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Theorem isinv 13990
Description: Value of the inverse relation. (Contributed by Mario Carneiro, 2-Jan-2017.)
Hypotheses
Ref Expression
invfval.b  |-  B  =  ( Base `  C
)
invfval.n  |-  N  =  (Inv `  C )
invfval.c  |-  ( ph  ->  C  e.  Cat )
invfval.x  |-  ( ph  ->  X  e.  B )
invfval.y  |-  ( ph  ->  Y  e.  B )
invfval.s  |-  S  =  (Sect `  C )
Assertion
Ref Expression
isinv  |-  ( ph  ->  ( F ( X N Y ) G  <-> 
( F ( X S Y ) G  /\  G ( Y S X ) F ) ) )

Proof of Theorem isinv
StepHypRef Expression
1 invfval.b . . . . 5  |-  B  =  ( Base `  C
)
2 invfval.n . . . . 5  |-  N  =  (Inv `  C )
3 invfval.c . . . . 5  |-  ( ph  ->  C  e.  Cat )
4 invfval.x . . . . 5  |-  ( ph  ->  X  e.  B )
5 invfval.y . . . . 5  |-  ( ph  ->  Y  e.  B )
6 invfval.s . . . . 5  |-  S  =  (Sect `  C )
71, 2, 3, 4, 5, 6invfval 13989 . . . 4  |-  ( ph  ->  ( X N Y )  =  ( ( X S Y )  i^i  `' ( Y S X ) ) )
87breqd 4226 . . 3  |-  ( ph  ->  ( F ( X N Y ) G  <-> 
F ( ( X S Y )  i^i  `' ( Y S X ) ) G ) )
9 brin 4262 . . 3  |-  ( F ( ( X S Y )  i^i  `' ( Y S X ) ) G  <->  ( F
( X S Y ) G  /\  F `' ( Y S X ) G ) )
108, 9syl6bb 254 . 2  |-  ( ph  ->  ( F ( X N Y ) G  <-> 
( F ( X S Y ) G  /\  F `' ( Y S X ) G ) ) )
11 eqid 2438 . . . . . 6  |-  (  Hom  `  C )  =  (  Hom  `  C )
12 eqid 2438 . . . . . 6  |-  (comp `  C )  =  (comp `  C )
13 eqid 2438 . . . . . 6  |-  ( Id
`  C )  =  ( Id `  C
)
141, 11, 12, 13, 6, 3, 5, 4sectss 13983 . . . . 5  |-  ( ph  ->  ( Y S X )  C_  ( ( Y (  Hom  `  C
) X )  X.  ( X (  Hom  `  C ) Y ) ) )
15 relxp 4986 . . . . 5  |-  Rel  (
( Y (  Hom  `  C ) X )  X.  ( X (  Hom  `  C ) Y ) )
16 relss 4966 . . . . 5  |-  ( ( Y S X ) 
C_  ( ( Y (  Hom  `  C
) X )  X.  ( X (  Hom  `  C ) Y ) )  ->  ( Rel  ( ( Y (  Hom  `  C ) X )  X.  ( X (  Hom  `  C
) Y ) )  ->  Rel  ( Y S X ) ) )
1714, 15, 16ee10 1386 . . . 4  |-  ( ph  ->  Rel  ( Y S X ) )
18 relbrcnvg 5246 . . . 4  |-  ( Rel  ( Y S X )  ->  ( F `' ( Y S X ) G  <->  G ( Y S X ) F ) )
1917, 18syl 16 . . 3  |-  ( ph  ->  ( F `' ( Y S X ) G  <->  G ( Y S X ) F ) )
2019anbi2d 686 . 2  |-  ( ph  ->  ( ( F ( X S Y ) G  /\  F `' ( Y S X ) G )  <->  ( F
( X S Y ) G  /\  G
( Y S X ) F ) ) )
2110, 20bitrd 246 1  |-  ( ph  ->  ( F ( X N Y ) G  <-> 
( F ( X S Y ) G  /\  G ( Y S X ) F ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    = wceq 1653    e. wcel 1726    i^i cin 3321    C_ wss 3322   class class class wbr 4215    X. cxp 4879   `'ccnv 4880   Rel wrel 4886   ` cfv 5457  (class class class)co 6084   Basecbs 13474    Hom chom 13545  compcco 13546   Catccat 13894   Idccid 13895  Sectcsect 13975  Invcinv 13976
This theorem is referenced by:  invsym  13992  invfun  13994  invco  14001  monsect  14009  funcinv  14075  fthinv  14128  fucinv  14175  invfuc  14176  setcinv  14250  catcisolem  14266  catciso  14267
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-sect 13978  df-inv 13979
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