MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cofurid Structured version   Unicode version

Theorem cofurid 15129
Description: The identity functor is a right identity for composition. (Contributed by Mario Carneiro, 3-Jan-2017.)
Hypotheses
Ref Expression
cofulid.g  |-  ( ph  ->  F  e.  ( C 
Func  D ) )
cofurid.1  |-  I  =  (idfunc `  C )
Assertion
Ref Expression
cofurid  |-  ( ph  ->  ( F  o.func  I )  =  F )

Proof of Theorem cofurid
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cofurid.1 . . . . . 6  |-  I  =  (idfunc `  C )
2 eqid 2441 . . . . . 6  |-  ( Base `  C )  =  (
Base `  C )
3 cofulid.g . . . . . . . 8  |-  ( ph  ->  F  e.  ( C 
Func  D ) )
4 funcrcl 15101 . . . . . . . 8  |-  ( F  e.  ( C  Func  D )  ->  ( C  e.  Cat  /\  D  e. 
Cat ) )
53, 4syl 16 . . . . . . 7  |-  ( ph  ->  ( C  e.  Cat  /\  D  e.  Cat )
)
65simpld 459 . . . . . 6  |-  ( ph  ->  C  e.  Cat )
71, 2, 6idfu1st 15117 . . . . 5  |-  ( ph  ->  ( 1st `  I
)  =  (  _I  |`  ( Base `  C
) ) )
87coeq2d 5151 . . . 4  |-  ( ph  ->  ( ( 1st `  F
)  o.  ( 1st `  I ) )  =  ( ( 1st `  F
)  o.  (  _I  |`  ( Base `  C
) ) ) )
9 eqid 2441 . . . . . 6  |-  ( Base `  D )  =  (
Base `  D )
10 relfunc 15100 . . . . . . 7  |-  Rel  ( C  Func  D )
11 1st2ndbr 6830 . . . . . . 7  |-  ( ( Rel  ( C  Func  D )  /\  F  e.  ( C  Func  D
) )  ->  ( 1st `  F ) ( C  Func  D )
( 2nd `  F
) )
1210, 3, 11sylancr 663 . . . . . 6  |-  ( ph  ->  ( 1st `  F
) ( C  Func  D ) ( 2nd `  F
) )
132, 9, 12funcf1 15104 . . . . 5  |-  ( ph  ->  ( 1st `  F
) : ( Base `  C ) --> ( Base `  D ) )
14 fcoi1 5745 . . . . 5  |-  ( ( 1st `  F ) : ( Base `  C
) --> ( Base `  D
)  ->  ( ( 1st `  F )  o.  (  _I  |`  ( Base `  C ) ) )  =  ( 1st `  F ) )
1513, 14syl 16 . . . 4  |-  ( ph  ->  ( ( 1st `  F
)  o.  (  _I  |`  ( Base `  C
) ) )  =  ( 1st `  F
) )
168, 15eqtrd 2482 . . 3  |-  ( ph  ->  ( ( 1st `  F
)  o.  ( 1st `  I ) )  =  ( 1st `  F
) )
1773ad2ant1 1016 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( Base `  C )  /\  y  e.  ( Base `  C ) )  ->  ( 1st `  I
)  =  (  _I  |`  ( Base `  C
) ) )
1817fveq1d 5854 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( Base `  C )  /\  y  e.  ( Base `  C ) )  ->  ( ( 1st `  I ) `  x
)  =  ( (  _I  |`  ( Base `  C ) ) `  x ) )
19 fvresi 6078 . . . . . . . . . 10  |-  ( x  e.  ( Base `  C
)  ->  ( (  _I  |`  ( Base `  C
) ) `  x
)  =  x )
20193ad2ant2 1017 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( Base `  C )  /\  y  e.  ( Base `  C ) )  ->  ( (  _I  |`  ( Base `  C
) ) `  x
)  =  x )
2118, 20eqtrd 2482 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( Base `  C )  /\  y  e.  ( Base `  C ) )  ->  ( ( 1st `  I ) `  x
)  =  x )
2217fveq1d 5854 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( Base `  C )  /\  y  e.  ( Base `  C ) )  ->  ( ( 1st `  I ) `  y
