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

Theorem funcinv 13763
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 2296 . . 3  |-  (Sect `  D )  =  (Sect `  D )
3 eqid 2296 . . 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 4040 . . . . . . . . 9  |-  ( F ( D  Func  E
) G  <->  <. F ,  G >.  e.  ( D 
Func  E ) )
104, 9sylib 188 . . . . . . . 8  |-  ( ph  -> 
<. F ,  G >.  e.  ( D  Func  E
) )
11 funcrcl 13753 . . . . . . . 8  |-  ( <. F ,  G >.  e.  ( D  Func  E
)  ->  ( D  e.  Cat  /\  E  e. 
Cat ) )
1210, 11syl 15 . . . . . . 7  |-  ( ph  ->  ( D  e.  Cat  /\  E  e.  Cat )
)
1312simpld 445 . . . . . 6  |-  ( ph  ->  D  e.  Cat )
141, 8, 13, 5, 6, 2isinv 13678 . . . . 5  |-  ( ph  ->  ( M ( X I Y ) N  <-> 
( M ( X (Sect `  D ) Y ) N  /\  N ( Y (Sect `  D ) X ) M ) ) )
157, 14mpbid 201 . . . 4  |-  ( ph  ->  ( M ( X (Sect `  D ) Y ) N  /\  N ( Y (Sect `  D ) X ) M ) )
1615simpld 445 . . 3  |-  ( ph  ->  M ( X (Sect `  D ) Y ) N )
171, 2, 3, 4, 5, 6, 16funcsect 13762 . 2  |-  ( ph  ->  ( ( X G Y ) `  M
) ( ( F `
 X ) (Sect `  E ) ( F `
 Y ) ) ( ( Y G X ) `  N
) )
1815simprd 449 . . 3  |-  ( ph  ->  N ( Y (Sect `  D ) X ) M )
191, 2, 3, 4, 6, 5, 18funcsect 13762 . 2  |-  ( ph  ->  ( ( Y G X ) `  N
) ( ( F `
 Y ) (Sect `  E ) ( F `
 X ) ) ( ( X G Y ) `  M
) )
20 eqid 2296 . . 3  |-  ( Base `  E )  =  (
Base `  E )
21 funcinv.t . . 3  |-  J  =  (Inv `  E )
2212simprd 449 . . 3  |-  ( ph  ->  E  e.  Cat )
231, 20, 4funcf1 13756 . . . 4  |-  ( ph  ->  F : B --> ( Base `  E ) )
2423, 5ffvelrnd 5682 . . 3  |-  ( ph  ->  ( F `  X
)  e.  ( Base `  E ) )
2523, 6ffvelrnd 5682 . . 3  |-  ( ph  ->  ( F `  Y
)  e.  ( Base `  E ) )
2620, 21, 22, 24, 25, 3isinv 13678 . 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 888 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 358    = wceq 1632    e. wcel 1696   <.cop 3656   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   Basecbs 13164   Catccat 13582  Sectcsect 13663  Invcinv 13664    Func cfunc 13744
This theorem is referenced by:  funciso  13764
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-map 6790  df-ixp 6834  df-sect 13666  df-inv 13667  df-func 13748
  Copyright terms: Public domain W3C validator