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Theorem funcoppc 15091
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 2460 . . 3  |-  ( Base `  C )  =  (
Base `  C )
31, 2oppcbas 14963 . 2  |-  ( Base `  C )  =  (
Base `  O )
4 funcoppc.p . . 3  |-  P  =  (oppCat `  D )
5 eqid 2460 . . 3  |-  ( Base `  D )  =  (
Base `  D )
64, 5oppcbas 14963 . 2  |-  ( Base `  D )  =  (
Base `  P )
7 eqid 2460 . 2  |-  ( Hom  `  O )  =  ( Hom  `  O )
8 eqid 2460 . 2  |-  ( Hom  `  P )  =  ( Hom  `  P )
9 eqid 2460 . 2  |-  ( Id
`  O )  =  ( Id `  O
)
10 eqid 2460 . 2  |-  ( Id
`  P )  =  ( Id `  P
)
11 eqid 2460 . 2  |-  (comp `  O )  =  (comp `  O )
12 eqid 2460 . 2  |-  (comp `  P )  =  (comp `  P )
13 funcoppc.f . . . . . 6  |-  ( ph  ->  F ( C  Func  D ) G )
14 df-br 4441 . . . . . 6  |-  ( F ( C  Func  D
) G  <->  <. F ,  G >.  e.  ( C 
Func  D ) )
1513, 14sylib 196 . . . . 5  |-  ( ph  -> 
<. F ,  G >.  e.  ( C  Func  D
) )
16 funcrcl 15079 . . . . 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 459 . . 3  |-  ( ph  ->  C  e.  Cat )
191oppccat 14967 . . 3  |-  ( C  e.  Cat  ->  O  e.  Cat )
2018, 19syl 16 . 2  |-  ( ph  ->  O  e.  Cat )
2117simprd 463 . . 3  |-  ( ph  ->  D  e.  Cat )
224oppccat 14967 . . 3  |-  ( D  e.  Cat  ->  P  e.  Cat )
2321, 22syl 16 . 2  |-  ( ph  ->  P  e.  Cat )
242, 5, 13funcf1 15082 . 2  |-  ( ph  ->  F : ( Base `  C ) --> ( Base `  D ) )
252, 13funcfn2 15085 . . 3  |-  ( ph  ->  G  Fn  ( (
Base `  C )  X.  ( Base `  C
) ) )
26 tposfn 6974 . . 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 2460 . . . 4  |-  ( Hom  `  C )  =  ( Hom  `  C )
29 eqid 2460 . . . 4  |-  ( Hom  `  D )  =  ( Hom  `  D )
3013adantr 465 . . . 4  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  F
( C  Func  D
) G )
31 simprr 756 . . . 4  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  y  e.  ( Base `  C
) )
32 simprl 755 . . . 4  |-  ( (
ph  /\  ( x  e.  ( Base `  C
)  /\  y  e.  ( Base `  C )
) )  ->  x  e.  ( Base `  C
) )
332, 28, 29, 30, 31, 32funcf2 15084 . . 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 6960 . . . . 5  |-  ( xtpos 
G y )  =  ( y G x )
3534feq1i 5714 . . . 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 14960 . . . . 5  |-  ( x ( Hom  `  O
) y )  =  ( y ( Hom  `  C ) x )
3729, 4oppchom 14960 . . . . 5  |-  ( ( F `  x ) ( Hom  `  P
) ( F `  y ) )  =  ( ( F `  y ) ( Hom  `  D ) ( F `
 x ) )
3836, 37feq23i 5716 . . . 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 249 . . 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 212 . 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 2460 . . . 4  |-  ( Id
`  C )  =  ( Id `  C
)
42 eqid 2460 . . . 4  |-  ( Id
`  D )  =  ( Id `  D
)
4313adantr 465 . . . 4  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  F ( C  Func  D ) G )
44 simpr 461 . . . 4  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  x  e.  ( Base `  C )
)
452, 41, 42, 43, 44funcid 15086 . . 3  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( (
x G x ) `
 ( ( Id
`  C ) `  x ) )  =  ( ( Id `  D ) `  ( F `  x )
) )
46 ovtpos 6960 . . . . 5  |-  ( xtpos 
G x )  =  ( x G x )
4746a1i 11 . . . 4  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( xtpos  G x )  =  ( x G x ) )
481, 41oppcid 14966 . . . . . . 7  |-  ( C  e.  Cat  ->  ( Id `  O )  =  ( Id `  C
) )
4918, 48syl 16 . . . . . 6  |-  ( ph  ->  ( Id `  O
)  =  ( Id
`  C ) )
5049adantr 465 . . . . 5  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( Id `  O )  =  ( Id `  C ) )
5150fveq1d 5859 . . . 4  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( ( Id `  O ) `  x )  =  ( ( Id `  C
) `  x )
)
5247, 51fveq12d 5863 . . 3  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( (
