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Theorem invss 15016
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 2467 . . . 4  |-  (Sect `  C )  =  (Sect `  C )
71, 2, 3, 4, 5, 6invfval 15014 . . 3  |-  ( ph  ->  ( X N Y )  =  ( ( X (Sect `  C
) Y )  i^i  `' ( Y (Sect `  C ) X ) ) )
8 inss1 3718 . . 3  |-  ( ( X (Sect `  C
) Y )  i^i  `' ( Y (Sect `  C ) X ) )  C_  ( X
(Sect `  C ) Y )
97, 8syl6eqss 3554 . 2  |-  ( ph  ->  ( X N Y )  C_  ( X
(Sect `  C ) Y ) )
10 invss.h . . 3  |-  H  =  ( Hom  `  C
)
11 eqid 2467 . . 3  |-  (comp `  C )  =  (comp `  C )
12 eqid 2467 . . 3  |-  ( Id
`  C )  =  ( Id `  C
)
131, 10, 11, 12, 6, 3, 4, 5sectss 15008 . 2  |-  ( ph  ->  ( X (Sect `  C ) Y ) 
C_  ( ( X H Y )  X.  ( Y H X ) ) )
149, 13sstrd 3514 1  |-  ( ph  ->  ( X N Y )  C_  ( ( X H Y )  X.  ( Y H X ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379    e. wcel 1767    i^i cin 3475    C_ wss 3476    X. cxp 4997   `'ccnv 4998   ` cfv 5588  (class class class)co 6284   Basecbs 14490   Hom chom 14566  compcco 14567   Catccat 14919   Idccid 14920  Sectcsect 15000  Invcinv 15001
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-1st 6784  df-2nd 6785  df-sect 15003  df-inv 15004
This theorem is referenced by:  invsym2  15018  invfun  15019  isohom  15027  invfuc  15201
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