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Theorem funcinv 14903
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 4402 . . . . . . . . 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 14893 . . . . . . . 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 459 . . . . . 6  |-  ( ph  ->  D  e.  Cat )
141, 8, 13, 5, 6, 2isinv 14818 . . . . 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 459 . . 3  |-  ( ph  ->  M ( X (Sect `  D ) Y ) N )
171, 2, 3, 4, 5, 6, 16funcsect 14902 . 2  |-  ( ph  ->  ( ( X G Y ) `  M
) ( ( F `
 X ) (Sect `  E ) ( F `
 Y ) ) ( ( Y G X ) `  N
) )
1815simprd 463 . . 3  |-  ( ph  ->  N ( Y (Sect `  D ) X ) M )
191, 2, 3, 4, 6, 5, 18funcsect 14902 . 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 463 . . 3  |-  ( ph  ->  E  e.  Cat )
231, 20, 4funcf1 14896 . . . 4  |-  ( ph  ->  F : B --> ( Base `  E ) )
2423, 5ffvelrnd 5954 . . 3  |-  ( ph  ->  ( F `  X
)  e.  ( Base `  E ) )
2523, 6ffvelrnd 5954 . . 3  |-  ( ph  ->  ( F `  Y
)  e.  ( Base `  E ) )
2620, 21, 22, 24, 25, 3isinv 14818 . 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 913 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 369    = wceq 1370    e. wcel 1758   <.cop 3992   class class class wbr 4401   ` cfv 5527  (class class class)co 6201   Basecbs 14293   Catccat 14722  Sectcsect 14803  Invcinv 14804    Func cfunc 14884
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-id 4745  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-1st 6688  df-2nd 6689  df-map 7327  df-ixp 7375  df-sect 14806  df-inv 14807  df-func 14888
This theorem is referenced by:  funciso  14904
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