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Theorem oppcsect 13989
Description: A section in the opposite category. (Contributed by Mario Carneiro, 3-Jan-2017.)
Hypotheses
Ref Expression
oppcsect.b  |-  B  =  ( Base `  C
)
oppcsect.o  |-  O  =  (oppCat `  C )
oppcsect.c  |-  ( ph  ->  C  e.  Cat )
oppcsect.x  |-  ( ph  ->  X  e.  B )
oppcsect.y  |-  ( ph  ->  Y  e.  B )
oppcsect.s  |-  S  =  (Sect `  C )
oppcsect.t  |-  T  =  (Sect `  O )
Assertion
Ref Expression
oppcsect  |-  ( ph  ->  ( F ( X T Y ) G  <-> 
G ( X S Y ) F ) )

Proof of Theorem oppcsect
StepHypRef Expression
1 oppcsect.b . . . . . 6  |-  B  =  ( Base `  C
)
2 eqid 2435 . . . . . 6  |-  (comp `  C )  =  (comp `  C )
3 oppcsect.o . . . . . 6  |-  O  =  (oppCat `  C )
4 oppcsect.x . . . . . . 7  |-  ( ph  ->  X  e.  B )
54adantr 452 . . . . . 6  |-  ( (
ph  /\  ( G  e.  ( X (  Hom  `  C ) Y )  /\  F  e.  ( Y (  Hom  `  C
) X ) ) )  ->  X  e.  B )
6 oppcsect.y . . . . . . 7  |-  ( ph  ->  Y  e.  B )
76adantr 452 . . . . . 6  |-  ( (
ph  /\  ( G  e.  ( X (  Hom  `  C ) Y )  /\  F  e.  ( Y (  Hom  `  C
) X ) ) )  ->  Y  e.  B )
81, 2, 3, 5, 7, 5oppcco 13933 . . . . 5  |-  ( (
ph  /\  ( G  e.  ( X (  Hom  `  C ) Y )  /\  F  e.  ( Y (  Hom  `  C
) X ) ) )  ->  ( G
( <. X ,  Y >. (comp `  O ) X ) F )  =  ( F (
<. X ,  Y >. (comp `  C ) X ) G ) )
9 oppcsect.c . . . . . . . 8  |-  ( ph  ->  C  e.  Cat )
109adantr 452 . . . . . . 7  |-  ( (
ph  /\  ( G  e.  ( X (  Hom  `  C ) Y )  /\  F  e.  ( Y (  Hom  `  C
) X ) ) )  ->  C  e.  Cat )
11 eqid 2435 . . . . . . . 8  |-  ( Id
`  C )  =  ( Id `  C
)
123, 11oppcid 13937 . . . . . . 7  |-  ( C  e.  Cat  ->  ( Id `  O )  =  ( Id `  C
) )
1310, 12syl 16 . . . . . 6  |-  ( (
ph  /\  ( G  e.  ( X (  Hom  `  C ) Y )  /\  F  e.  ( Y (  Hom  `  C
) X ) ) )  ->  ( Id `  O )  =  ( Id `  C ) )
1413fveq1d 5722 . . . . 5  |-  ( (
ph  /\  ( G  e.  ( X (  Hom  `  C ) Y )  /\  F  e.  ( Y (  Hom  `  C
) X ) ) )  ->  ( ( Id `  O ) `  X )  =  ( ( Id `  C
) `  X )
)
158, 14eqeq12d 2449 . . . 4  |-  ( (
ph  /\  ( G  e.  ( X (  Hom  `  C ) Y )  /\  F  e.  ( Y (  Hom  `  C
) X ) ) )  ->  ( ( G ( <. X ,  Y >. (comp `  O
) X ) F )  =  ( ( Id `  O ) `
 X )  <->  ( F
( <. X ,  Y >. (comp `  C ) X ) G )  =  ( ( Id
`  C ) `  X ) ) )
1615pm5.32da 623 . . 3  |-  ( ph  ->  ( ( ( G  e.  ( X (  Hom  `  C ) Y )  /\  F  e.  ( Y (  Hom  `  C ) X ) )  /\  ( G ( <. X ,  Y >. (comp `  O ) X ) F )  =  ( ( Id
