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Theorem oppcsect 13919
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 2380 . . . . . 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 13863 . . . . 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 2380 . . . . . . . 8  |-  ( Id
`  C )  =  ( Id `  C
)
123, 11oppcid 13867 . . . . . . 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 5663 . . . . 5  |-  ( (
ph  /\  ( G  e.  ( X (  Hom  `  C ) Y )  /\  F  e.  ( Y (  Hom  `  C
) X ) ) )  ->  ( ( Id `  O ) `  X )  =  ( ( Id `  C
) `  X )
)
158, 14eqeq12d 2394 . . . 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 2380 . . . . . . . 8  |-  (  Hom  `  C )  =  (  Hom  `  C )
1918, 3oppchom 13861 . . . . . . 7  |-  ( X (  Hom  `  O
) Y )  =  ( Y (  Hom  `  C ) X )
2019eleq2i 2444 . . . . . 6  |-  ( F  e.  ( X (  Hom  `  O ) Y )  <->  F  e.  ( Y (  Hom  `  C
) X ) )
2118, 3oppchom 13861 . . . . . . 7  |-  ( Y (  Hom  `  O
) X )  =  ( X (  Hom  `  C ) Y )
2221eleq2i 2444 . . . . . 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 13864 . . 3  |-  B  =  ( Base `  O
)
29 eqid 2380 . . 3  |-  (  Hom  `  O )  =  (  Hom  `  O )
30 eqid 2380 . . 3  |-  (comp `  O )  =  (comp `  O )
31 eqid 2380 . . 3  |-  ( Id
`  O )  =  ( Id `  O
)
32 oppcsect.t . . 3  |-  T  =  (Sect `  O )
333oppccat 13868 . . . 4  |-  ( C  e.  Cat  ->  O  e.  Cat )
349, 33syl 16 . . 3  |-  ( ph  ->  O  e.  Cat )
3528, 29, 30, 31, 32, 34, 4, 6issect 13899 . 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 13899 . 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 1649    e. wcel 1717   <.cop 3753   class class class wbr 4146   ` cfv 5387  (class class class)co 6013   Basecbs 13389    Hom chom 13460  compcco 13461   Catccat 13809   Idccid 13810  oppCatcoppc 13857  Sectcsect 13890
This theorem is referenced by:  oppcsect2  13920  sectepi  13925  episect  13926
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-cnex 8972  ax-resscn 8973  ax-1cn 8974  ax-icn 8975  ax-addcl 8976  ax-addrcl 8977  ax-mulcl 8978  ax-mulrcl 8979  ax-mulcom 8980  ax-addass 8981  ax-mulass 8982  ax-distr 8983  ax-i2m1 8984  ax-1ne0 8985  ax-1rid 8986  ax-rnegex 8987  ax-rrecex 8988  ax-cnre 8989  ax-pre-lttri 8990  ax-pre-lttrn 8991  ax-pre-ltadd 8992  ax-pre-mulgt0 8993
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 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-reu 2649  df-rmo 2650  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-uni 3951  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-tr 4237  df-eprel 4428  df-id 4432  df-po 4437  df-so 4438  df-fr 4475  df-we 4477  df-ord 4518  df-on 4519  df-lim 4520  df-suc 4521  df-om 4779  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-1st 6281  df-2nd 6282  df-tpos 6408  df-riota 6478  df-recs 6562  df-rdg 6597  df-er 6834  df-en 7039  df-dom 7040  df-sdom 7041  df-pnf 9048  df-mnf 9049  df-xr 9050  df-ltxr 9051  df-le 9052  df-sub 9218  df-neg 9219  df-nn 9926  df-2 9983  df-3 9984  df-4 9985  df-5 9986  df-6 9987  df-7 9988  df-8 9989  df-9 9990  df-10 9991  df-n0 10147  df-z 10208  df-dec 10308  df-ndx 13392  df-slot 13393  df-base 13394  df-sets 13395  df-hom 13473  df-cco 13474  df-cat 13813  df-cid 13814  df-oppc 13858  df-sect 13893
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