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Theorem funciso 13748
Description: The image of an isomorphism under a functor is an isomorphism. (Contributed by Mario Carneiro, 3-Jan-2017.)
Hypotheses
Ref Expression
funciso.b  |-  B  =  ( Base `  D
)
funciso.s  |-  I  =  (  Iso  `  D
)
funciso.t  |-  J  =  (  Iso  `  E
)
funciso.f  |-  ( ph  ->  F ( D  Func  E ) G )
funciso.x  |-  ( ph  ->  X  e.  B )
funciso.y  |-  ( ph  ->  Y  e.  B )
funciso.m  |-  ( ph  ->  M  e.  ( X I Y ) )
Assertion
Ref Expression
funciso  |-  ( ph  ->  ( ( X G Y ) `  M
)  e.  ( ( F `  X ) J ( F `  Y ) ) )

Proof of Theorem funciso
StepHypRef Expression
1 eqid 2283 . 2  |-  ( Base `  E )  =  (
Base `  E )
2 eqid 2283 . 2  |-  (Inv `  E )  =  (Inv
`  E )
3 funciso.f . . . . 5  |-  ( ph  ->  F ( D  Func  E ) G )
4 df-br 4024 . . . . 5  |-  ( F ( D  Func  E
) G  <->  <. F ,  G >.  e.  ( D 
Func  E ) )
53, 4sylib 188 . . . 4  |-  ( ph  -> 
<. F ,  G >.  e.  ( D  Func  E
) )
6 funcrcl 13737 . . . 4  |-  ( <. F ,  G >.  e.  ( D  Func  E
)  ->  ( D  e.  Cat  /\  E  e. 
Cat ) )
75, 6syl 15 . . 3  |-  ( ph  ->  ( D  e.  Cat  /\  E  e.  Cat )
)
87simprd 449 . 2  |-  ( ph  ->  E  e.  Cat )
9 funciso.b . . . 4  |-  B  =  ( Base `  D
)
109, 1, 3funcf1 13740 . . 3  |-  ( ph  ->  F : B --> ( Base `  E ) )
11 funciso.x . . 3  |-  ( ph  ->  X  e.  B )
1210, 11ffvelrnd 5666 . 2  |-  ( ph  ->  ( F `  X
)  e.  ( Base `  E ) )
13 funciso.y . . 3  |-  ( ph  ->  Y  e.  B )
1410, 13ffvelrnd 5666 . 2  |-  ( ph  ->  ( F `  Y
)  e.  ( Base `  E ) )
15 funciso.t . 2  |-  J  =  (  Iso  `  E
)
16 eqid 2283 . . 3  |-  (Inv `  D )  =  (Inv
`  D )
17 funciso.m . . . . 5  |-  ( ph  ->  M  e.  ( X I Y ) )
187simpld 445 . . . . . 6  |-  ( ph  ->  D  e.  Cat )
19 funciso.s . . . . . 6  |-  I  =  (  Iso  `  D
)
209, 16, 18, 11, 13, 19isoval 13667 . . . . 5  |-  ( ph  ->  ( X I Y )  =  dom  ( X (Inv `  D ) Y ) )
2117, 20eleqtrd 2359 . . . 4  |-  ( ph  ->  M  e.  dom  ( X (Inv `  D ) Y ) )
229, 16, 18, 11, 13invfun 13666 . . . . 5  |-  ( ph  ->  Fun  ( X (Inv
`  D ) Y ) )
23 funfvbrb 5638 . . . . 5  |-  ( Fun  ( X (Inv `  D ) Y )  ->  ( M  e. 
dom  ( X (Inv
`  D ) Y )  <->  M ( X (Inv
`  D ) Y ) ( ( X (Inv `  D ) Y ) `  M
) ) )
2422, 23syl 15 . . . 4  |-  ( ph  ->  ( M  e.  dom  ( X (Inv `  D
) Y )  <->  M ( X (Inv `  D ) Y ) ( ( X (Inv `  D
) Y ) `  M ) ) )
2521, 24mpbid 201 . . 3  |-  ( ph  ->  M ( X (Inv
`  D ) Y ) ( ( X (Inv `  D ) Y ) `  M
) )
269, 16, 2, 3, 11, 13, 25funcinv 13747 . 2  |-  ( ph  ->  ( ( X G Y ) `  M
) ( ( F `
 X ) (Inv
`  E ) ( F `  Y ) ) ( ( Y G X ) `  ( ( X (Inv
`  D ) Y ) `  M ) ) )
271, 2, 8, 12, 14, 15, 26inviso1 13668 1  |-  ( ph  ->  ( ( X G Y ) `  M
)  e.  ( ( F `  X ) J ( F `  Y ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   <.cop 3643   class class class wbr 4023   dom cdm 4689   Fun wfun 5249   ` cfv 5255  (class class class)co 5858   Basecbs 13148   Catccat 13566  Invcinv 13648    Iso ciso 13649    Func cfunc 13728
This theorem is referenced by:  ffthiso  13803
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-map 6774  df-ixp 6818  df-cat 13570  df-cid 13571  df-sect 13650  df-inv 13651  df-iso 13652  df-func 13732
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