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Theorem funciso 13764
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 2296 . 2  |-  ( Base `  E )  =  (
Base `  E )
2 eqid 2296 . 2  |-  (Inv `  E )  =  (Inv
`  E )
3 funciso.f . . . . 5  |-  ( ph  ->  F ( D  Func  E ) G )
4 df-br 4040 . . . . 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 13753 . . . 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 13756 . . 3  |-  ( ph  ->  F : B --> ( Base `  E ) )
11 funciso.x . . 3  |-  ( ph  ->  X  e.  B )
1210, 11ffvelrnd 5682 . 2  |-  ( ph  ->  ( F `  X
)  e.  ( Base `  E ) )
13 funciso.y . . 3  |-  ( ph  ->  Y  e.  B )
1410, 13ffvelrnd 5682 . 2  |-  ( ph  ->  ( F `  Y
)  e.  ( Base `  E ) )
15 funciso.t . 2  |-  J  =  (  Iso  `  E
)
16 eqid 2296 . . 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 13683 . . . . 5  |-  ( ph  ->  ( X I Y )  =  dom  ( X (Inv `  D ) Y ) )
2117, 20eleqtrd 2372 . . . 4  |-  ( ph  ->  M  e.  dom  ( X (Inv `  D ) Y ) )
229, 16, 18, 11, 13invfun 13682 . . . . 5  |-  ( ph  ->  Fun  ( X (Inv
`  D ) Y ) )
23 funfvbrb 5654 . . . . 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 13763 . 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 13684 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 1632    e. wcel 1696   <.cop 3656   class class class wbr 4039   dom cdm 4705   Fun wfun 5265   ` cfv 5271  (class class class)co 5874   Basecbs 13164   Catccat 13582  Invcinv 13664    Iso ciso 13665    Func cfunc 13744
This theorem is referenced by:  ffthiso  13819
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-rmo 2564  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-riota 6320  df-map 6790  df-ixp 6834  df-cat 13586  df-cid 13587  df-sect 13666  df-inv 13667  df-iso 13668  df-func 13748
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