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Theorem invss 13913
Description: The inverse relation is a relation between morphisms  F : X --> Y and their inverses  G : Y --> X. (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 )
invss.h  |-  H  =  (  Hom  `  C
)
Assertion
Ref Expression
invss  |-  ( ph  ->  ( X N Y )  C_  ( ( X H Y )  X.  ( Y H X ) ) )

Proof of Theorem invss
StepHypRef Expression
1 invfval.b . . . 4  |-  B  =  ( Base `  C
)
2 invfval.n . . . 4  |-  N  =  (Inv `  C )
3 invfval.c . . . 4  |-  ( ph  ->  C  e.  Cat )
4 invfval.x . . . 4  |-  ( ph  ->  X  e.  B )
5 invfval.y . . . 4  |-  ( ph  ->  Y  e.  B )
6 eqid 2387 . . . 4  |-  (Sect `  C )  =  (Sect `  C )
71, 2, 3, 4, 5, 6invfval 13911 . . 3  |-  ( ph  ->  ( X N Y )  =  ( ( X (Sect `  C
) Y )  i^i  `' ( Y (Sect `  C ) X ) ) )
8 inss1 3504 . . 3  |-  ( ( X (Sect `  C
) Y )  i^i  `' ( Y (Sect `  C ) X ) )  C_  ( X
(Sect `  C ) Y )
97, 8syl6eqss 3341 . 2  |-  ( ph  ->  ( X N Y )  C_  ( X
(Sect `  C ) Y ) )
10 invss.h . . 3  |-  H  =  (  Hom  `  C
)
11 eqid 2387 . . 3  |-  (comp `  C )  =  (comp `  C )
12 eqid 2387 . . 3  |-  ( Id
`  C )  =  ( Id `  C
)
131, 10, 11, 12, 6, 3, 4, 5sectss 13905 . 2  |-  ( ph  ->  ( X (Sect `  C ) Y ) 
C_  ( ( X H Y )  X.  ( Y H X ) ) )
149, 13sstrd 3301 1  |-  ( ph  ->  ( X N Y )  C_  ( ( X H Y )  X.  ( Y H X ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1717    i^i cin 3262    C_ wss 3263    X. cxp 4816   `'ccnv 4817   ` cfv 5394  (class class class)co 6020   Basecbs 13396    Hom chom 13467  compcco 13468   Catccat 13816   Idccid 13817  Sectcsect 13897  Invcinv 13898
This theorem is referenced by:  invsym2  13915  invfun  13916  isohom  13924  invfuc  14098
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
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 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-reu 2656  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-1st 6288  df-2nd 6289  df-sect 13900  df-inv 13901
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