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Theorem funcoppc 13848
Description: A functor on categories yields a functor on the opposite categories (in the same direction). (Contributed by Mario Carneiro, 4-Jan-2017.)
Hypotheses
Ref Expression
funcoppc.o  |-  O  =  (oppCat `  C )
funcoppc.p  |-  P  =  (oppCat `  D )
funcoppc.f  |-  ( ph  ->  F ( C  Func  D ) G )
Assertion
Ref Expression
funcoppc  |-  ( ph  ->  F ( O  Func  P )tpos  G )

Proof of Theorem funcoppc
Dummy variables  f 
g  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 funcoppc.o . . 3  |-  O  =  (oppCat `  C )
2 eqid 2358 . . 3  |-  ( Base `  C )  =  (
Base `  C )
31, 2oppcbas 13720 . 2  |-  ( Base `  C )  =  (
Base `  O )
4 funcoppc.p . . 3  |-  P  =  (oppCat `  D )
5 eqid 2358 . . 3  |-  ( Base `  D )  =  (
Base `  D )
64, 5oppcbas 13720 . 2  |-  ( Base `  D )  =  (
Base `  P )
7 eqid 2358 . 2  |-  (  Hom  `  O )  =  (  Hom  `  O )
8 eqid 2358 . 2  |-  (  Hom  `  P )  =  (  Hom  `  P )
9 eqid 2358 . 2  |-  ( Id
`  O )  =  ( Id `  O
)
10 eqid 2358 . 2  |-  ( Id
`  P )  =  ( Id `  P
)
11 eqid 2358 . 2  |-  (comp `  O )  =  (comp `  O )
12 eqid 2358 . 2  |-  (comp `  P )  =  (comp `  P )
13 funcoppc.f . . . . . 6  |-  ( ph  ->  F ( C  Func  D ) G )
14 df-br 4105 . . . . . 6  |-  ( F ( C  Func  D
) G  <->  <. F ,  G >.  e.  ( C 
Func  D ) )
1513, 14sylib 188 . . . . 5  |-  ( ph  -> 
<. F ,  G >.  e.  ( C  Func  D
) )
16 funcrcl 13836 . . . . 5  |-  ( <. F ,  G >.  e.  ( C  Func  D
)  ->  ( C  e.  Cat  /\  D  e. 
Cat ) )
1715, 16syl 15 . . . 4  |-  ( ph  ->  ( C  e.  Cat  /\  D  e.  Cat )
)
1817simpld 445 . . 3  |-  ( ph  ->  C  e.  Cat )
191oppccat 13724 . . 3  |-  ( C  e.  Cat  ->  O  e.  Cat )
2018, 19syl 15 . 2  |-  ( ph  ->  O  e.  Cat )
2117simprd 449 . . 3  |-  ( ph  ->  D  e.  Cat )
224oppccat 13724 . . 3  |-  ( D  e.  Cat  ->  P  e.  Cat )
2321, 22syl 15 . 2  |-  ( ph  ->  P  e.  Cat )
242, 5, 13funcf1 13839 . 2  |-  ( ph  ->  F : ( Base `  C ) --> ( Base `  D ) )
252, 13funcfn2 13842 . . 3  |-  ( ph  ->  G  Fn  ( (
Base `  C )  X.  ( Base `  C
) ) )
26 tposfn 6350 . . 3  |-  ( G  Fn  ( ( Base `  C )  X.  ( Base `  C ) )  -> tpos  G  Fn  (
( Base `  C )  X.  ( Base `  C
) ) )
2725, 26syl 15 . 2  |-  ( ph  -> tpos  G  Fn  ( (
Base `  C )  X.  ( Base `  C
) ) )
28 eqid 2358 . . . 4  |-  (  Hom  `  C )  =  (  Hom  `  C )
29 eqid 2358 . . . 4  |-  (  Hom  `  D )  =  (  Hom  `  D )
3013adantr 451 . . . 4  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  F
( C  Func  D
) G )
31 simprr 733 . . . 4  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  y  e.  ( Base `  C
) )
32 simprl 732 . . . 4  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  x  e.  ( Base `  C
) )
332, 28, 29, 30, 31, 32funcf2 13841 . . 3  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  (
y G x ) : ( y (  Hom  `  C )
x ) --> ( ( F `  y ) (  Hom  `  D
) ( F `  x ) ) )
34 ovtpos 6336 . . . . 5  |-  ( xtpos 
G y )  =  ( y G x )
3534feq1i 5466 . . . 4  |-  ( ( xtpos  G y ) : ( x (  Hom  `  O )
y ) --> ( ( F `  x ) (  Hom  `  P
) ( F `  y ) )  <->  ( y G x ) : ( x (  Hom  `  O ) y ) --> ( ( F `  x ) (  Hom  `  P ) ( F `
 y ) ) )
3628, 1oppchom 13717 . . . . 5  |-  ( x (  Hom  `  O
) y )  =  ( y (  Hom  `  C ) x )
3729, 4oppchom 13717 . . . . 5  |-  ( ( F `  x ) (  Hom  `  P
