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Theorem isinv 13944
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 13943 . . . 4  |-  ( ph  ->  ( X N Y )  =  ( ( X S Y )  i^i  `' ( Y S X ) ) )
87breqd 4187 . . 3  |-  ( ph  ->  ( F ( X N Y ) G  <-> 
F ( ( X S Y )  i^i  `' ( Y S X ) ) G ) )
9 brin 4223 . . 3  |-  ( F ( ( X S Y )  i^i  `' ( Y S X ) ) G  <->  ( F
( X S Y ) G  /\  F `' ( Y S X ) G ) )
108, 9syl6bb 253 . 2  |-  ( ph  ->  ( F ( X N Y ) G  <-> 
( F ( X S Y ) G  /\  F `' ( Y S X ) G ) ) )
11 eqid 2408 . . . . . 6  |-  (  Hom  `  C )  =  (  Hom  `  C )
12 eqid 2408 . . . . . 6  |-  (comp `  C )  =  (comp `  C )
13 eqid 2408 . . . . . 6  |-  ( Id
`  C )  =  ( Id `  C
)
141, 11, 12, 13, 6, 3, 5, 4sectss 13937 . . . . 5  |-  ( ph  ->  ( Y S X )  C_  ( ( Y (  Hom  `  C
) X )  X.  ( X (  Hom  `  C ) Y ) ) )
15 relxp 4946 . . . . 5  |-  Rel  (
( Y (  Hom  `  C ) X )  X.  ( X (  Hom  `  C ) Y ) )
16 relss 4926 . . . . 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 1382 . . . 4  |-  ( ph  ->  Rel  ( Y S X ) )
18 relbrcnvg 5206 . . . 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 685 . 2  |-  ( ph  ->  ( ( F ( X S Y ) G  /\  F `' ( Y S X ) G )  <->  ( F
( X S Y ) G  /\  G
( Y S X ) F ) ) )
2110, 20bitrd 245 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 177    /\ wa 359    = wceq 1649    e. wcel 1721    i^i cin 3283    C_ wss 3284   class class class wbr 4176    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  Sectcsect 13929  Invcinv 13930
This theorem is referenced by:  invsym  13946  invfun  13948  invco  13955  monsect  13963  funcinv  14029  fthinv  14082  fucinv  14129  invfuc  14130  setcinv  14204  catcisolem  14220  catciso  14221
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
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  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-ral 2675  df-rex 2676  df-reu 2677  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-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-id 4462  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-sect 13932  df-inv 13933
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