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Theorem oppcsect 14800
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 2450 . . . . . 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 14744 . . . . 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 2450 . . . . . . . 8  |-  ( Id
`  C )  =  ( Id `  C
)
123, 11oppcid 14748 . . . . . . 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 5777 . . . . 5  |-  ( (
ph  /\  ( G  e.  ( X ( Hom  `  C ) Y )  /\  F  e.  ( Y ( Hom  `  C
) X ) ) )  ->  ( ( Id `  O ) `  X )  =  ( ( Id `  C
) `  X )
)
158, 14eqeq12d 2471 . . . 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 967 . . . 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 2450 . . . . . . . 8  |-  ( Hom  `  C )  =  ( Hom  `  C )
1918, 3oppchom 14742 . . . . . . 7  |-  ( X ( Hom  `  O
) Y )  =  ( Y ( Hom  `  C ) X )
2019eleq2i 2526 . . . . . 6  |-  ( F  e.  ( X ( Hom  `  O ) Y )  <->  F  e.  ( Y ( Hom  `  C
) X ) )
2118, 3oppchom 14742 . . . . . . 7  |-  ( Y ( Hom  `  O
) X )  =  ( X ( Hom  `  C ) Y )
2221eleq2i 2526 . . . . . 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 967 . . 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 14745 . . 3  |-  B  =  ( Base `  O
)
29 eqid 2450 . . 3  |-  ( Hom  `  O )  =  ( Hom  `  O )
30 eqid 2450 . . 3  |-  (comp `  O )  =  (comp `  O )
31 eqid 2450 . . 3  |-  ( Id
`  O )  =  ( Id `  O
)
32 oppcsect.t . . 3  |-  T  =  (Sect `  O )
333oppccat 14749 . . . 4  |-  ( C  e.  Cat  ->  O  e.  Cat )
349, 33syl 16 . . 3  |-  ( ph  ->  O  e.  Cat )
3528, 29, 30, 31, 32, 34, 4, 6issect 14780 . 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 14780 . 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 965    = wceq 1370    e. wcel 1757   <.cop 3967   class class class wbr 4376   ` cfv 5502  (class class class)co 6176   Basecbs 14262   Hom chom 14337  compcco 14338   Catccat 14690   Idccid 14691  oppCatcoppc 14738  Sectcsect 14771
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1709  ax-7 1729  ax-8 1759  ax-9 1761  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2429  ax-rep 4487  ax-sep 4497  ax-nul 4505  ax-pow 4554  ax-pr 4615  ax-un 6458  ax-cnex 9425  ax-resscn 9426  ax-1cn 9427  ax-icn 9428  ax-addcl 9429  ax-addrcl 9430  ax-mulcl 9431  ax-mulrcl 9432  ax-mulcom 9433  ax-addass 9434  ax-mulass 9435  ax-distr 9436  ax-i2m1 9437  ax-1ne0 9438  ax-1rid 9439  ax-rnegex 9440  ax-rrecex 9441  ax-cnre 9442  ax-pre-lttri 9443  ax-pre-lttrn 9444  ax-pre-ltadd 9445  ax-pre-mulgt0 9446
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1702  df-eu 2263  df-mo 2264  df-clab 2436  df-cleq 2442  df-clel 2445  df-nfc 2598  df-ne 2643  df-nel 2644  df-ral 2797  df-rex 2798  df-reu 2799  df-rmo 2800  df-rab 2801  df-v 3056  df-sbc 3271  df-csb 3373  df-dif 3415  df-un 3417  df-in 3419  df-ss 3426  df-pss 3428  df-nul 3722  df-if 3876  df-pw 3946  df-sn 3962  df-pr 3964  df-tp 3966  df-op 3968  df-uni 4176  df-iun 4257  df-br 4377  df-opab 4435  df-mpt 4436  df-tr 4470  df-eprel 4716  df-id 4720  df-po 4725  df-so 4726  df-fr 4763  df-we 4765  df-ord 4806  df-on 4807  df-lim 4808  df-suc 4809  df-xp 4930  df-rel 4931  df-cnv 4932  df-co 4933  df-dm 4934  df-rn 4935  df-res 4936  df-ima 4937  df-iota 5465  df-fun 5504  df-fn 5505  df-f 5506  df-f1 5507  df-fo 5508  df-f1o 5509  df-fv 5510  df-riota 6137  df-ov 6179  df-oprab 6180  df-mpt2 6181  df-om 6563  df-1st 6663  df-2nd 6664  df-tpos 6831  df-recs 6918  df-rdg 6952  df-er 7187  df-en 7397  df-dom 7398  df-sdom 7399  df-pnf 9507  df-mnf 9508  df-xr 9509  df-ltxr 9510  df-le 9511  df-sub 9684  df-neg 9685  df-nn 10410  df-2 10467  df-3 10468  df-4 10469  df-5 10470  df-6 10471  df-7 10472  df-8 10473  df-9 10474  df-10 10475  df-n0 10667  df-z 10734  df-dec 10843  df-ndx 14265  df-slot 14266  df-base 14267  df-sets 14268  df-hom 14350  df-cco 14351  df-cat 14694  df-cid 14695  df-oppc 14739  df-sect 14774
This theorem is referenced by:  oppcsect2  14801  sectepi  14806  episect  14807
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