) ( F `  y ) )  =  ( ( F `  y ) (  Hom  `  D ) ( F `
 x ) )
3836, 37feq23i 5468 . . . 4  |-  ( ( y G x ) : ( x (  Hom  `  O )
y ) --> ( ( F `  x ) (  Hom  `  P
) ( F `  y ) )  <->  ( y G x ) : ( y (  Hom  `  C ) x ) --> ( ( F `  y ) (  Hom  `  D ) ( F `
 x ) ) )
3935, 38bitri 240 . . 3  |-  ( ( xtpos  G y ) : ( x (  Hom  `  O )
y ) --> ( ( F `  x ) (  Hom  `  P
) ( F `  y ) )  <->  ( y G x ) : ( y (  Hom  `  C ) x ) --> ( ( F `  y ) (  Hom  `  D ) ( F `
 x ) ) )
4033, 39sylibr 203 . 2  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  (
xtpos  G y ) : ( x (  Hom  `  O )
y ) --> ( ( F `  x ) (  Hom  `  P
) ( F `  y ) ) )
41 eqid 2358 . . . 4  |-  ( Id
`  C )  =  ( Id `  C
)
42 eqid 2358 . . . 4  |-  ( Id
`  D )  =  ( Id `  D
)
4313adantr 451 . . . 4  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  F ( C  Func  D ) G )
44 simpr 447 . . . 4  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  x  e.  ( Base `  C )
)
452, 41, 42, 43, 44funcid 13843 . . 3  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( (
x G x ) `
 ( ( Id
`  C ) `  x ) )  =  ( ( Id `  D ) `  ( F `  x )
) )
46 ovtpos 6336 . . . . 5  |-  ( xtpos 
G x )  =  ( x G x )
4746a1i 10 . . . 4  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( xtpos  G x )  =  ( x G x ) )
481, 41oppcid 13723 . . . . . . 7  |-  ( C  e.  Cat  ->  ( Id `  O )  =  ( Id `  C
) )
4918, 48syl 15 . . . . . 6  |-  ( ph  ->  ( Id `  O
)  =  ( Id
`  C ) )
5049adantr 451 . . . . 5  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( Id `  O )  =  ( Id `  C ) )
5150fveq1d 5610 . . . 4  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( ( Id `  O ) `  x )  =  ( ( Id `  C
) `  x )
)
5247, 51fveq12d 5614 . . 3  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( (
xtpos  G x ) `
 ( ( Id
`  O ) `  x ) )  =  ( ( x G x ) `  (
( Id `  C
) `  x )
) )
534, 42oppcid 13723 . . . . . 6  |-  ( D  e.  Cat  ->  ( Id `  P )  =  ( Id `  D
) )
5421, 53syl 15 . . . . 5  |-  ( ph  ->  ( Id `  P
)  =  ( Id
`  D ) )
5554adantr 451 . . . 4  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( Id `  P )  =  ( Id `  D ) )
5655fveq1d 5610 . . 3  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( ( Id `  P ) `  ( F `  x ) )  =  ( ( Id `  D ) `
 ( F `  x ) ) )
5745, 52, 563eqtr4d 2400 . 2  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( (
xtpos  G x ) `
 ( ( Id
`  O ) `  x ) )  =  ( ( Id `  P ) `  ( F `  x )
) )
58 eqid 2358 . . . . 5  |-  (comp `  C )  =  (comp `  C )
59 eqid 2358 . . . . 5  |-  (comp `  D )  =  (comp `  D )
60133ad2ant1 976 . . . . 5  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  z  e.  ( Base `  C ) )  /\  ( f  e.  ( x (  Hom  `  O ) y )  /\  g  e.  ( y (  Hom  `  O
) z ) ) )  ->  F ( C  Func  D ) G )
61 simp23 990 . . . . 5  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  z  e.  ( Base `  C ) )  /\  ( f  e.  ( x (  Hom  `  O ) y )  /\  g  e.  ( y (  Hom  `  O
) z ) ) )  ->  z  e.  ( Base `  C )
)
62 simp22 989 . . . . 5  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  z  e.  ( Base `  C ) )  /\  ( f  e.  ( x (  Hom  `  O ) y )  /\  g  e.  ( y (  Hom  `  O
) z ) ) )  ->  y  e.  ( Base `  C )
)
63 simp21 988 . . . . 5  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  z  e.  ( Base `  C ) )  /\  ( f  e.  ( x (  Hom  `  O ) y )  /\  g  e.  ( y (  Hom  `  O
