MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  funcoppc Unicode version

Theorem funcoppc 14027
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 2404 . . 3  |-  ( Base `  C )  =  (
Base `  C )
31, 2oppcbas 13899 . 2  |-  ( Base `  C )  =  (
Base `  O )
4 funcoppc.p . . 3  |-  P  =  (oppCat `  D )
5 eqid 2404 . . 3  |-  ( Base `  D )  =  (
Base `  D )
64, 5oppcbas 13899 . 2  |-  ( Base `  D )  =  (
Base `  P )
7 eqid 2404 . 2  |-  (  Hom  `  O )  =  (  Hom  `  O )
8 eqid 2404 . 2  |-  (  Hom  `  P )  =  (  Hom  `  P )
9 eqid 2404 . 2  |-  ( Id
`  O )  =  ( Id `  O
)
10 eqid 2404 . 2  |-  ( Id
`  P )  =  ( Id `  P
)
11 eqid 2404 . 2  |-  (comp `  O )  =  (comp `  O )
12 eqid 2404 . 2  |-  (comp `  P )  =  (comp `  P )
13 funcoppc.f . . . . . 6  |-  ( ph  ->  F ( C  Func  D ) G )
14 df-br 4173 . . . . . 6  |-  ( F ( C  Func  D
) G  <->  <. F ,  G >.  e.  ( C 
Func  D ) )
1513, 14sylib 189 . . . . 5  |-  ( ph  -> 
<. F ,  G >.  e.  ( C  Func  D
) )
16 funcrcl 14015 . . . . 5  |-  ( <. F ,  G >.  e.  ( C  Func  D
)  ->  ( C  e.  Cat  /\  D  e. 
Cat ) )
1715, 16syl 16 . . . 4  |-  ( ph  ->  ( C  e.  Cat  /\  D  e.  Cat )
)
1817simpld 446 . . 3  |-  ( ph  ->  C  e.  Cat )
191oppccat 13903 . . 3  |-  ( C  e.  Cat  ->  O  e.  Cat )
2018, 19syl 16 . 2  |-  ( ph  ->  O  e.  Cat )
2117simprd 450 . . 3  |-  ( ph  ->  D  e.  Cat )
224oppccat 13903 . . 3  |-  ( D  e.  Cat  ->  P  e.  Cat )
2321, 22syl 16 . 2  |-  ( ph  ->  P  e.  Cat )
242, 5, 13funcf1 14018 . 2  |-  ( ph  ->  F : ( Base `  C ) --> ( Base `  D ) )
252, 13funcfn2 14021 . . 3  |-  ( ph  ->  G  Fn  ( (
Base `  C )  X.  ( Base `  C
) ) )
26 tposfn 6467 . . 3  |-  ( G  Fn  ( ( Base `  C )  X.  ( Base `  C ) )  -> tpos  G  Fn  (
( Base `  C )  X.  ( Base `  C
) ) )
2725, 26syl 16 . 2  |-  ( ph  -> tpos  G  Fn  ( (
Base `  C )  X.  ( Base `  C
) ) )
28 eqid 2404 . . . 4  |-  (  Hom  `  C )  =  (  Hom  `  C )
29 eqid 2404 . . . 4  |-  (  Hom  `  D )  =  (  Hom  `  D )
3013adantr 452 . . . 4  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  F
( C  Func  D
) G )
31 simprr 734 . . . 4  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  y  e.  ( Base `  C
) )
32 simprl 733 . . . 4  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  x  e.  ( Base `  C
) )
332, 28, 29, 30, 31, 32funcf2 14020 . . 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 6453 . . . . 5  |-  ( xtpos 
G y )  =  ( y G x )
3534feq1i 5544 . . . 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 13896 . . . . 5  |-  ( x (  Hom  `  O
) y )  =  ( y (  Hom  `  C ) x )
3729, 4oppchom 13896 . . . . 5  |-  ( ( F `  x ) (  Hom  `  P
) ( F `  y ) )  =  ( ( F `  y ) (  Hom  `  D ) ( F `
 x ) )
3836, 37feq23i 5546 . . . 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 241 . . 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 204 . 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 2404 . . . 4  |-  ( Id
`  C )  =  ( Id `  C
)
42 eqid 2404 . . . 4  |-  ( Id
`  D )  =  ( Id `  D
)
4313adantr 452 . . . 4  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  F ( C  Func  D ) G )
44 simpr 448 . . . 4  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  x  e.  ( Base `  C )
)
452, 41, 42, 43, 44funcid 14022 . . 3  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( (
x G x ) `
 ( ( Id
`  C ) `  x ) )  =  ( ( Id `  D ) `  ( F `  x )
) )
46 ovtpos 6453 . . . . 5  |-  ( xtpos 
G x )  =  ( x G x )
4746a1i 11 . . . 4  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( xtpos  G x )  =  ( x G x ) )
481, 41oppcid 13902 . . . . . . 7  |-  ( C  e.  Cat  ->  ( Id `  O )  =  ( Id `  C
) )
4918, 48syl 16 . . . . . 6  |-  ( ph  ->  ( Id `  O
)  =  ( Id
`  C ) )
5049adantr 452 . . . . 5  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( Id `  O )  =  ( Id `  C ) )