`  O ) `  X ) )  <->  ( ( G  e.  ( X
(  Hom  `  C ) Y )  /\  F  e.  ( Y (  Hom  `  C ) X ) )  /\  ( F ( <. X ,  Y >. (comp `  C ) X ) G )  =  ( ( Id
`  C ) `  X ) ) ) )
17 df-3an 938 . . . 4  |-  ( ( F  e.  ( X (  Hom  `  O
) Y )  /\  G  e.  ( Y
(  Hom  `  O ) X )  /\  ( G ( <. X ,  Y >. (comp `  O
) X ) F )  =  ( ( Id `  O ) `
 X ) )  <-> 
( ( F  e.  ( X (  Hom  `  O ) Y )  /\  G  e.  ( Y (  Hom  `  O
) X ) )  /\  ( G (
<. X ,  Y >. (comp `  O ) X ) F )  =  ( ( Id `  O
) `  X )
) )
18 eqid 2435 . . . . . . . 8  |-  (  Hom  `  C )  =  (  Hom  `  C )
1918, 3oppchom 13931 . . . . . . 7  |-  ( X (  Hom  `  O
) Y )  =  ( Y (  Hom  `  C ) X )
2019eleq2i 2499 . . . . . 6  |-  ( F  e.  ( X (  Hom  `  O ) Y )  <->  F  e.  ( Y (  Hom  `  C
) X ) )
2118, 3oppchom 13931 . . . . . . 7  |-  ( Y (  Hom  `  O
) X )  =  ( X (  Hom  `  C ) Y )
2221eleq2i 2499 . . . . . 6  |-  ( G  e.  ( Y (  Hom  `  O ) X )  <->  G  e.  ( X (  Hom  `  C
) Y ) )
2320, 22anbi12ci 680 . . . . 5  |-  ( ( F  e.  ( X (  Hom  `  O
) Y )  /\  G  e.  ( Y
(  Hom  `  O ) X ) )  <->  ( G  e.  ( X (  Hom  `  C ) Y )  /\  F  e.  ( Y (  Hom  `  C
) X ) ) )
2423anbi1i 677 . . . 4  |-  ( ( ( F  e.  ( X (  Hom  `  O
) Y )  /\  G  e.  ( Y
(  Hom  `  O ) X ) )  /\  ( G ( <. X ,  Y >. (comp `  O
) X ) F )  =  ( ( Id `  O ) `
 X ) )  <-> 
( ( G  e.  ( X (  Hom  `  C ) Y )  /\  F  e.  ( Y (  Hom  `  C
) X ) )  /\  ( G (
<. X ,  Y >. (comp `  O ) X ) F )  =  ( ( Id `  O
) `  X )
) )
2517, 24bitri 241 . . 3  |-  ( ( F  e.  ( X (  Hom  `  O
) Y )  /\  G  e.  ( Y
(  Hom  `  O ) X )  /\  ( G ( <. X ,  Y >. (comp `  O
) X ) F )  =  ( ( Id `  O ) `
 X ) )  <-> 
( ( G  e.  ( X (  Hom  `  C ) Y )  /\  F  e.  ( Y (  Hom  `  C
) X ) )  /\  ( G (
<. X ,  Y >. (comp `  O ) X ) F )  =  ( ( Id `  O
) `  X )
) )
26 df-3an 938 . . 3  |-  ( ( G  e.  ( X (  Hom  `  C
) Y )  /\  F  e.  ( Y
(  Hom  `  C ) X )  /\  ( F ( <. X ,  Y >. (comp `  C
) X ) G )  =  ( ( Id `  C ) `
 X ) )  <-> 
( ( G  e.  ( X (  Hom  `  C ) Y )  /\  F  e.  ( Y (  Hom  `  C
) X ) )  /\  ( F (
<. X ,  Y >. (comp `  C ) X ) G )  =  ( ( Id `  C
) `  X )
) )
2716, 25, 263bitr4g 280 . 2  |-  ( ph  ->  ( ( F  e.  ( X (  Hom  `  O ) Y )  /\  G  e.  ( Y (  Hom  `  O
) X )  /\  ( G ( <. X ,  Y >. (comp `  O
) X ) F )  =  ( ( Id `  O ) `
 X ) )  <-> 
( G  e.  ( X (  Hom  `  C
) Y )  /\  F  e.  ( Y
