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Theorem oppcsect 13692
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 2296 . . . . . 6  |-  (comp `  C )  =  (comp `  C )
3 oppcsect.o . . . . . 6  |-  O  =  (oppCat `  C )
4 oppcsect.x . . . . . . 7  |-  ( ph  ->  X  e.  B )
54adantr 451 . . . . . 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 451 . . . . . 6  |-  ( (
ph  /\  ( G  e.  ( X (  Hom  `  C ) Y )  /\  F  e.  ( Y (  Hom  `  C
) X ) ) )  ->  Y  e.  B )
81, 2, 3, 5, 7, 5oppcco 13636 . . . . 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 451 . . . . . . 7  |-  ( (
ph  /\  ( G  e.  ( X (  Hom  `  C ) Y )  /\  F  e.  ( Y (  Hom  `  C
) X ) ) )  ->  C  e.  Cat )
11 eqid 2296 . . . . . . . 8  |-  ( Id
`  C )  =  ( Id `  C
)
123, 11oppcid 13640 . . . . . . 7  |-  ( C  e.  Cat  ->  ( Id `  O )  =  ( Id `  C
) )
1310, 12syl 15 . . . . . 6  |-  ( (
ph  /\  ( G  e.  ( X (  Hom  `  C ) Y )  /\  F  e.  ( Y (  Hom  `  C
) X ) ) )  ->  ( Id `  O )  =  ( Id `  C ) )
1413fveq1d 5543 . . . . 5  |-  ( (
ph  /\  ( G  e.  ( X (  Hom  `  C ) Y )  /\  F  e.  ( Y (  Hom  `  C
) X ) ) )  ->  ( ( Id `  O ) `  X )  =  ( ( Id `  C
) `  X )
)
158, 14eqeq12d 2310 . . . 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 622 . . 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 936 . . . 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 2296 . . . . . . . 8  |-  (  Hom  `  C )  =  (  Hom  `  C )
1918, 3oppchom 13634 . . . . . . 7  |-  ( X (  Hom  `  O
) Y )  =  ( Y (  Hom  `  C ) X )
2019eleq2i 2360 . . . . . 6  |-  ( F  e.  ( X (  Hom  `  O ) Y )  <->  F  e.  ( Y (  Hom  `  C
) X ) )
2118, 3oppchom 13634 . . . . . . 7  |-  ( Y (  Hom  `  O
) X )  =  ( X (  Hom  `  C ) Y )
2221eleq2i 2360 . . . . . 6  |-  ( G  e.  ( Y (  Hom  `  O ) X )  <->  G  e.  ( X (  Hom  `  C
) Y ) )
2320, 22anbi12ci 679 . . . . 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 676 . . . 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 240 . . 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 936 . . 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 279 . 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 13637 . . 3  |-  B  =  ( Base `  O
)
29 eqid 2296 . . 3  |-  (  Hom  `  O )  =  (  Hom  `  O )
30 eqid 2296 . . 3  |-  (comp `  O )  =  (comp `  O )
31 eqid 2296 . . 3  |-  ( Id
`  O )  =  ( Id `  O
)
32 oppcsect.t . . 3  |-  T  =  (Sect `  O )
333oppccat 13641 . . . 4  |-  ( C  e.  Cat  ->  O  e.  Cat )
349, 33syl 15 . . 3  |-  ( ph  ->  O  e.  Cat )
3528, 29, 30, 31, 32, 34, 4, 6issect 13672 . 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 13672 . 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 276 1  |-  ( ph  ->  ( F ( X T Y ) G  <-> 
G ( X S Y ) F ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   <.cop 3656   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   Basecbs 13164    Hom chom 13235  compcco 13236   Catccat 13582   Idccid 13583  oppCatcoppc 13630  Sectcsect 13663
This theorem is referenced by:  oppcsect2  13693  sectepi  13698  episect  13699
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-tpos 6250  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-3 9821  df-4 9822  df-5 9823  df-6 9824  df-7 9825  df-8 9826  df-9 9827  df-10 9828  df-n0 9982  df-z 10041  df-dec 10141  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-hom 13248  df-cco 13249  df-cat 13586  df-cid 13587  df-oppc 13631  df-sect 13666
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