)  =  ( (  _I  |`  ( Base `  C ) ) `  y ) )
23 fvresi 6078 . . . . . . . . . 10  |-  ( y  e.  ( Base `  C
)  ->  ( (  _I  |`  ( Base `  C
) ) `  y
)  =  y )
24233ad2ant3 1018 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( Base `  C )  /\  y  e.  ( Base `  C ) )  ->  ( (  _I  |`  ( Base `  C
) ) `  y
)  =  y )
2522, 24eqtrd 2482 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( Base `  C )  /\  y  e.  ( Base `  C ) )  ->  ( ( 1st `  I ) `  y
)  =  y )
2621, 25oveq12d 6295 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( Base `  C )  /\  y  e.  ( Base `  C ) )  ->  ( ( ( 1st `  I ) `
 x ) ( 2nd `  F ) ( ( 1st `  I
) `  y )
)  =  ( x ( 2nd `  F
) y ) )
2763ad2ant1 1016 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( Base `  C )  /\  y  e.  ( Base `  C ) )  ->  C  e.  Cat )
28 eqid 2441 . . . . . . . 8  |-  ( Hom  `  C )  =  ( Hom  `  C )
29 simp2 996 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( Base `  C )  /\  y  e.  ( Base `  C ) )  ->  x  e.  (
Base `  C )
)
30 simp3 997 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( Base `  C )  /\  y  e.  ( Base `  C ) )  ->  y  e.  (
Base `  C )
)
311, 2, 27, 28, 29, 30idfu2nd 15115 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( Base `  C )  /\  y  e.  ( Base `  C ) )  ->  ( x ( 2nd `  I ) y )  =  (  _I  |`  ( x
( Hom  `  C ) y ) ) )
3226, 31coeq12d 5153 . . . . . 6  |-  ( (
ph  /\  x  e.  ( Base `  C )  /\  y  e.  ( Base `  C ) )  ->  ( ( ( ( 1st `  I
) `  x )
( 2nd `  F
) ( ( 1st `  I ) `  y
) )  o.  (
x ( 2nd `  I
) y ) )  =  ( ( x ( 2nd `  F
) y )  o.  (  _I  |`  (
x ( Hom  `  C
) y ) ) ) )
33 eqid 2441 . . . . . . . 8  |-  ( Hom  `  D )  =  ( Hom  `  D )
34123ad2ant1 1016 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( Base `  C )  /\  y  e.  ( Base `  C ) )  ->  ( 1st `  F
) ( C  Func  D ) ( 2nd `  F
) )
352, 28, 33, 34, 29, 30funcf2 15106 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( Base `  C )  /\  y  e.  ( Base `  C ) )  ->  ( x ( 2nd `  F ) y ) : ( x ( Hom  `  C
) y ) --> ( ( ( 1st `  F
) `  x )
( Hom  `  D ) ( ( 1st `  F
) `  y )
) )
36 fcoi1 5745 . . . . . . 7  |-  ( ( x ( 2nd `  F
) y ) : ( x ( Hom  `  C ) y ) --> ( ( ( 1st `  F ) `  x
) ( Hom  `  D
) ( ( 1st `  F ) `  y
) )  ->  (
( x ( 2nd `  F ) y )  o.  (  _I  |`  (
x ( Hom  `  C
) y ) ) )  =  ( x ( 2nd `  F
) y ) )
3735, 36syl 16 . . . . . 6  |-  ( (
ph  /\  x  e.  ( Base `  C )  /\  y  e.  ( Base `  C ) )  ->  ( ( x ( 2nd `  F
) y )  o.  (  _I  |`  (
x ( Hom  `  C
) y ) ) )  =  ( x ( 2nd `  F
) y ) )
3832, 37eqtrd 2482 . . . . 5  |-  ( (
ph  /\  x  e.  ( Base `  C )  /\  y  e.  ( Base `  C ) )  ->  ( ( ( ( 1st `  I
) `  x )
( 2nd `  F
) ( ( 1st `  I ) `  y
) )  o.  (
x ( 2nd `  I
) y ) )  =  ( x ( 2nd `  F ) y ) )
