MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  invinv Unicode version

Theorem invinv 13915
Description: The inverse of the inverse of an isomorphism is itself. Proposition 3.14(1) of [Adamek] p. 29. (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 )
isoval.n  |-  I  =  (  Iso  `  C
)
invinv.f  |-  ( ph  ->  F  e.  ( X I Y ) )
Assertion
Ref Expression
invinv  |-  ( ph  ->  ( ( Y N X ) `  (
( X N Y ) `  F ) )  =  F )

Proof of Theorem invinv
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 )
61, 2, 3, 4, 5invsym2 13908 . . 3  |-  ( ph  ->  `' ( X N Y )  =  ( Y N X ) )
76fveq1d 5663 . 2  |-  ( ph  ->  ( `' ( X N Y ) `  ( ( X N Y ) `  F
) )  =  ( ( Y N X ) `  ( ( X N Y ) `
 F ) ) )
8 isoval.n . . . 4  |-  I  =  (  Iso  `  C
)
91, 2, 3, 4, 5, 8invf1o 13914 . . 3  |-  ( ph  ->  ( X N Y ) : ( X I Y ) -1-1-onto-> ( Y I X ) )
10 invinv.f . . 3  |-  ( ph  ->  F  e.  ( X I Y ) )
11 f1ocnvfv1 5946 . . 3  |-  ( ( ( X N Y ) : ( X I Y ) -1-1-onto-> ( Y I X )  /\  F  e.  ( X I Y ) )  -> 
( `' ( X N Y ) `  ( ( X N Y ) `  F
) )  =  F )
129, 10, 11syl2anc 643 . 2  |-  ( ph  ->  ( `' ( X N Y ) `  ( ( X N Y ) `  F
) )  =  F )
137, 12eqtr3d 2414 1  |-  ( ph  ->  ( ( Y N X ) `  (
( X N Y ) `  F ) )  =  F )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1717   `'ccnv 4810   -1-1-onto->wf1o 5386   ` cfv 5387  (class class class)co 6013   Basecbs 13389   Catccat 13809  Invcinv 13891    Iso ciso 13892
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 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634
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 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-reu 2649  df-rmo 2650  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-id 4432  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-1st 6281  df-2nd 6282  df-riota 6478  df-cat 13813  df-cid 13814  df-sect 13893  df-inv 13894  df-iso 13895
  Copyright terms: Public domain W3C validator