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Theorem isinv 14719
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 14718 . . . 4  |-  ( ph  ->  ( X N Y )  =  ( ( X S Y )  i^i  `' ( Y S X ) ) )
87breqd 4324 . . 3  |-  ( ph  ->  ( F ( X N Y ) G  <-> 
F ( ( X S Y )  i^i  `' ( Y S X ) ) G ) )
9 brin 4362 . . 3  |-  ( F ( ( X S Y )  i^i  `' ( Y S X ) ) G  <->  ( F
( X S Y ) G  /\  F `' ( Y S X ) G ) )
108, 9syl6bb 261 . 2  |-  ( ph  ->  ( F ( X N Y ) G  <-> 
( F ( X S Y ) G  /\  F `' ( Y S X ) G ) ) )
11 eqid 2443 . . . . . 6  |-  ( Hom  `  C )  =  ( Hom  `  C )
12 eqid 2443 . . . . . 6  |-  (comp `  C )  =  (comp `  C )
13 eqid 2443 . . . . . 6  |-  ( Id
`  C )  =  ( Id `  C
)
141, 11, 12, 13, 6, 3, 5, 4sectss 14712 . . . . 5  |-  ( ph  ->  ( Y S X )  C_  ( ( Y ( Hom  `  C
) X )  X.  ( X ( Hom  `  C ) Y ) ) )
15 relxp 4968 . . . . 5  |-  Rel  (
( Y ( Hom  `  C ) X )  X.  ( X ( Hom  `  C ) Y ) )
16 relss 4948 . . . . 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, 16mpisyl 18 . . . 4  |-  ( ph  ->  Rel  ( Y S X ) )
18 relbrcnvg 5228 . . . 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 703 . 2  |-  ( ph  ->  ( ( F ( X S Y ) G  /\  F `' ( Y S X ) G )  <->  ( F
( X S Y ) G  /\  G
( Y S X ) F ) ) )
2110, 20bitrd 253 1  |-  ( ph  ->  ( F ( X N Y ) G  <-> 
( F ( X S Y ) G  /\  G ( Y S X ) F ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756    i^i cin 3348    C_ wss 3349   class class class wbr 4313    X. cxp 4859   `'ccnv 4860   Rel wrel 4866   ` cfv 5439  (class class class)co 6112   Basecbs 14195   Hom chom 14270  compcco 14271   Catccat 14623   Idccid 14624  Sectcsect 14704  Invcinv 14705
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4424  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2741  df-rex 2742  df-reu 2743  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-op 3905  df-uni 4113  df-iun 4194  df-br 4314  df-opab 4372  df-mpt 4373  df-id 4657  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-1st 6598  df-2nd 6599  df-sect 14707  df-inv 14708
This theorem is referenced by:  invsym  14721  invfun  14723  invco  14730  monsect  14738  funcinv  14804  fthinv  14857  fucinv  14904  invfuc  14905  setcinv  14979  catcisolem  14995  catciso  14996
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