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Theorem oppcval 15324
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 3067 . . 3  |-  ( C  e.  V  ->  C  e.  _V )
3 id 22 . . . . . 6  |-  ( c  =  C  ->  c  =  C )
4 fveq2 5848 . . . . . . . . 9  |-  ( c  =  C  ->  ( Hom  `  c )  =  ( Hom  `  C
) )
5 oppcval.h . . . . . . . . 9  |-  H  =  ( Hom  `  C
)
64, 5syl6eqr 2461 . . . . . . . 8  |-  ( c  =  C  ->  ( Hom  `  c )  =  H )
76tposeqd 6960 . . . . . . 7  |-  ( c  =  C  -> tpos  ( Hom  `  c )  = tpos  H
)
87opeq2d 4165 . . . . . 6  |-  ( c  =  C  ->  <. ( Hom  `  ndx ) , tpos  ( Hom  `  c
) >.  =  <. ( Hom  `  ndx ) , tpos 
H >. )
93, 8oveq12d 6295 . . . . 5  |-  ( c  =  C  ->  (
c sSet  <. ( Hom  `  ndx ) , tpos  ( Hom  `  c ) >. )  =  ( C sSet  <. ( Hom  `  ndx ) , tpos 
H >. ) )
10 fveq2 5848 . . . . . . . . 9  |-  ( c  =  C  ->  ( Base `  c )  =  ( Base `  C
) )
11 oppcval.b . . . . . . . . 9  |-  B  =  ( Base `  C
)
1210, 11syl6eqr 2461 . . . . . . . 8  |-  ( c  =  C  ->  ( Base `  c )  =  B )
1312sqxpeqd 4848 . . . . . . 7  |-  ( c  =  C  ->  (
( Base `  c )  X.  ( Base `  c
) )  =  ( B  X.  B ) )
14 fveq2 5848 . . . . . . . . . 10  |-  ( c  =  C  ->  (comp `  c )  =  (comp `  C ) )
15 oppcval.x . . . . . . . . . 10  |-  .x.  =  (comp `  C )
1614, 15syl6eqr 2461 . . . . . . . . 9  |-  ( c  =  C  ->  (comp `  c )  =  .x.  )
1716oveqd 6294 . . . . . . . 8  |-  ( c  =  C  ->  ( <. z ,  ( 2nd `  u ) >. (comp `  c ) ( 1st `  u ) )  =  ( <. z ,  ( 2nd `  u )
>.  .x.  ( 1st `  u
) ) )
1817tposeqd 6960 . . . . . . 7  |-  ( c  =  C  -> tpos  ( <.
z ,  ( 2nd `  u ) >. (comp `  c ) ( 1st `  u ) )  = tpos  ( <. z ,  ( 2nd `  u )
>.  .x.  ( 1st `  u
) ) )
1913, 12, 18mpt2eq123dv 6339 . . . . . 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 4165 . . . . 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 6295 . . . 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 15323 . . . 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 6305 . . . 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 5931 . . 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 17 . 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 2455 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 1405    e. wcel 1842   _Vcvv 3058   <.cop 3977    X. cxp 4820   ` cfv 5568  (class class class)co 6277    |-> cmpt2 6279   1stc1st 6781   2ndc2nd 6782  tpos ctpos 6956   ndxcnx 14836   sSet csts 14837   Basecbs 14839   Hom chom 14918  compcco 14919  oppCatcoppc 15322
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pr 4629
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-sbc 3277  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-res 4834  df-iota 5532  df-fun 5570  df-fv 5576  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-tpos 6957  df-oppc 15323
This theorem is referenced by:  oppchomfval  15325  oppccofval  15327  oppcbas  15329  catcoppccl  15709
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