) z ) ) )  ->  x  e.  ( Base `  C )
)
64 simp3r 984 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  z  e.  ( Base `  C ) )  /\  ( f  e.  ( x (  Hom  `  O ) y )  /\  g  e.  ( y (  Hom  `  O
) z ) ) )  ->  g  e.  ( y (  Hom  `  O ) z ) )
6528, 1oppchom 13717 . . . . . 6  |-  ( y (  Hom  `  O
) z )  =  ( z (  Hom  `  C ) y )
6664, 65syl6eleq 2448 . . . . 5  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  z  e.  ( Base `  C ) )  /\  ( f  e.  ( x (  Hom  `  O ) y )  /\  g  e.  ( y (  Hom  `  O
) z ) ) )  ->  g  e.  ( z (  Hom  `  C ) y ) )
67 simp3l 983 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  z  e.  ( Base `  C ) )  /\  ( f  e.  ( x (  Hom  `  O ) y )  /\  g  e.  ( y (  Hom  `  O
) z ) ) )  ->  f  e.  ( x (  Hom  `  O ) y ) )
6867, 36syl6eleq 2448 . . . . 5  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  z  e.  ( Base `  C ) )  /\  ( f  e.  ( x (  Hom  `  O ) y )  /\  g  e.  ( y (  Hom  `  O
) z ) ) )  ->  f  e.  ( y (  Hom  `  C ) x ) )
692, 28, 58, 59, 60, 61, 62, 63, 66, 68funcco 13844 . . . 4  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  z  e.  ( Base `  C ) )  /\  ( f  e.  ( x (  Hom  `  O ) y )  /\  g  e.  ( y (  Hom  `  O
) z ) ) )  ->  ( (
z G x ) `
 ( f (
<. z ,  y >.
(comp `  C )
x ) g ) )  =  ( ( ( y G x ) `  f ) ( <. ( F `  z ) ,  ( F `  y )
>. (comp `  D )
( F `  x
) ) ( ( z G y ) `
 g ) ) )
702, 58, 1, 63, 62, 61oppcco 13719 . . . . 5  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  z  e.  ( Base `  C ) )  /\  ( f  e.  ( x (  Hom  `  O ) y )  /\  g  e.  ( y (  Hom  `  O
) z ) ) )  ->  ( g
( <. x ,  y
>. (comp `  O )
z ) f )  =  ( f (
<. z ,  y >.
(comp `  C )
x ) g ) )
7170fveq2d 5612 . . . 4  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  z  e.  ( Base `  C ) )  /\  ( f  e.  ( x (  Hom  `  O ) y )  /\  g  e.  ( y (  Hom  `  O
) z ) ) )  ->  ( (
z G x ) `
 ( g (
<. x ,  y >.
(comp `  O )
z ) f ) )  =  ( ( z G x ) `
 ( f (
<. z ,  y >.
(comp `  C )
x ) g ) ) )
72243ad2ant1 976 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  z  e.  ( Base `  C ) )  /\  ( f  e.  ( x (  Hom  `  O ) y )  /\  g  e.  ( y (  Hom  `  O
) z ) ) )  ->  F :
( Base `  C ) --> ( Base `  D )
)
7372, 63ffvelrnd 5749 . . . . 5  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  z  e.  ( Base `  C ) )  /\  ( f  e.  ( x (  Hom  `  O ) y )  /\  g  e.  ( y (  Hom  `  O
) z ) ) )  ->  ( F `  x )  e.  (
Base `  D )
)
7472, 62ffvelrnd 5749 . . . . 5  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  z  e.  ( Base `  C ) )  /\  ( f  e.  ( x (  Hom  `  O ) y )  /\  g  e.  ( y (  Hom  `  O
) z ) ) )  ->  ( F `  y )  e.  (
Base `  D )
)
7572, 61ffvelrnd 5749 . . . . 5  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  z  e.  ( Base `  C ) )  /\  ( f  e.  ( x (  Hom  `  O ) y )  /\  g  e.  ( y (  Hom  `  O
) z ) ) )  ->  ( F `  z )  e.  (
Base `  D )
)
765, 59, 4, 73, 74, 75oppcco 13719 . . . 4  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  z  e.  ( Base `  C ) )  /\  ( f  e.  ( x (  Hom  `  O ) y )  /\  g  e.  ( y (  Hom  `  O
) z ) ) )  ->  ( (
( z G y ) `  g ) ( <. ( F `  x ) ,  ( F `  y )
>. (comp `  P )
( F `  z
) ) ( ( y G x ) `
 f ) )  =  ( ( ( y G x ) `
 f ) (
<. ( F `  z
) ,  ( F `
 y ) >.