3938mpt2eq3dva 6342 . . . 4  |-  ( ph  ->  ( x  e.  (
Base `  C ) ,  y  e.  ( Base `  C )  |->  ( ( ( ( 1st `  I ) `  x
) ( 2nd `  F
) ( ( 1st `  I ) `  y
) )  o.  (
x ( 2nd `  I
) y ) ) )  =  ( x  e.  ( Base `  C
) ,  y  e.  ( Base `  C
)  |->  ( x ( 2nd `  F ) y ) ) )
402, 12funcfn2 15107 . . . . 5  |-  ( ph  ->  ( 2nd `  F
)  Fn  ( (
Base `  C )  X.  ( Base `  C
) ) )
41 fnov 6391 . . . . 5  |-  ( ( 2nd `  F )  Fn  ( ( Base `  C )  X.  ( Base `  C ) )  <-> 
( 2nd `  F
)  =  ( x  e.  ( Base `  C
) ,  y  e.  ( Base `  C
)  |->  ( x ( 2nd `  F ) y ) ) )
4240, 41sylib 196 . . . 4  |-  ( ph  ->  ( 2nd `  F
)  =  ( x  e.  ( Base `  C
) ,  y  e.  ( Base `  C
)  |->  ( x ( 2nd `  F ) y ) ) )
4339, 42eqtr4d 2485 . . 3  |-  ( ph  ->  ( x  e.  (
Base `  C ) ,  y  e.  ( Base `  C )  |->  ( ( ( ( 1st `  I ) `  x
) ( 2nd `  F
) ( ( 1st `  I ) `  y
) )  o.  (
x ( 2nd `  I
) y ) ) )  =  ( 2nd `  F ) )
4416, 43opeq12d 4206 . 2  |-  ( ph  -> 
<. ( ( 1st `  F
)  o.  ( 1st `  I ) ) ,  ( x  e.  (
Base `  C ) ,  y  e.  ( Base `  C )  |->  ( ( ( ( 1st `  I ) `  x
) ( 2nd `  F
) ( ( 1st `  I ) `  y
) )  o.  (
x ( 2nd `  I
) y ) ) ) >.  =  <. ( 1st `  F ) ,  ( 2nd `  F
) >. )
451idfucl 15119 . . . 4  |-  ( C  e.  Cat  ->  I  e.  ( C  Func  C
) )
466, 45syl 16 . . 3  |-  ( ph  ->  I  e.  ( C 
Func  C ) )
472, 46, 3cofuval 15120 . 2  |-  ( ph  ->  ( F  o.func  I )  =  <. ( ( 1st `  F )  o.  ( 1st `  I ) ) ,  ( x  e.  ( Base `  C
) ,  y  e.  ( Base `  C
)  |->  ( ( ( ( 1st `  I
) `  x )
( 2nd `  F
) ( ( 1st `  I ) `  y
) )  o.  (
x ( 2nd `  I
) y ) ) ) >. )
48 1st2nd 6827 . . 3  |-  ( ( Rel  ( C  Func  D )  /\  F  e.  ( C  Func  D
) )  ->  F  =  <. ( 1st `  F
) ,  ( 2nd `  F ) >. )
4910, 3, 48sylancr 663 . 2  |-  ( ph  ->  F  =  <. ( 1st `  F ) ,  ( 2nd `  F
) >. )
5044, 47, 493eqtr4d 2492 1  |-  ( ph  ->  ( F  o.func  I )  =  F )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 972    = wceq 1381    e. wcel 1802   <.cop 4016   class class class wbr 4433    _I cid 4776    X. cxp 4983    |` cres 4987    o. ccom 4989   Rel wrel 4990    Fn wfn 5569   -->wf 5570   ` cfv 5574  (class class class)co 6277    |-> cmpt2 6279   1stc1st 6779   2ndc2nd 6780   Basecbs 14504   Hom chom 14580   Catccat 14933    Func cfunc 15092  idfunccidfu 15093    o.func ccofu 15094
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4544  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-reu 2798  df-rmo 2799  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-op 4017  df-uni 4231  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-id 4781  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-1st 6781  df-2nd 6782  df-map 7420  df-ixp 7468  df-cat 14937  df-cid 14938  df-func 15096  df-idfu 15097  df-cofu 15098
This theorem is referenced by:  catccatid  15298
  Copyright terms: Public domain W3C validator