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Theorem oppcsect 15194
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 2457 . . . . . 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 15133 . . . . 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 2457 . . . . . . . 8  |-  ( Id
`  C )  =  ( Id `  C
)
123, 11oppcid 15137 . . . . . . 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 5874 . . . . 5  |-  ( (
ph  /\  ( G  e.  ( X ( Hom  `  C ) Y )  /\  F  e.  ( Y ( Hom  `  C
) X ) ) )  ->  ( ( Id `  O ) `  X )  =  ( ( Id `  C
) `  X )
)
158, 14eqeq12d 2479 . . . 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 2457 . . . . . . . 8  |-  ( Hom  `  C )  =  ( Hom  `  C )
1918, 3oppchom 15131 . . . . . . 7  |-  ( X ( Hom  `  O
) Y )  =  ( Y ( Hom  `  C ) X )
2019eleq2i 2535 . . . . . 6  |-  ( F  e.  ( X ( Hom  `  O ) Y )  <->  F  e.  ( Y ( Hom  `  C
) X ) )
2118, 3oppchom 15131 . . . . . . 7  |-  ( Y ( Hom  `  O
) X )  =  ( X ( Hom  `  C ) Y )
2221eleq2i 2535 . . . . . 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 15134 . . 3  |-  B  =  ( Base `  O
)
29 eqid 2457 . . 3  |-  ( Hom  `  O )  =  ( Hom  `  O )
30 eqid 2457 . . 3  |-  (comp `  O )  =  (comp `  O )
31 eqid 2457 . . 3  |-  ( Id
`  O )  =  ( Id `  O
)
32 oppcsect.t . . 3  |-  T  =  (Sect `  O )
333oppccat 15138 . . . 4  |-  ( C  e.  Cat  ->  O  e.  Cat )
349, 33syl 16 . . 3  |-  ( ph  ->  O  e.  Cat )
3528, 29, 30, 31, 32, 34, 4, 6issect 15169 . 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 15169 . 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 1395    e. wcel 1819   <.cop 4038   class class class wbr 4456   ` cfv 5594  (class class class)co 6296   Basecbs 14644   Hom chom 14723  compcco 14724   Catccat 15081   Idccid 15082  oppCatcoppc 15127  Sectcsect 15160
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-tpos 6973  df-recs 7060  df-rdg 7094  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-2 10615  df-3 10616  df-4 10617  df-5 10618  df-6 10619  df-7 10620  df-8 10621  df-9 10622  df-10 10623  df-n0 10817  df-z 10886  df-dec 11001  df-ndx 14647  df-slot 14648  df-base 14649  df-sets 14650  df-hom 14736  df-cco 14737  df-cat 15085  df-cid 15086  df-oppc 15128  df-sect 15163
This theorem is referenced by:  oppcsect2  15195  sectepi  15200  episect  15201
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