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Theorem oppcinv 13777
Description: An inverse 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 )
oppcinv.s  |-  I  =  (Inv `  C )
oppcinv.t  |-  J  =  (Inv `  O )
Assertion
Ref Expression
oppcinv  |-  ( ph  ->  ( X J Y )  =  ( Y I X ) )

Proof of Theorem oppcinv
StepHypRef Expression
1 incom 3437 . . 3  |-  ( ( X (Sect `  O
) Y )  i^i  `' ( Y (Sect `  O ) X ) )  =  ( `' ( Y (Sect `  O ) X )  i^i  ( X (Sect `  O ) Y ) )
2 oppcsect.b . . . . . . 7  |-  B  =  ( Base `  C
)
3 oppcsect.o . . . . . . 7  |-  O  =  (oppCat `  C )
4 oppcsect.c . . . . . . 7  |-  ( ph  ->  C  e.  Cat )
5 oppcsect.y . . . . . . 7  |-  ( ph  ->  Y  e.  B )
6 oppcsect.x . . . . . . 7  |-  ( ph  ->  X  e.  B )
7 eqid 2358 . . . . . . 7  |-  (Sect `  C )  =  (Sect `  C )
8 eqid 2358 . . . . . . 7  |-  (Sect `  O )  =  (Sect `  O )
92, 3, 4, 5, 6, 7, 8oppcsect2 13776 . . . . . 6  |-  ( ph  ->  ( Y (Sect `  O ) X )  =  `' ( Y (Sect `  C ) X ) )
109cnveqd 4939 . . . . 5  |-  ( ph  ->  `' ( Y (Sect `  O ) X )  =  `' `' ( Y (Sect `  C
) X ) )
11 eqid 2358 . . . . . . . 8  |-  (  Hom  `  C )  =  (  Hom  `  C )
12 eqid 2358 . . . . . . . 8  |-  (comp `  C )  =  (comp `  C )
13 eqid 2358 . . . . . . . 8  |-  ( Id
`  C )  =  ( Id `  C
)
142, 11, 12, 13, 7, 4, 5, 6sectss 13754 . . . . . . 7  |-  ( ph  ->  ( Y (Sect `  C ) X ) 
C_  ( ( Y (  Hom  `  C
) X )  X.  ( X (  Hom  `  C ) Y ) ) )
15 relxp 4876 . . . . . . 7  |-  Rel  (
( Y (  Hom  `  C ) X )  X.  ( X (  Hom  `  C ) Y ) )
16 relss 4857 . . . . . . 7  |-  ( ( Y (Sect `  C
) X )  C_  ( ( Y (  Hom  `  C ) X )  X.  ( X (  Hom  `  C
) Y ) )  ->  ( Rel  (
( Y (  Hom  `  C ) X )  X.  ( X (  Hom  `  C ) Y ) )  ->  Rel  ( Y (Sect `  C ) X ) ) )
1714, 15, 16ee10 1376 . . . . . 6  |-  ( ph  ->  Rel  ( Y (Sect `  C ) X ) )
18 dfrel2 5206 . . . . . 6  |-  ( Rel  ( Y (Sect `  C ) X )  <->  `' `' ( Y (Sect `  C ) X )  =  ( Y (Sect `  C ) X ) )
1917, 18sylib 188 . . . . 5  |-  ( ph  ->  `' `' ( Y (Sect `  C ) X )  =  ( Y (Sect `  C ) X ) )
2010, 19eqtrd 2390 . . . 4  |-  ( ph  ->  `' ( Y (Sect `  O ) X )  =  ( Y (Sect `  C ) X ) )
212, 3, 4, 6, 5, 7, 8oppcsect2 13776 . . . 4  |-  ( ph  ->  ( X (Sect `  O ) Y )  =  `' ( X (Sect `  C ) Y ) )
2220, 21ineq12d 3447 . . 3  |-  ( ph  ->  ( `' ( Y (Sect `  O ) X )  i^i  ( X (Sect `  O ) Y ) )  =  ( ( Y (Sect `  C ) X )  i^i  `' ( X (Sect `  C ) Y ) ) )
231, 22syl5eq 2402 . 2  |-  ( ph  ->  ( ( X (Sect `  O ) Y )  i^i  `' ( Y (Sect `  O ) X ) )  =  ( ( Y (Sect `  C ) X )  i^i  `' ( X (Sect `  C ) Y ) ) )
243, 2oppcbas 13720 . . 3  |-  B  =  ( Base `  O
)
25 oppcinv.t . . 3  |-  J  =  (Inv `  O )
263oppccat 13724 . . . 4  |-  ( C  e.  Cat  ->  O  e.  Cat )
274, 26syl 15 . . 3  |-  ( ph  ->  O  e.  Cat )
2824, 25, 27, 6, 5, 8invfval 13760 . 2  |-  ( ph  ->  ( X J Y )  =  ( ( X (Sect `  O
) Y )  i^i  `' ( Y (Sect `  O ) X ) ) )
29 oppcinv.s . . 3  |-  I  =  (Inv `  C )
302, 29, 4, 5, 6, 7invfval 13760 . 2  |-  ( ph  ->  ( Y I X )  =  ( ( Y (Sect `  C
) X )  i^i  `' ( X (Sect `  C ) Y ) ) )
3123, 28, 303eqtr4d 2400 1  |-  ( ph  ->  ( X J Y )  =  ( Y I X ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1642    e. wcel 1710    i^i cin 3227    C_ wss 3228    X. cxp 4769   `'ccnv 4770   Rel wrel 4776   ` cfv 5337  (class class class)co 5945   Basecbs 13245    Hom chom 13316  compcco 13317   Catccat 13665   Idccid 13666  oppCatcoppc 13713  Sectcsect 13746  Invcinv 13747
This theorem is referenced by:  oppciso  13778  episect  13782
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4212  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594  ax-cnex 8883  ax-resscn 8884  ax-1cn 8885  ax-icn 8886  ax-addcl 8887  ax-addrcl 8888  ax-mulcl 8889  ax-mulrcl 8890  ax-mulcom 8891  ax-addass 8892  ax-mulass 8893  ax-distr 8894  ax-i2m1 8895  ax-1ne0 8896  ax-1rid 8897  ax-rnegex 8898  ax-rrecex 8899  ax-cnre 8900  ax-pre-lttri 8901  ax-pre-lttrn 8902  ax-pre-ltadd 8903  ax-pre-mulgt0 8904
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3909  df-iun 3988  df-br 4105  df-opab 4159  df-mpt 4160  df-tr 4195  df-eprel 4387  df-id 4391  df-po 4396  df-so 4397  df-fr 4434  df-we 4436  df-ord 4477  df-on 4478  df-lim 4479  df-suc 4480  df-om 4739  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-1st 6209  df-2nd 6210  df-tpos 6321  df-riota 6391  df-recs 6475  df-rdg 6510  df-er 6747  df-en 6952  df-dom 6953  df-sdom 6954  df-pnf 8959  df-mnf 8960  df-xr 8961  df-ltxr 8962  df-le 8963  df-sub 9129  df-neg 9130  df-nn 9837  df-2 9894  df-3 9895  df-4 9896  df-5 9897  df-6 9898  df-7 9899  df-8 9900  df-9 9901  df-10 9902  df-n0 10058  df-z 10117  df-dec 10217  df-ndx 13248  df-slot 13249  df-base 13250  df-sets 13251  df-hom 13329  df-cco 13330  df-cat 13669  df-cid 13670  df-oppc 13714  df-sect 13749  df-inv 13750
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