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Theorem oppcinv 13960
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 3497 . . 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 2408 . . . . . . 7  |-  (Sect `  C )  =  (Sect `  C )
8 eqid 2408 . . . . . . 7  |-  (Sect `  O )  =  (Sect `  O )
92, 3, 4, 5, 6, 7, 8oppcsect2 13959 . . . . . 6  |-  ( ph  ->  ( Y (Sect `  O ) X )  =  `' ( Y (Sect `  C ) X ) )
109cnveqd 5011 . . . . 5  |-  ( ph  ->  `' ( Y (Sect `  O ) X )  =  `' `' ( Y (Sect `  C
) X ) )
11 eqid 2408 . . . . . . . 8  |-  (  Hom  `  C )  =  (  Hom  `  C )
12 eqid 2408 . . . . . . . 8  |-  (comp `  C )  =  (comp `  C )
13 eqid 2408 . . . . . . . 8  |-  ( Id
`  C )  =  ( Id `  C
)
142, 11, 12, 13, 7, 4, 5, 6sectss 13937 . . . . . . 7  |-  ( ph  ->  ( Y (Sect `  C ) X ) 
C_  ( ( Y (  Hom  `  C
) X )  X.  ( X (  Hom  `  C ) Y ) ) )
15 relxp 4946 . . . . . . 7  |-  Rel  (
( Y (  Hom  `  C ) X )  X.  ( X (  Hom  `  C ) Y ) )
16 relss 4926 . . . . . . 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 1382 . . . . . 6  |-  ( ph  ->  Rel  ( Y (Sect `  C ) X ) )
18 dfrel2 5284 . . . . . 6  |-  ( Rel  ( Y (Sect `  C ) X )  <->  `' `' ( Y (Sect `  C ) X )  =  ( Y (Sect `  C ) X ) )
1917, 18sylib 189 . . . . 5  |-  ( ph  ->  `' `' ( Y (Sect `  C ) X )  =  ( Y (Sect `  C ) X ) )
2010, 19eqtrd 2440 . . . 4  |-  ( ph  ->  `' ( Y (Sect `  O ) X )  =  ( Y (Sect `  C ) X ) )
212, 3, 4, 6, 5, 7, 8oppcsect2 13959 . . . 4  |-  ( ph  ->  ( X (Sect `  O ) Y )  =  `' ( X (Sect `  C ) Y ) )
2220, 21ineq12d 3507 . . 3  |-  ( ph  ->  ( `' ( Y (Sect `  O ) X )  i^i  ( X (Sect `  O ) Y ) )  =  ( ( Y (Sect `  C ) X )  i^i  `' ( X (Sect `  C ) Y ) ) )
231, 22syl5eq 2452 . 2  |-  ( ph  ->  ( ( X (Sect `  O ) Y )  i^i  `' ( Y (Sect `  O ) X ) )  =  ( ( Y (Sect `  C ) X )  i^i  `' ( X (Sect `  C ) Y ) ) )
243, 2oppcbas 13903 . . 3  |-  B  =  ( Base `  O
)
25 oppcinv.t . . 3  |-  J  =  (Inv `  O )
263oppccat 13907 . . . 4  |-  ( C  e.  Cat  ->  O  e.  Cat )
274, 26syl 16 . . 3  |-  ( ph  ->  O  e.  Cat )
2824, 25, 27, 6, 5, 8invfval 13943 . 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 13943 . 2  |-  ( ph  ->  ( Y I X )  =  ( ( Y (Sect `  C
) X )  i^i  `' ( X (Sect `  C ) Y ) ) )
3123, 28, 303eqtr4d 2450 1  |-  ( ph  ->  ( X J Y )  =  ( Y I X ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1721    i^i cin 3283    C_ wss 3284    X. cxp 4839   `'ccnv 4840   Rel wrel 4846   ` cfv 5417  (class class class)co 6044   Basecbs 13428    Hom chom 13499  compcco 13500   Catccat 13848   Idccid 13849  oppCatcoppc 13896  Sectcsect 13929  Invcinv 13930
This theorem is referenced by:  oppciso  13961  episect  13965
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664  ax-cnex 9006  ax-resscn 9007  ax-1cn 9008  ax-icn 9009  ax-addcl 9010  ax-addrcl 9011  ax-mulcl 9012  ax-mulrcl 9013  ax-mulcom 9014  ax-addass 9015  ax-mulass 9016  ax-distr 9017  ax-i2m1 9018  ax-1ne0 9019  ax-1rid 9020  ax-rnegex 9021  ax-rrecex 9022  ax-cnre 9023  ax-pre-lttri 9024  ax-pre-lttrn 9025  ax-pre-ltadd 9026  ax-pre-mulgt0 9027
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-nel 2574  df-ral 2675  df-rex 2676  df-reu 2677  df-rmo 2678  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-pss 3300  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-tp 3786  df-op 3787  df-uni 3980  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-tr 4267  df-eprel 4458  df-id 4462  df-po 4467  df-so 4468  df-fr 4505  df-we 4507  df-ord 4548  df-on 4549  df-lim 4550  df-suc 4551  df-om 4809  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-1st 6312  df-2nd 6313  df-tpos 6442  df-riota 6512  df-recs 6596  df-rdg 6631  df-er 6868  df-en 7073  df-dom 7074  df-sdom 7075  df-pnf 9082  df-mnf 9083  df-xr 9084  df-ltxr 9085  df-le 9086  df-sub 9253  df-neg 9254  df-nn 9961  df-2 10018  df-3 10019  df-4 10020  df-5 10021  df-6 10022  df-7 10023  df-8 10024  df-9 10025  df-10 10026  df-n0 10182  df-z 10243  df-dec 10343  df-ndx 13431  df-slot 13432  df-base 13433  df-sets 13434  df-hom 13512  df-cco 13513  df-cat 13852  df-cid 13853  df-oppc 13897  df-sect 13932  df-inv 13933
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