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Theorem funcinv 13990
Description: The image of an inverse under a functor is an inverse. (Contributed by Mario Carneiro, 3-Jan-2017.)
Hypotheses
Ref Expression
funcinv.b  |-  B  =  ( Base `  D
)
funcinv.s  |-  I  =  (Inv `  D )
funcinv.t  |-  J  =  (Inv `  E )
funcinv.f  |-  ( ph  ->  F ( D  Func  E ) G )
funcinv.x  |-  ( ph  ->  X  e.  B )
funcinv.y  |-  ( ph  ->  Y  e.  B )
funcinv.m  |-  ( ph  ->  M ( X I Y ) N )
Assertion
Ref Expression
funcinv  |-  ( ph  ->  ( ( X G Y ) `  M
) ( ( F `
 X ) J ( F `  Y
) ) ( ( Y G X ) `
 N ) )

Proof of Theorem funcinv
StepHypRef Expression
1 funcinv.b . . 3  |-  B  =  ( Base `  D
)
2 eqid 2380 . . 3  |-  (Sect `  D )  =  (Sect `  D )
3 eqid 2380 . . 3  |-  (Sect `  E )  =  (Sect `  E )
4 funcinv.f . . 3  |-  ( ph  ->  F ( D  Func  E ) G )
5 funcinv.x . . 3  |-  ( ph  ->  X  e.  B )
6 funcinv.y . . 3  |-  ( ph  ->  Y  e.  B )
7 funcinv.m . . . . 5  |-  ( ph  ->  M ( X I Y ) N )
8 funcinv.s . . . . . 6  |-  I  =  (Inv `  D )
9 df-br 4147 . . . . . . . . 9  |-  ( F ( D  Func  E
) G  <->  <. F ,  G >.  e.  ( D 
Func  E ) )
104, 9sylib 189 . . . . . . . 8  |-  ( ph  -> 
<. F ,  G >.  e.  ( D  Func  E
) )
11 funcrcl 13980 . . . . . . . 8  |-  ( <. F ,  G >.  e.  ( D  Func  E
)  ->  ( D  e.  Cat  /\  E  e. 
Cat ) )
1210, 11syl 16 . . . . . . 7  |-  ( ph  ->  ( D  e.  Cat  /\  E  e.  Cat )
)
1312simpld 446 . . . . . 6  |-  ( ph  ->  D  e.  Cat )
141, 8, 13, 5, 6, 2isinv 13905 . . . . 5  |-  ( ph  ->  ( M ( X I Y ) N  <-> 
( M ( X (Sect `  D ) Y ) N  /\  N ( Y (Sect `  D ) X ) M ) ) )
157, 14mpbid 202 . . . 4  |-  ( ph  ->  ( M ( X (Sect `  D ) Y ) N  /\  N ( Y (Sect `  D ) X ) M ) )
1615simpld 446 . . 3  |-  ( ph  ->  M ( X (Sect `  D ) Y ) N )
171, 2, 3, 4, 5, 6, 16funcsect 13989 . 2  |-  ( ph  ->  ( ( X G Y ) `  M
) ( ( F `
 X ) (Sect `  E ) ( F `
 Y ) ) ( ( Y G X ) `  N
) )
1815simprd 450 . . 3  |-  ( ph  ->  N ( Y (Sect `  D ) X ) M )
191, 2, 3, 4, 6, 5, 18funcsect 13989 . 2  |-  ( ph  ->  ( ( Y G X ) `  N
) ( ( F `
 Y ) (Sect `  E ) ( F `
 X ) ) ( ( X G Y ) `  M
) )
20 eqid 2380 . . 3  |-  ( Base `  E )  =  (
Base `  E )
21 funcinv.t . . 3  |-  J  =  (Inv `  E )
2212simprd 450 . . 3  |-  ( ph  ->  E  e.  Cat )
231, 20, 4funcf1 13983 . . . 4  |-  ( ph  ->  F : B --> ( Base `  E ) )
2423, 5ffvelrnd 5803 . . 3  |-  ( ph  ->  ( F `  X
)  e.  ( Base `  E ) )
2523, 6ffvelrnd 5803 . . 3  |-  ( ph  ->  ( F `  Y
)  e.  ( Base `  E ) )
2620, 21, 22, 24, 25, 3isinv 13905 . 2  |-  ( ph  ->  ( ( ( X G Y ) `  M ) ( ( F `  X ) J ( F `  Y ) ) ( ( Y G X ) `  N )  <-> 
( ( ( X G Y ) `  M ) ( ( F `  X ) (Sect `  E )
( F `  Y
) ) ( ( Y G X ) `
 N )  /\  ( ( Y G X ) `  N
) ( ( F `
 Y ) (Sect `  E ) ( F `
 X ) ) ( ( X G Y ) `  M
) ) ) )
2717, 19, 26mpbir2and 889 1  |-  ( ph  ->  ( ( X G Y ) `  M
) ( ( F `
 X ) J ( F `  Y
) ) ( ( Y G X ) `
 N ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   <.cop 3753   class class class wbr 4146   ` cfv 5387  (class class class)co 6013   Basecbs 13389   Catccat 13809  Sectcsect 13890  Invcinv 13891    Func cfunc 13971
This theorem is referenced by:  funciso  13991
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-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-map 6949  df-ixp 6993  df-sect 13893  df-inv 13894  df-func 13975
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