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Theorem isinv 13755
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 13754 . . . 4  |-  ( ph  ->  ( X N Y )  =  ( ( X S Y )  i^i  `' ( Y S X ) ) )
87breqd 4113 . . 3  |-  ( ph  ->  ( F ( X N Y ) G  <-> 
F ( ( X S Y )  i^i  `' ( Y S X ) ) G ) )
9 brin 4149 . . 3  |-  ( F ( ( X S Y )  i^i  `' ( Y S X ) ) G  <->  ( F
( X S Y ) G  /\  F `' ( Y S X ) G ) )
108, 9syl6bb 252 . 2  |-  ( ph  ->  ( F ( X N Y ) G  <-> 
( F ( X S Y ) G  /\  F `' ( Y S X ) G ) ) )
11 eqid 2358 . . . . . 6  |-  (  Hom  `  C )  =  (  Hom  `  C )
12 eqid 2358 . . . . . 6  |-  (comp `  C )  =  (comp `  C )
13 eqid 2358 . . . . . 6  |-  ( Id
`  C )  =  ( Id `  C
)
141, 11, 12, 13, 6, 3, 5, 4sectss 13748 . . . . 5  |-  ( ph  ->  ( Y S X )  C_  ( ( Y (  Hom  `  C
) X )  X.  ( X (  Hom  `  C ) Y ) ) )
15 relxp 4873 . . . . 5  |-  Rel  (
( Y (  Hom  `  C ) X )  X.  ( X (  Hom  `  C ) Y ) )
16 relss 4854 . . . . 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 1376 . . . 4  |-  ( ph  ->  Rel  ( Y S X ) )
18 relbrcnvg 5131 . . . 4  |-  ( Rel  ( Y S X )  ->  ( F `' ( Y S X ) G  <->  G ( Y S X ) F ) )
1917, 18syl 15 . . 3  |-  ( ph  ->  ( F `' ( Y S X ) G  <->  G ( Y S X ) F ) )
2019anbi2d 684 . 2  |-  ( ph  ->  ( ( F ( X S Y ) G  /\  F `' ( Y S X ) G )  <->  ( F
( X S Y ) G  /\  G
( Y S X ) F ) ) )
2110, 20bitrd 244 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 176    /\ wa 358    = wceq 1642    e. wcel 1710    i^i cin 3227    C_ wss 3228   class class class wbr 4102    X. cxp 4766   `'ccnv 4767   Rel wrel 4773   ` cfv 5334  (class class class)co 5942   Basecbs 13239    Hom chom 13310  compcco 13311   Catccat 13659   Idccid 13660  Sectcsect 13740  Invcinv 13741
This theorem is referenced by:  invsym  13757  invfun  13759  invco  13766  monsect  13774  funcinv  13840  fthinv  13893  fucinv  13940  invfuc  13941  setcinv  14015  catcisolem  14031  catciso  14032
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 4210  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  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-ral 2624  df-rex 2625  df-reu 2626  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-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3907  df-iun 3986  df-br 4103  df-opab 4157  df-mpt 4158  df-id 4388  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-f1 5339  df-fo 5340  df-f1o 5341  df-fv 5342  df-ov 5945  df-oprab 5946  df-mpt2 5947  df-1st 6206  df-2nd 6207  df-sect 13743  df-inv 13744
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