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Theorem oppcval 14664
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 2993 . . 3  |-  ( C  e.  V  ->  C  e.  _V )
3 id 22 . . . . . 6  |-  ( c  =  C  ->  c  =  C )
4 fveq2 5703 . . . . . . . . 9  |-  ( c  =  C  ->  ( Hom  `  c )  =  ( Hom  `  C
) )
5 oppcval.h . . . . . . . . 9  |-  H  =  ( Hom  `  C
)
64, 5syl6eqr 2493 . . . . . . . 8  |-  ( c  =  C  ->  ( Hom  `  c )  =  H )
76tposeqd 6760 . . . . . . 7  |-  ( c  =  C  -> tpos  ( Hom  `  c )  = tpos  H
)
87opeq2d 4078 . . . . . 6  |-  ( c  =  C  ->  <. ( Hom  `  ndx ) , tpos  ( Hom  `  c
) >.  =  <. ( Hom  `  ndx ) , tpos 
H >. )
93, 8oveq12d 6121 . . . . 5  |-  ( c  =  C  ->  (
c sSet  <. ( Hom  `  ndx ) , tpos  ( Hom  `  c ) >. )  =  ( C sSet  <. ( Hom  `  ndx ) , tpos 
H >. ) )
10 fveq2 5703 . . . . . . . . 9  |-  ( c  =  C  ->  ( Base `  c )  =  ( Base `  C
) )
11 oppcval.b . . . . . . . . 9  |-  B  =  ( Base `  C
)
1210, 11syl6eqr 2493 . . . . . . . 8  |-  ( c  =  C  ->  ( Base `  c )  =  B )
1312, 12xpeq12d 4877 . . . . . . 7  |-  ( c  =  C  ->  (
( Base `  c )  X.  ( Base `  c
) )  =  ( B  X.  B ) )
14 fveq2 5703 . . . . . . . . . 10  |-  ( c  =  C  ->  (comp `  c )  =  (comp `  C ) )
15 oppcval.x . . . . . . . . . 10  |-  .x.  =  (comp `  C )
1614, 15syl6eqr 2493 . . . . . . . . 9  |-  ( c  =  C  ->  (comp `  c )  =  .x.  )
1716oveqd 6120 . . . . . . . 8  |-  ( c  =  C  ->  ( <. z ,  ( 2nd `  u ) >. (comp `  c ) ( 1st `  u ) )  =  ( <. z ,  ( 2nd `  u )
>.  .x.  ( 1st `  u
) ) )
1817tposeqd 6760 . . . . . . 7  |-  ( c  =  C  -> tpos  ( <.
z ,  ( 2nd `  u ) >. (comp `  c ) ( 1st `  u ) )  = tpos  ( <. z ,  ( 2nd `  u )
>.  .x.  ( 1st `  u
) ) )
1913, 12, 18mpt2eq123dv 6160 . . . . . 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
) ) ) )
2019opeq2d 4078 . . . . 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
) ) ) >.
)
219, 20oveq12d 6121 . . . 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
) ) ) >.
) )
22 df-oppc 14663 . . . 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
) ) ) >.
) )
23 ovex 6128 . . . 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
2421, 22, 23fvmpt 5786 . . 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 ) ) ) >. ) )
252, 24syl 16 . 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 ) ) ) >. ) )
261, 25syl5eq 2487 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 setvar class
Syntax hints:    -> wi 4    = wceq 1369    e. wcel 1756   _Vcvv 2984   <.cop 3895    X. cxp 4850   ` cfv 5430  (class class class)co 6103    e. cmpt2 6105   1stc1st 6587   2ndc2nd 6588  tpos ctpos 6756   ndxcnx 14183   sSet csts 14184   Basecbs 14186   Hom chom 14261  compcco 14262  oppCatcoppc 14662
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4425  ax-nul 4433  ax-pr 4543
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-ral 2732  df-rex 2733  df-rab 2736  df-v 2986  df-sbc 3199  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-nul 3650  df-if 3804  df-sn 3890  df-pr 3892  df-op 3896  df-uni 4104  df-br 4305  df-opab 4363  df-mpt 4364  df-id 4648  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-res 4864  df-iota 5393  df-fun 5432  df-fv 5438  df-ov 6106  df-oprab 6107  df-mpt2 6108  df-tpos 6757  df-oppc 14663
This theorem is referenced by:  oppchomfval  14665  oppccofval  14667  oppcbas  14669  catcoppccl  14988
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