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Theorem oppcval 13894
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 2924 . . 3  |-  ( C  e.  V  ->  C  e.  _V )
3 id 20 . . . . . 6  |-  ( c  =  C  ->  c  =  C )
4 fveq2 5687 . . . . . . . . 9  |-  ( c  =  C  ->  (  Hom  `  c )  =  (  Hom  `  C
) )
5 oppcval.h . . . . . . . . 9  |-  H  =  (  Hom  `  C
)
64, 5syl6eqr 2454 . . . . . . . 8  |-  ( c  =  C  ->  (  Hom  `  c )  =  H )
76tposeqd 6441 . . . . . . 7  |-  ( c  =  C  -> tpos  (  Hom  `  c )  = tpos  H
)
87opeq2d 3951 . . . . . 6  |-  ( c  =  C  ->  <. (  Hom  `  ndx ) , tpos  (  Hom  `  c
) >.  =  <. (  Hom  `  ndx ) , tpos 
H >. )
93, 8oveq12d 6058 . . . . 5  |-  ( c  =  C  ->  (
c sSet  <. (  Hom  `  ndx ) , tpos  (  Hom  `  c
) >. )  =  ( C sSet  <. (  Hom  `  ndx ) , tpos  H >. )
)
10 fveq2 5687 . . . . . . . . 9  |-  ( c  =  C  ->  ( Base `  c )  =  ( Base `  C
) )
11 oppcval.b . . . . . . . . 9  |-  B  =  ( Base `  C
)
1210, 11syl6eqr 2454 . . . . . . . 8  |-  ( c  =  C  ->  ( Base `  c )  =  B )
1312, 12xpeq12d 4862 . . . . . . 7  |-  ( c  =  C  ->  (
( Base `  c )  X.  ( Base `  c
) )  =  ( B  X.  B ) )
14 fveq2 5687 . . . . . . . . . 10  |-  ( c  =  C  ->  (comp `  c )  =  (comp `  C ) )
15 oppcval.x . . . . . . . . . 10  |-  .x.  =  (comp `  C )
1614, 15syl6eqr 2454 . . . . . . . . 9  |-  ( c  =  C  ->  (comp `  c )  =  .x.  )
1716oveqd 6057 . . . . . . . 8  |-  ( c  =  C  ->  ( <. z ,  ( 2nd `  u ) >. (comp `  c ) ( 1st `  u ) )  =  ( <. z ,  ( 2nd `  u )
>.  .x.  ( 1st `  u
) ) )
1817tposeqd 6441 . . . . . . 7  |-  ( c  =  C  -> tpos  ( <.
z ,  ( 2nd `  u ) >. (comp `  c ) ( 1st `  u ) )  = tpos  ( <. z ,  ( 2nd `  u )
>.  .x.  ( 1st `  u
) ) )
1913, 12, 18mpt2eq123dv 6095 . . . . . 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 3951 . . . . 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 6058 . . . 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 13893 . . . 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 6065 . . . 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 5765 . . 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 2448 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 1649    e. wcel 1721   _Vcvv 2916   <.cop 3777    X. cxp 4835   ` cfv 5413  (class class class)co 6040    e. cmpt2 6042   1stc1st 6306   2ndc2nd 6307  tpos ctpos 6437   ndxcnx 13421   sSet csts 13422   Basecbs 13424    Hom chom 13495  compcco 13496  oppCatcoppc 13892
This theorem is referenced by:  oppchomfval  13895  oppccofval  13897  oppcbas  13899  catcoppccl  14218
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-res 4849  df-iota 5377  df-fun 5415  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-tpos 6438  df-oppc 13893
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