5150fveq1d 5689 . . . 4  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( ( Id `  O ) `  x )  =  ( ( Id `  C
) `  x )
)
5247, 51fveq12d 5693 . . 3  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( (
xtpos  G x ) `
 ( ( Id
`  O ) `  x ) )  =  ( ( x G x ) `  (
( Id `  C
) `  x )
) )
534, 42oppcid 13902 . . . . . 6  |-  ( D  e.  Cat  ->  ( Id `  P )  =  ( Id `  D
) )
5421, 53syl 16 . . . . 5  |-  ( ph  ->  ( Id `  P
)  =  ( Id
`  D ) )
5554adantr 452 . . . 4  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( Id `  P )  =  ( Id `  D ) )
5655fveq1d 5689 . . 3  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( ( Id `  P ) `  ( F `  x ) )  =  ( ( Id `  D ) `
 ( F `  x ) ) )
5745, 52, 563eqtr4d 2446 . 2  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( (
xtpos  G x ) `
 ( ( Id
`  O ) `  x ) )  =  ( ( Id `  P ) `  ( F `  x )
) )
58 eqid 2404 . . . . 5  |-  (comp `  C )  =  (comp `  C )
59 eqid 2404 . . . . 5  |-  (comp `  D )  =  (comp `  D )
60133ad2ant1 978 . . . . 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 992 . . . . 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 991 . . . . 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 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 ) ) )  ->  x  e.  ( Base `  C )
)
64 simp3r 986 . . . . . 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 13896 . . . . . 6  |-  ( y (  Hom  `  O
) z )  =  ( z (  Hom  `  C ) y )
6664, 65syl6eleq 2494 . . . . 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 985 . . . . . 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 2494 . . . . 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 14023 . . . 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 13898 . . . . 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 5691 . . . 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 978 . . . . . 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 5830 . . . . 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 5830 . . . . 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 5830 . . . . 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 13898 . . . 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 2446 . . 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 6453 . . . 4  |-  ( xtpos 
G z )  =  ( z G x )
7978fveq1i 5688 . . 3  |-  ( ( xtpos  G z ) `
 ( g (
<. x ,  y >.
(comp `  O )
z ) f ) )  =  ( ( z G x ) `
 ( g (
<. x ,  y >.
(comp `  O )
z ) f ) )
80 ovtpos 6453 . . . . 5  |-  ( ytpos 
G z )  =  ( z G y )
8180fveq1i 5688 . . . 4  |-  ( ( ytpos  G z ) `
 g )  =  ( ( z G y ) `  g
)
8234fveq1i 5688 . . . 4  |-  ( ( xtpos  G y ) `
 f )  =  ( ( y G x ) `  f
)
8381, 82oveq12i 6052 . . 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 2461 . 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 14017 1  |-  ( ph  ->  F ( O  Func  P )tpos  G )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   <.cop 3777   class class class wbr 4172    X. cxp 4835    Fn wfn 5408   -->wf 5409   ` cfv 5413  (class class class)co 6040  tpos ctpos 6437   Basecbs 13424    Hom chom 13495  compcco 13496   Catccat 13844   Idccid 13845  oppCatcoppc 13892    Func cfunc 14006
This theorem is referenced by:  fulloppc  14074  fthoppc  14075  yonedalem1  14324  yonedalem21  14325  yonedalem22  14330
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-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  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-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-tpos 6438  df-riota 6508  df-recs 6592  df-rdg 6627  df-er 6864  df-map 6979  df-ixp 7023  df-en 7069  df-dom 7070  df-sdom 7071  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-dec 10339  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-hom 13508  df-cco 13509  df-cat 13848  df-cid 13849  df-oppc 13893  df-func 14010
  Copyright terms: Public domain W3C validator