(  Hom  `  C ) X )  /\  ( F ( <. X ,  Y >. (comp `  C
) X ) G )  =  ( ( Id `  C ) `
 X ) ) ) )
283, 1oppcbas 13934 . . 3  |-  B  =  ( Base `  O
)
29 eqid 2435 . . 3  |-  (  Hom  `  O )  =  (  Hom  `  O )
30 eqid 2435 . . 3  |-  (comp `  O )  =  (comp `  O )
31 eqid 2435 . . 3  |-  ( Id
`  O )  =  ( Id `  O
)
32 oppcsect.t . . 3  |-  T  =  (Sect `  O )
333oppccat 13938 . . . 4  |-  ( C  e.  Cat  ->  O  e.  Cat )
349, 33syl 16 . . 3  |-  ( ph  ->  O  e.  Cat )
3528, 29, 30, 31, 32, 34, 4, 6issect 13969 . 2  |-  ( ph  ->  ( F ( X T Y ) G  <-> 
( F  e.  ( X (  Hom  `  O
) Y )  /\  G  e.  ( Y
(  Hom  `  O ) X )  /\  ( G ( <. X ,  Y >. (comp `  O
) X ) F )  =  ( ( Id `  O ) `
 X ) ) ) )
36 oppcsect.s . . 3  |-  S  =  (Sect `  C )
371, 18, 2, 11, 36, 9, 4, 6issect 13969 . 2  |-  ( ph  ->  ( G ( X S Y ) F  <-> 
( G  e.  ( X (  Hom  `  C
) Y )  /\  F  e.  ( Y
(  Hom  `  C ) X )  /\  ( F ( <. X ,  Y >. (comp `  C
) X ) G )  =  ( ( Id `  C ) `
 X ) ) ) )
3827, 35, 373bitr4d 277 1  |-  ( ph  ->  ( F ( X T Y ) G  <-> 
G ( X S Y ) F ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   <.cop 3809   class class class wbr 4204   ` cfv 5446  (class class class)co 6073   Basecbs 13459    Hom chom 13530  compcco 13531   Catccat 13879   Idccid 13880  oppCatcoppc 13927  Sectcsect 13960
This theorem is referenced by:  oppcsect2  13990  sectepi  13995  episect  13996
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9036  ax-resscn 9037  ax-1cn 9038  ax-icn 9039  ax-addcl 9040  ax-addrcl 9041  ax-mulcl 9042  ax-mulrcl 9043  ax-mulcom 9044  ax-addass 9045  ax-mulass 9046  ax-distr 9047  ax-i2m1 9048  ax-1ne0 9049  ax-1rid 9050  ax-rnegex 9051  ax-rrecex 9052  ax-cnre 9053  ax-pre-lttri 9054  ax-pre-lttrn 9055  ax-pre-ltadd 9056  ax-pre-mulgt0 9057
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-tpos 6471  df-riota 6541  df-recs 6625  df-rdg 6660  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-pnf 9112  df-mnf 9113  df-xr 9114  df-ltxr 9115  df-le 9116  df-sub 9283  df-neg 9284  df-nn 9991  df-2 10048  df-3 10049  df-4 10050  df-5 10051  df-6 10052  df-7 10053  df-8 10054  df-9 10055  df-10 10056  df-n0 10212  df-z 10273  df-dec 10373  df-ndx 13462  df-slot 13463  df-base 13464  df-sets 13465  df-hom 13543  df-cco 13544  df-cat 13883  df-cid 13884  df-oppc 13928  df-sect 13963
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