xtpos  G x ) `
 ( ( Id
`  O ) `  x ) )  =  ( ( x G x ) `  (
( Id `  C
) `  x )
) )
534, 42oppcid 14966 . . . . . 6  |-  ( D  e.  Cat  ->  ( Id `  P )  =  ( Id `  D
) )
5421, 53syl 16 . . . . 5  |-  ( ph  ->  ( Id `  P
)  =  ( Id
`  D ) )
5554adantr 465 . . . 4  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( Id `  P )  =  ( Id `  D ) )
5655fveq1d 5859 . . 3  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( ( Id `  P ) `  ( F `  x ) )  =  ( ( Id `  D ) `
 ( F `  x ) ) )
5745, 52, 563eqtr4d 2511 . 2  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( (
xtpos  G x ) `
 ( ( Id
`  O ) `  x ) )  =  ( ( Id `  P ) `  ( F `  x )
) )
58 eqid 2460 . . . . 5  |-  (comp `  C )  =  (comp `  C )
59 eqid 2460 . . . . 5  |-  (comp `  D )  =  (comp `  D )
60133ad2ant1 1012 . . . . 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 1026 . . . . 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 1025 . . . . 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 1024 . . . . 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 1020 . . . . . 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 14960 . . . . . 6  |-  ( y ( Hom  `  O
) z )  =  ( z ( Hom  `  C ) y )
6664, 65syl6eleq 2558 . . . . 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 1019 . . . . . 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 2558 . . . . 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 15087 . . . 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 14962 . . . . 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 5861 . . . 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 1012 . . . . . 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 6013 . . . . 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 6013 . . . . 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 6013 . . . . 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 14962 . . . 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 2511 . . 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 6960 . . . 4  |-  ( xtpos 
G z )  =  ( z G x )
7978fveq1i 5858 . . 3  |-  ( ( xtpos  G z ) `
 ( g (
<. x ,  y >.
(comp `  O )
z ) f ) )  =  ( ( z G x ) `
 ( g (
<. x ,  y >.
(comp `  O )
z ) f ) )
80 ovtpos 6960 . . . . 5  |-  ( ytpos 
G z )  =  ( z G y )
8180fveq1i 5858 . . . 4  |-  ( ( ytpos  G z ) `
 g )  =  ( ( z G y ) `  g
)
8234fveq1i 5858 . . . 4  |-  ( ( xtpos  G y ) `
 f )  =  ( ( y G x ) `  f
)
8381, 82oveq12i 6287 . . 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 2526 . 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 15081 1  |-  ( ph  ->  F ( O  Func  P )tpos  G )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762   <.cop 4026   class class class wbr 4440    X. cxp 4990    Fn wfn 5574   -->wf 5575   ` cfv 5579  (class class class)co 6275  tpos ctpos 6944   Basecbs 14479   Hom chom 14555  compcco 14556   Catccat 14908   Idccid 14909  oppCatcoppc 14956    Func cfunc 15070
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-1st 6774  df-2nd 6775  df-tpos 6945  df-recs 7032  df-rdg 7066  df-er 7301  df-map 7412  df-ixp 7460  df-en 7507  df-dom 7508  df-sdom 7509  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-nn 10526  df-2 10583  df-3 10584  df-4 10585  df-5 10586  df-6 10587  df-7 10588  df-8 10589  df-9 10590  df-10 10591  df-n0 10785  df-z 10854  df-dec 10966  df-ndx 14482  df-slot 14483  df-base 14484  df-sets 14485  df-hom 14568  df-cco 14569  df-cat 14912  df-cid 14913  df-oppc 14957  df-func 15074
This theorem is referenced by:  fulloppc  15138  fthoppc  15139  yonedalem1  15388  yonedalem21  15389  yonedalem22  15394
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