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Theorem invss 15617
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 2429 . . . 4  |-  (Sect `  C )  =  (Sect `  C )
71, 2, 3, 4, 5, 6invfval 15615 . . 3  |-  ( ph  ->  ( X N Y )  =  ( ( X (Sect `  C
) Y )  i^i  `' ( Y (Sect `  C ) X ) ) )
8 inss1 3688 . . 3  |-  ( ( X (Sect `  C
) Y )  i^i  `' ( Y (Sect `  C ) X ) )  C_  ( X
(Sect `  C ) Y )
97, 8syl6eqss 3520 . 2  |-  ( ph  ->  ( X N Y )  C_  ( X
(Sect `  C ) Y ) )
10 invss.h . . 3  |-  H  =  ( Hom  `  C
)
11 eqid 2429 . . 3  |-  (comp `  C )  =  (comp `  C )
12 eqid 2429 . . 3  |-  ( Id
`  C )  =  ( Id `  C
)
131, 10, 11, 12, 6, 3, 4, 5sectss 15608 . 2  |-  ( ph  ->  ( X (Sect `  C ) Y ) 
C_  ( ( X H Y )  X.  ( Y H X ) ) )
149, 13sstrd 3480 1  |-  ( ph  ->  ( X N Y )  C_  ( ( X H Y )  X.  ( Y H X ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1437    e. wcel 1870    i^i cin 3441    C_ wss 3442    X. cxp 4852   `'ccnv 4853   ` cfv 5601  (class class class)co 6305   Basecbs 15084   Hom chom 15163  compcco 15164   Catccat 15521   Idccid 15522  Sectcsect 15600  Invcinv 15601
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-reu 2789  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-1st 6807  df-2nd 6808  df-sect 15603  df-inv 15604
This theorem is referenced by:  invsym2  15619  invfun  15620  isohom  15632  invfuc  15830
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