(comp `  D )
( F `  x
) ) ( ( z G y ) `
 g ) ) )
7769, 71, 763eqtr4d 2400 . . 3  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  z  e.  ( Base `  C ) )  /\  ( f  e.  ( x (  Hom  `  O ) y )  /\  g  e.  ( y (  Hom  `  O
) z ) ) )  ->  ( (
z G x ) `
 ( g (
<. x ,  y >.
(comp `  O )
z ) f ) )  =  ( ( ( z G y ) `  g ) ( <. ( F `  x ) ,  ( F `  y )
>. (comp `  P )
( F `  z
) ) ( ( y G x ) `
 f ) ) )
78 ovtpos 6336 . . . 4  |-  ( xtpos 
G z )  =  ( z G x )
7978fveq1i 5609 . . 3  |-  ( ( xtpos  G z ) `
 ( g (
<. x ,  y >.
(comp `  O )
z ) f ) )  =  ( ( z G x ) `
 ( g (
<. x ,  y >.
(comp `  O )
z ) f ) )
80 ovtpos 6336 . . . . 5  |-  ( ytpos 
G z )  =  ( z G y )
8180fveq1i 5609 . . . 4  |-  ( ( ytpos  G z ) `
 g )  =  ( ( z G y ) `  g
)
8234fveq1i 5609 . . . 4  |-  ( ( xtpos  G y ) `
 f )  =  ( ( y G x ) `  f
)
8381, 82oveq12i 5957 . . 3  |-  ( ( ( ytpos  G z ) `  g ) ( <. ( F `  x ) ,  ( F `  y )
>. (comp `  P )
( F `  z
) ) ( ( xtpos  G y ) `
 f ) )  =  ( ( ( z G y ) `
 g ) (
<. ( F `  x
) ,  ( F `
 y ) >.
(comp `  P )
( F `  z
) ) ( ( y G x ) `
 f ) )
8477, 79, 833eqtr4g 2415 . 2  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )  /\  z  e.  ( Base `  C ) )  /\  ( f  e.  ( x (  Hom  `  O ) y )  /\  g  e.  ( y (  Hom  `  O
) z ) ) )  ->  ( (
xtpos  G z ) `
 ( g (
<. x ,  y >.
(comp `  O )
z ) f ) )  =  ( ( ( ytpos  G z ) `  g ) ( <. ( F `  x ) ,  ( F `  y )
>. (comp `  P )
( F `  z
) ) ( ( xtpos  G y ) `
 f ) ) )
853, 6, 7, 8, 9, 10, 11, 12, 20, 23, 24, 27, 40, 57, 84isfuncd 13838 1  |-  ( ph  ->  F ( O  Func  P )tpos  G )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1642    e. wcel 1710   <.cop 3719   class class class wbr 4104    X. cxp 4769    Fn wfn 5332   -->wf 5333   ` cfv 5337  (class class class)co 5945  tpos ctpos 6320   Basecbs 13245    Hom chom 13316  compcco 13317   Catccat 13665   Idccid 13666  oppCatcoppc 13713    Func cfunc 13827
This theorem is referenced by:  fulloppc  13895  fthoppc  13896  yonedalem1  14145  yonedalem21  14146  yonedalem22  14151
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4212  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594  ax-cnex 8883  ax-resscn 8884  ax-1cn 8885  ax-icn 8886  ax-addcl 8887  ax-addrcl 8888  ax-mulcl 8889  ax-mulrcl 8890  ax-mulcom 8891  ax-addass 8892  ax-mulass 8893  ax-distr 8894  ax-i2m1 8895  ax-1ne0 8896  ax-1rid 8897  ax-rnegex 8898  ax-rrecex 8899  ax-cnre 8900  ax-pre-lttri 8901  ax-pre-lttrn 8902  ax-pre-ltadd 8903  ax-pre-mulgt0 8904
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3909  df-iun 3988  df-br 4105  df-opab 4159  df-mpt 4160  df-tr 4195  df-eprel 4387  df-id 4391  df-po 4396  df-so 4397  df-fr 4434  df-we 4436  df-ord 4477  df-on 4478  df-lim 4479  df-suc 4480  df-om 4739  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-1st 6209  df-2nd 6210  df-tpos 6321  df-riota 6391  df-recs 6475  df-rdg 6510  df-er 6747  df-map 6862  df-ixp 6906  df-en 6952  df-dom 6953  df-sdom 6954  df-pnf 8959  df-mnf 8960  df-xr 8961  df-ltxr 8962  df-le 8963  df-sub 9129  df-neg 9130  df-nn 9837  df-2 9894  df-3 9895  df-4 9896  df-5 9897  df-6 9898  df-7 9899  df-8 9900  df-9 9901  df-10 9902  df-n0 10058  df-z 10117  df-dec 10217  df-ndx 13248  df-slot 13249  df-base 13250  df-sets 13251  df-hom 13329  df-cco 13330  df-cat 13669  df-cid 13670  df-oppc 13714  df-func 13831
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