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Theorem funcid 14059
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 2435 . . . . . 6  |-  ( Base `  E )  =  (
Base `  E )
5 eqid 2435 . . . . . 6  |-  (  Hom  `  D )  =  (  Hom  `  D )
6 eqid 2435 . . . . . 6  |-  (  Hom  `  E )  =  (  Hom  `  E )
7 funcid.1 . . . . . 6  |-  .1.  =  ( Id `  D )
8 funcid.i . . . . . 6  |-  I  =  ( Id `  E
)
9 eqid 2435 . . . . . 6  |-  (comp `  D )  =  (comp `  D )
10 eqid 2435 . . . . . 6  |-  (comp `  E )  =  (comp `  E )
11 df-br 4205 . . . . . . . . 9  |-  ( F ( D  Func  E
) G  <->  <. F ,  G >.  e.  ( D 
Func  E ) )
122, 11sylib 189 . . . . . . . 8  |-  ( ph  -> 
<. F ,  G >.  e.  ( D  Func  E
) )
13 funcrcl 14052 . . . . . . . 8  |-  ( <. F ,  G >.  e.  ( D  Func  E
)  ->  ( D  e.  Cat  /\  E  e. 
Cat ) )
1412, 13syl 16 . . . . . . 7  |-  ( ph  ->  ( D  e.  Cat  /\  E  e.  Cat )
)
1514simpld 446 . . . . . 6  |-  ( ph  ->  D  e.  Cat )
1614simprd 450 . . . . . 6  |-  ( ph  ->  E  e.  Cat )
173, 4, 5, 6, 7, 8, 9, 10, 15, 16isfunc 14053 . . . . 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 202 . . . 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 971 . . 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 444 . . . 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 2773 . . 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 16 . 2  |-  ( ph  ->  A. x  e.  B  ( ( x G x ) `  (  .1.  `  x ) )  =  ( I `  ( F `  x ) ) )
23 id 20 . . . . . 6  |-  ( x  =  X  ->  x  =  X )
2423, 23oveq12d 6091 . . . . 5  |-  ( x  =  X  ->  (
x G x )  =  ( X G X ) )
25 fveq2 5720 . . . . 5  |-  ( x  =  X  ->  (  .1.  `  x )  =  (  .1.  `  X
) )
2624, 25fveq12d 5726 . . . 4  |-  ( x  =  X  ->  (
( x G x ) `  (  .1.  `  x ) )  =  ( ( X G X ) `  (  .1.  `  X ) ) )
27 fveq2 5720 . . . . 5  |-  ( x  =  X  ->  ( F `  x )  =  ( F `  X ) )
2827fveq2d 5724 . . . 4  |-  ( x  =  X  ->  (
I `  ( F `  x ) )  =  ( I `  ( F `  X )
) )
2926, 28eqeq12d 2449 . . 3  |-  ( x  =  X  ->  (
( ( x G x ) `  (  .1.  `  x ) )  =  ( I `  ( F `  x ) )  <->  ( ( X G X ) `  (  .1.  `  X )
)  =  ( I `
 ( F `  X ) ) ) )
3029rspcv 3040 . 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 58 1  |-  ( ph  ->  ( ( X G X ) `  (  .1.  `  X ) )  =  ( I `  ( F `  X ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   A.wral 2697   <.cop 3809   class class class wbr 4204    X. cxp 4868   -->wf 5442   ` cfv 5446  (class class class)co 6073   1stc1st 6339   2ndc2nd 6340    ^m cmap 7010   X_cixp 7055   Basecbs 13461    Hom chom 13532  compcco 13533   Catccat 13881   Idccid 13882    Func cfunc 14043
This theorem is referenced by:  funcsect  14061  funcoppc  14064  cofucl  14077  funcres  14085  fthsect  14114  catcisolem  14253  prfcl  14292  evlfcl  14311  curf1cl  14317  curfcl  14321  curfuncf  14327  yonedainv  14370
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 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-map 7012  df-ixp 7056  df-func 14047
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