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Theorem funcid 13744
Description: A functor maps each identity to the corresponding identity in the target category. (Contributed by Mario Carneiro, 2-Jan-2017.)
Hypotheses
Ref Expression
funcid.b  |-  B  =  ( Base `  D
)
funcid.1  |-  .1.  =  ( Id `  D )
funcid.i  |-  I  =  ( Id `  E
)
funcid.f  |-  ( ph  ->  F ( D  Func  E ) G )
funcid.x  |-  ( ph  ->  X  e.  B )
Assertion
Ref Expression
funcid  |-  ( ph  ->  ( ( X G X ) `  (  .1.  `  X ) )  =  ( I `  ( F `  X ) ) )

Proof of Theorem funcid
Dummy variables  m  n  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 funcid.x . 2  |-  ( ph  ->  X  e.  B )
2 funcid.f . . . . 5  |-  ( ph  ->  F ( D  Func  E ) G )
3 funcid.b . . . . . 6  |-  B  =  ( Base `  D
)
4 eqid 2283 . . . . . 6  |-  ( Base `  E )  =  (
Base `  E )
5 eqid 2283 . . . . . 6  |-  (  Hom  `  D )  =  (  Hom  `  D )
6 eqid 2283 . . . . . 6  |-  (  Hom  `  E )  =  (  Hom  `  E )
7 funcid.1 . . . . . 6  |-  .1.  =  ( Id `  D )
8 funcid.i . . . . . 6  |-  I  =  ( Id `  E
)
9 eqid 2283 . . . . . 6  |-  (comp `  D )  =  (comp `  D )
10 eqid 2283 . . . . . 6  |-  (comp `  E )  =  (comp `  E )
11 df-br 4024 . . . . . . . . 9  |-  ( F ( D  Func  E
) G  <->  <. F ,  G >.  e.  ( D 
Func  E ) )
122, 11sylib 188 . . . . . . . 8  |-  ( ph  -> 
<. F ,  G >.  e.  ( D  Func  E
) )
13 funcrcl 13737 . . . . . . . 8  |-  ( <. F ,  G >.  e.  ( D  Func  E
)  ->  ( D  e.  Cat  /\  E  e. 
Cat ) )
1412, 13syl 15 . . . . . . 7  |-  ( ph  ->  ( D  e.  Cat  /\  E  e.  Cat )
)
1514simpld 445 . . . . . 6  |-  ( ph  ->  D  e.  Cat )
1614simprd 449 . . . . . 6  |-  ( ph  ->  E  e.  Cat )
173, 4, 5, 6, 7, 8, 9, 10, 15, 16isfunc 13738 . . . . 5  |-  ( ph  ->  ( F ( D 
Func  E ) G  <->  ( F : B --> ( Base `  E
)  /\  G  e.  X_ z  e.  ( B  X.  B ) ( ( ( F `  ( 1st `  z ) ) (  Hom  `  E
) ( F `  ( 2nd `  z ) ) )  ^m  (
(  Hom  `  D ) `
 z ) )  /\  A. x  e.  B  ( ( ( x G x ) `
 (  .1.  `  x ) )  =  ( I `  ( F `  x )
)  /\  A. y  e.  B  A. z  e.  B  A. m  e.  ( x (  Hom  `  D ) y ) A. n  e.  ( y (  Hom  `  D
) z ) ( ( x G z ) `  ( n ( <. x ,  y
>. (comp `  D )
z ) m ) )  =  ( ( ( y G z ) `  n ) ( <. ( F `  x ) ,  ( F `  y )
>. (comp `  E )
( F `  z
) ) ( ( x G y ) `
 m ) ) ) ) ) )
182, 17mpbid 201 . . . 4  |-  ( ph  ->  ( F : B --> ( Base `  E )  /\  G  e.  X_ z  e.  ( B  X.  B
) ( ( ( F `  ( 1st `  z ) ) (  Hom  `  E )
( F `  ( 2nd `  z ) ) )  ^m  ( (  Hom  `  D ) `  z ) )  /\  A. x  e.  B  ( ( ( x G x ) `  (  .1.  `  x ) )  =  ( I `  ( F `  x ) )  /\  A. y  e.  B  A. z  e.  B  A. m  e.  ( x (  Hom  `  D ) y ) A. n  e.  ( y (  Hom  `  D
) z ) ( ( x G z ) `  ( n ( <. x ,  y
>. (comp `  D )
z ) m ) )  =  ( ( ( y G z ) `  n ) ( <. ( F `  x ) ,  ( F `  y )
>. (comp `  E )
( F `  z
) ) ( ( x G y ) `
 m ) ) ) ) )
1918simp3d 969 . . 3  |-  ( ph  ->  A. x  e.  B  ( ( ( x G x ) `  (  .1.  `  x )
)  =  ( I `
 ( F `  x ) )  /\  A. y  e.  B  A. z  e.  B  A. m  e.  ( x
(  Hom  `  D ) y ) A. n  e.  ( y (  Hom  `  D ) z ) ( ( x G z ) `  (
n ( <. x ,  y >. (comp `  D ) z ) m ) )  =  ( ( ( y G z ) `  n ) ( <.
