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Theorem funciso 14071
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 2436 . 2  |-  ( Base `  E )  =  (
Base `  E )
2 eqid 2436 . 2  |-  (Inv `  E )  =  (Inv
`  E )
3 funciso.f . . . . 5  |-  ( ph  ->  F ( D  Func  E ) G )
4 df-br 4213 . . . . 5  |-  ( F ( D  Func  E
) G  <->  <. F ,  G >.  e.  ( D 
Func  E ) )
53, 4sylib 189 . . . 4  |-  ( ph  -> 
<. F ,  G >.  e.  ( D  Func  E
) )
6 funcrcl 14060 . . . 4  |-  ( <. F ,  G >.  e.  ( D  Func  E
)  ->  ( D  e.  Cat  /\  E  e. 
Cat ) )
75, 6syl 16 . . 3  |-  ( ph  ->  ( D  e.  Cat  /\  E  e.  Cat )
)
87simprd 450 . 2  |-  ( ph  ->  E  e.  Cat )
9 funciso.b . . . 4  |-  B  =  ( Base `  D
)
109, 1, 3funcf1 14063 . . 3  |-  ( ph  ->  F : B --> ( Base `  E ) )
11 funciso.x . . 3  |-  ( ph  ->  X  e.  B )
1210, 11ffvelrnd 5871 . 2  |-  ( ph  ->  ( F `  X
)  e.  ( Base `  E ) )
13 funciso.y . . 3  |-  ( ph  ->  Y  e.  B )
1410, 13ffvelrnd 5871 . 2  |-  ( ph  ->  ( F `  Y
)  e.  ( Base `  E ) )
15 funciso.t . 2  |-  J  =  (  Iso  `  E
)
16 eqid 2436 . . 3  |-  (Inv `  D )  =  (Inv
`  D )
17 funciso.m . . . . 5  |-  ( ph  ->  M  e.  ( X I Y ) )
187simpld 446 . . . . . 6  |-  ( ph  ->  D  e.  Cat )
19 funciso.s . . . . . 6  |-  I  =  (  Iso  `  D
)
209, 16, 18, 11, 13, 19isoval 13990 . . . . 5  |-  ( ph  ->  ( X I Y )  =  dom  ( X (Inv `  D ) Y ) )
2117, 20eleqtrd 2512 . . . 4  |-  ( ph  ->  M  e.  dom  ( X (Inv `  D ) Y ) )
229, 16, 18, 11, 13invfun 13989 . . . . 5  |-  ( ph  ->  Fun  ( X (Inv
`  D ) Y ) )
23 funfvbrb 5843 . . . . 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 16 . . . 4  |-  ( ph  ->  ( M  e.  dom  ( X (Inv `  D
) Y )  <->  M ( X (Inv `  D ) Y ) ( ( X (Inv `  D
) Y ) `  M ) ) )
2521, 24mpbid 202 . . 3  |-  ( ph  ->  M ( X (Inv
`  D ) Y ) ( ( X (Inv `  D ) Y ) `  M
) )
269, 16, 2, 3, 11, 13, 25funcinv 14070 . 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 13991 1  |-  ( ph  ->  ( ( X G Y ) `  M
)  e.  ( ( F `  X ) J ( F `  Y ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   <.cop 3817   class class class wbr 4212   dom cdm 4878   Fun wfun 5448   ` cfv 5454  (class class class)co 6081   Basecbs 13469   Catccat 13889  Invcinv 13971    Iso ciso 13972    Func cfunc 14051
This theorem is referenced by:  ffthiso  14126
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-map 7020  df-ixp 7064  df-cat 13893  df-cid 13894  df-sect 13973  df-inv 13974  df-iso 13975  df-func 14055
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