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Theorem funcinv 15361
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 2454 . . 3  |-  (Sect `  D )  =  (Sect `  D )
3 eqid 2454 . . 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 4440 . . . . . . . . 9  |-  ( F ( D  Func  E
) G  <->  <. F ,  G >.  e.  ( D 
Func  E ) )
104, 9sylib 196 . . . . . . . 8  |-  ( ph  -> 
<. F ,  G >.  e.  ( D  Func  E
) )
11 funcrcl 15351 . . . . . . . 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 457 . . . . . 6  |-  ( ph  ->  D  e.  Cat )
141, 8, 13, 5, 6, 2isinv 15248 . . . . 5  |-  ( ph  ->  ( M ( X I Y ) N  <-> 
( M ( X (Sect `  D ) Y ) N  /\  N ( Y (Sect `  D ) X ) M ) ) )
157, 14mpbid 210 . . . 4  |-  ( ph  ->  ( M ( X (Sect `  D ) Y ) N  /\  N ( Y (Sect `  D ) X ) M ) )
1615simpld 457 . . 3  |-  ( ph  ->  M ( X (Sect `  D ) Y ) N )
171, 2, 3, 4, 5, 6, 16funcsect 15360 . 2  |-  ( ph  ->  ( ( X G Y ) `  M
) ( ( F `
 X ) (Sect `  E ) ( F `
 Y ) ) ( ( Y G X ) `  N
) )
1815simprd 461 . . 3  |-  ( ph  ->  N ( Y (Sect `  D ) X ) M )
191, 2, 3, 4, 6, 5, 18funcsect 15360 . 2  |-  ( ph  ->  ( ( Y G X ) `  N
) ( ( F `
 Y ) (Sect `  E ) ( F `
 X ) ) ( ( X G Y ) `  M
) )
20 eqid 2454 . . 3  |-  ( Base `  E )  =  (
Base `  E )
21 funcinv.t . . 3  |-  J  =  (Inv `  E )
2212simprd 461 . . 3  |-  ( ph  ->  E  e.  Cat )
231, 20, 4funcf1 15354 . . . 4  |-  ( ph  ->  F : B --> ( Base `  E ) )
2423, 5ffvelrnd 6008 . . 3  |-  ( ph  ->  ( F `  X
)  e.  ( Base `  E ) )
2523, 6ffvelrnd 6008 . . 3  |-  ( ph  ->  ( F `  Y
)  e.  ( Base `  E ) )
2620, 21, 22, 24, 25, 3isinv 15248 . 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 920 1  |-  ( ph  ->  ( ( X G Y ) `  M
) ( ( F `
 X ) J ( F `  Y
) ) ( ( Y G X ) `
 N ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1398    e. wcel 1823   <.cop 4022   class class class wbr 4439   ` cfv 5570  (class class class)co 6270   Basecbs 14716   Catccat 15153  Sectcsect 15232  Invcinv 15233    Func cfunc 15342
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-1st 6773  df-2nd 6774  df-map 7414  df-ixp 7463  df-sect 15235  df-inv 15236  df-func 15346
This theorem is referenced by:  funciso  15362
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