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Theorem oppcval 13944
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 2966 . . 3  |-  ( C  e.  V  ->  C  e.  _V )
3 id 21 . . . . . 6  |-  ( c  =  C  ->  c  =  C )
4 fveq2 5731 . . . . . . . . 9  |-  ( c  =  C  ->  (  Hom  `  c )  =  (  Hom  `  C
) )
5 oppcval.h . . . . . . . . 9  |-  H  =  (  Hom  `  C
)
64, 5syl6eqr 2488 . . . . . . . 8  |-  ( c  =  C  ->  (  Hom  `  c )  =  H )
76tposeqd 6485 . . . . . . 7  |-  ( c  =  C  -> tpos  (  Hom  `  c )  = tpos  H
)
87opeq2d 3993 . . . . . 6  |-  ( c  =  C  ->  <. (  Hom  `  ndx ) , tpos  (  Hom  `  c
) >.  =  <. (  Hom  `  ndx ) , tpos 
H >. )
93, 8oveq12d 6102 . . . . 5  |-  ( c  =  C  ->  (
c sSet  <. (  Hom  `  ndx ) , tpos  (  Hom  `  c
) >. )  =  ( C sSet  <. (  Hom  `  ndx ) , tpos  H >. )
)
10 fveq2 5731 . . . . . . . . 9  |-  ( c  =  C  ->  ( Base `  c )  =  ( Base `  C
) )
11 oppcval.b . . . . . . . . 9  |-  B  =  ( Base `  C
)
1210, 11syl6eqr 2488 . . . . . . . 8  |-  ( c  =  C  ->  ( Base `  c )  =  B )
1312, 12xpeq12d 4906 . . . . . . 7  |-  ( c  =  C  ->  (
( Base `  c )  X.  ( Base `  c
) )  =  ( B  X.  B ) )
14 fveq2 5731 . . . . . . . . . 10  |-  ( c  =  C  ->  (comp `  c )  =  (comp `  C ) )
15 oppcval.x . . . . . . . . . 10  |-  .x.  =  (comp `  C )
1614, 15syl6eqr 2488 . . . . . . . . 9  |-  ( c  =  C  ->  (comp `  c )  =  .x.  )
1716oveqd 6101 . . . . . . . 8  |-  ( c  =  C  ->  ( <. z ,  ( 2nd `  u ) >. (comp `  c ) ( 1st `  u ) )  =  ( <. z ,  ( 2nd `  u )
>.  .x.  ( 1st `  u
) ) )
1817tposeqd 6485 . . . . . . 7  |-  ( c  =  C  -> tpos  ( <.
z ,  ( 2nd `  u ) >. (comp `  c ) ( 1st `  u ) )  = tpos  ( <. z ,  ( 2nd `  u )
>.  .x.  ( 1st `  u
) ) )
1913, 12, 18mpt2eq123dv 6139 . . . . . 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 3993 . . . . 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 6102 . . . 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 13943 . . . 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 6109 . . . 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 5809 . . 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 2482 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 1653    e. wcel 1726   _Vcvv 2958   <.cop 3819    X. cxp 4879   ` cfv 5457  (class class class)co 6084    e. cmpt2 6086   1stc1st 6350   2ndc2nd 6351  tpos ctpos 6481   ndxcnx 13471   sSet csts 13472   Basecbs 13474    Hom chom 13545  compcco 13546  oppCatcoppc 13942
This theorem is referenced by:  oppchomfval  13945  oppccofval  13947  oppcbas  13949  catcoppccl  14268
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pr 4406
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-res 4893  df-iota 5421  df-fun 5459  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-tpos 6482  df-oppc 13943
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