( C sSet  <. (  Hom  `  ndx ) , tpos  H >. ) sSet  <.
(comp `  ndx ) ,  ( u  e.  ( B  X.  B ) ,  z  e.  B  |-> tpos  ( <. z ,  ( 2nd `  u )
>.  .x.  ( 1st `  u
) ) ) >.
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1642    e. wcel 1710   _Vcvv 2864   <.cop 3719    X. cxp 4769   ` cfv 5337  (class class class)co 5945    e. cmpt2 5947   1stc1st 6207   2ndc2nd 6208  tpos ctpos 6320   ndxcnx 13242   sSet csts 13243   Basecbs 13245    Hom 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
  Copyright terms: Public domain W3C validator