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Theorem oppcsect 15022
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 2467 . . . . . 6  |-  (comp `  C )  =  (comp `  C )
3 oppcsect.o . . . . . 6  |-  O  =  (oppCat `  C )
4 oppcsect.x . . . . . . 7  |-  ( ph  ->  X  e.  B )
54adantr 465 . . . . . 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 465 . . . . . 6  |-  ( (
ph  /\  ( G  e.  ( X ( Hom  `  C ) Y )  /\  F  e.  ( Y ( Hom  `  C
) X ) ) )  ->  Y  e.  B )
81, 2, 3, 5, 7, 5oppcco 14966 . . . . 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 465 . . . . . . 7  |-  ( (
ph  /\  ( G  e.  ( X ( Hom  `  C ) Y )  /\  F  e.  ( Y ( Hom  `  C
) X ) ) )  ->  C  e.  Cat )
11 eqid 2467 . . . . . . . 8  |-  ( Id
`  C )  =  ( Id `  C
)
123, 11oppcid 14970 . . . . . . 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 5866 . . . . 5  |-  ( (
ph  /\  ( G  e.  ( X ( Hom  `  C ) Y )  /\  F  e.  ( Y ( Hom  `  C
) X ) ) )  ->  ( ( Id `  O ) `  X )  =  ( ( Id `  C
) `  X )
)
158, 14eqeq12d 2489 . . . 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 641 . . 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 975 . . . 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 2467 . . . . . . . 8  |-  ( Hom  `  C )  =  ( Hom  `  C )
1918, 3oppchom 14964 . . . . . . 7  |-  ( X ( Hom  `  O
) Y )  =  ( Y ( Hom  `  C ) X )
2019eleq2i 2545 . . . . . 6  |-  ( F  e.  ( X ( Hom  `  O ) Y )  <->  F  e.  ( Y ( Hom  `  C
) X ) )
2118, 3oppchom 14964 . . . . . . 7  |-  ( Y ( Hom  `  O
) X )  =  ( X ( Hom  `  C ) Y )
2221eleq2i 2545 . . . . . 6  |-  ( G  e.  ( Y ( Hom  `  O ) X )  <->  G  e.  ( X ( Hom  `  C
) Y ) )
2320, 22anbi12ci 698 . . . . 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 695 . . . 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 249 . . 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 975 . . 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 288 . 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 14967 . . 3  |-  B  =  ( Base `  O
)
29 eqid 2467 . . 3  |-  ( Hom  `  O )  =  ( Hom  `  O )
30 eqid 2467 . . 3  |-  (comp `  O )  =  (comp `  O )
31 eqid 2467 . . 3  |-  ( Id
`  O )  =  ( Id `  O
)
32 oppcsect.t . . 3  |-  T  =  (Sect `  O )
333oppccat 14971 . . . 4  |-  ( C  e.  Cat  ->  O  e.  Cat )
349, 33syl 16 . . 3  |-  ( ph  ->  O  e.  Cat )
3528, 29, 30, 31, 32, 34, 4, 6issect 15002 . 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 15002 . 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 285 1  |-  ( ph  ->  ( F ( X T Y ) G  <-> 
G ( X S Y ) F ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   <.cop 4033   class class class wbr 4447   ` cfv 5586  (class class class)co 6282   Basecbs 14483   Hom chom 14559  compcco 14560   Catccat 14912   Idccid 14913  oppCatcoppc 14960  Sectcsect 14993
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-tpos 6952  df-recs 7039  df-rdg 7073  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-nn 10533  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-10 10598  df-n0 10792  df-z 10861  df-dec 10973  df-ndx 14486  df-slot 14487  df-base 14488  df-sets 14489  df-hom 14572  df-cco 14573  df-cat 14916  df-cid 14917  df-oppc 14961  df-sect 14996
This theorem is referenced by:  oppcsect2  15023  sectepi  15028  episect  15029
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