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Theorem oppcval 14962
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 3122 . . 3  |-  ( C  e.  V  ->  C  e.  _V )
3 id 22 . . . . . 6  |-  ( c  =  C  ->  c  =  C )
4 fveq2 5864 . . . . . . . . 9  |-  ( c  =  C  ->  ( Hom  `  c )  =  ( Hom  `  C
) )
5 oppcval.h . . . . . . . . 9  |-  H  =  ( Hom  `  C
)
64, 5syl6eqr 2526 . . . . . . . 8  |-  ( c  =  C  ->  ( Hom  `  c )  =  H )
76tposeqd 6955 . . . . . . 7  |-  ( c  =  C  -> tpos  ( Hom  `  c )  = tpos  H
)
87opeq2d 4220 . . . . . 6  |-  ( c  =  C  ->  <. ( Hom  `  ndx ) , tpos  ( Hom  `  c
) >.  =  <. ( Hom  `  ndx ) , tpos 
H >. )
93, 8oveq12d 6300 . . . . 5  |-  ( c  =  C  ->  (
c sSet  <. ( Hom  `  ndx ) , tpos  ( Hom  `  c ) >. )  =  ( C sSet  <. ( Hom  `  ndx ) , tpos 
H >. ) )
10 fveq2 5864 . . . . . . . . 9  |-  ( c  =  C  ->  ( Base `  c )  =  ( Base `  C
) )
11 oppcval.b . . . . . . . . 9  |-  B  =  ( Base `  C
)
1210, 11syl6eqr 2526 . . . . . . . 8  |-  ( c  =  C  ->  ( Base `  c )  =  B )
1312, 12xpeq12d 5024 . . . . . . 7  |-  ( c  =  C  ->  (
( Base `  c )  X.  ( Base `  c
) )  =  ( B  X.  B ) )
14 fveq2 5864 . . . . . . . . . 10  |-  ( c  =  C  ->  (comp `  c )  =  (comp `  C ) )
15 oppcval.x . . . . . . . . . 10  |-  .x.  =  (comp `  C )
1614, 15syl6eqr 2526 . . . . . . . . 9  |-  ( c  =  C  ->  (comp `  c )  =  .x.  )
1716oveqd 6299 . . . . . . . 8  |-  ( c  =  C  ->  ( <. z ,  ( 2nd `  u ) >. (comp `  c ) ( 1st `  u ) )  =  ( <. z ,  ( 2nd `  u )
>.  .x.  ( 1st `  u
) ) )
1817tposeqd 6955 . . . . . . 7  |-  ( c  =  C  -> tpos  ( <.
z ,  ( 2nd `  u ) >. (comp `  c ) ( 1st `  u ) )  = tpos  ( <. z ,  ( 2nd `  u )
>.  .x.  ( 1st `  u
) ) )
1913, 12, 18mpt2eq123dv 6341 . . . . . 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 4220 . . . . 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 6300 . . . 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 14961 . . . 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 6307 . . . 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 5948 . . 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 2520 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 1379    e. wcel 1767   _Vcvv 3113   <.cop 4033    X. cxp 4997   ` cfv 5586  (class class class)co 6282    |-> cmpt2 6284   1stc1st 6779   2ndc2nd 6780  tpos ctpos 6951   ndxcnx 14480   sSet csts 14481   Basecbs 14483   Hom chom 14559  compcco 14560  oppCatcoppc 14960
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-res 5011  df-iota 5549  df-fun 5588  df-fv 5594  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-tpos 6952  df-oppc 14961
This theorem is referenced by:  oppchomfval  14963  oppccofval  14965  oppcbas  14967  catcoppccl  15286
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