( F `  x
) ,  ( F `
 y ) >.
(comp `  E )
( F `  z
) ) ( ( x G y ) `
 m ) ) ) )
20 simpl 443 . . . 4  |-  ( ( ( ( x G x ) `  (  .1.  `  x ) )  =  ( I `  ( F `  x ) )  /\  A. y  e.  B  A. z  e.  B  A. m  e.  ( x (  Hom  `  D ) y ) A. n  e.  ( y (  Hom  `  D
) z ) ( ( x G z ) `  ( n ( <. x ,  y
>. (comp `  D )
z ) m ) )  =  ( ( ( y G z ) `  n ) ( <. ( F `  x ) ,  ( F `  y )
>. (comp `  E )
( F `  z
) ) ( ( x G y ) `
 m ) ) )  ->  ( (
x G x ) `
 (  .1.  `  x ) )  =  ( I `  ( F `  x )
) )
2120ralimi 2618 . . 3  |-  ( A. x  e.  B  (
( ( x G x ) `  (  .1.  `  x ) )  =  ( I `  ( F `  x ) )  /\  A. y  e.  B  A. z  e.  B  A. m  e.  ( x (  Hom  `  D ) y ) A. n  e.  ( y (  Hom  `  D
) z ) ( ( x G z ) `  ( n ( <. x ,  y
>. (comp `  D )
z ) m ) )  =  ( ( ( y G z ) `  n ) ( <. ( F `  x ) ,  ( F `  y )
>. (comp `  E )
( F `  z
) ) ( ( x G y ) `
 m ) ) )  ->  A. x  e.  B  ( (
x G x ) `
 (  .1.  `  x ) )  =  ( I `  ( F `  x )
) )
2219, 21syl 15 . 2  |-  ( ph  ->  A. x  e.  B  ( ( x G x ) `  (  .1.  `  x ) )  =  ( I `  ( F `  x ) ) )
23 id 19 . . . . . 6  |-  ( x  =  X  ->  x  =  X )
2423, 23oveq12d 5876 . . . . 5  |-  ( x  =  X  ->  (
x G x )  =  ( X G X ) )
25 fveq2 5525 . . . . 5  |-  ( x  =  X  ->  (  .1.  `  x )  =  (  .1.  `  X
) )
2624, 25fveq12d 5531 . . . 4  |-  ( x  =  X  ->  (
( x G x ) `  (  .1.  `  x ) )  =  ( ( X G X ) `  (  .1.  `  X ) ) )
27 fveq2 5525 . . . . 5  |-  ( x  =  X  ->  ( F `  x )  =  ( F `  X ) )
2827fveq2d 5529 . . . 4  |-  ( x  =  X  ->  (
I `  ( F `  x ) )  =  ( I `  ( F `  X )
) )
2926, 28eqeq12d 2297 . . 3  |-  ( x  =  X  ->  (
( ( x G x ) `  (  .1.  `  x ) )  =  ( I `  ( F `  x ) )  <->  ( ( X G X ) `  (  .1.  `  X )
)  =  ( I `
 ( F `  X ) ) ) )
3029rspcv 2880 . 2  |-  ( X  e.  B  ->  ( A. x  e.  B  ( ( x G x ) `  (  .1.  `  x ) )  =  ( I `  ( F `  x ) )  ->  ( ( X G X ) `  (  .1.  `  X )
)  =  ( I `
 ( F `  X ) ) ) )
311, 22, 30sylc 56 1  |-  ( ph  ->  ( ( X G X ) `  (  .1.  `  X ) )  =  ( I `  ( F `  X ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543   <.cop 3643   class class class wbr 4023    X. cxp 4687   -->wf 5251   ` cfv 5255  (class class class)co 5858   1stc1st 6120   2ndc2nd 6121    ^m cmap 6772   X_cixp 6817   Basecbs 13148    Hom chom 13219  compcco 13220   Catccat 13566   Idccid 13567    Func cfunc 13728
This theorem is referenced by:  funcsect  13746  funcoppc  13749  cofucl  13762  funcres  13770  fthsect  13799  catcisolem  13938  prfcl  13977  evlfcl  13996  curf1cl  14002  curfcl  14006  curfuncf  14012  yonedainv  14055
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-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-map 6774  df-ixp 6818  df-func 13732
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