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

Theorem cofulid 13780
Description: The identity functor is a left identity for composition. (Contributed by Mario Carneiro, 3-Jan-2017.)
Hypotheses
Ref Expression
cofulid.g  |-  ( ph  ->  F  e.  ( C 
Func  D ) )
cofulid.1  |-  I  =  (idfunc `  D )
Assertion
Ref Expression
cofulid  |-  ( ph  ->  ( I  o.func  F )  =  F )

Proof of Theorem cofulid
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cofulid.1 . . . . . 6  |-  I  =  (idfunc `  D )
2 eqid 2296 . . . . . 6  |-  ( Base `  D )  =  (
Base `  D )
3 cofulid.g . . . . . . . 8  |-  ( ph  ->  F  e.  ( C 
Func  D ) )
4 funcrcl 13753 . . . . . . . 8  |-  ( F  e.  ( C  Func  D )  ->  ( C  e.  Cat  /\  D  e. 
Cat ) )
53, 4syl 15 . . . . . . 7  |-  ( ph  ->  ( C  e.  Cat  /\  D  e.  Cat )
)
65simprd 449 . . . . . 6  |-  ( ph  ->  D  e.  Cat )
71, 2, 6idfu1st 13769 . . . . 5  |-  ( ph  ->  ( 1st `  I
)  =  (  _I  |`  ( Base `  D
) ) )
87coeq1d 4861 . . . 4  |-  ( ph  ->  ( ( 1st `  I
)  o.  ( 1st `  F ) )  =  ( (  _I  |`  ( Base `  D ) )  o.  ( 1st `  F
) ) )
9 eqid 2296 . . . . . 6  |-  ( Base `  C )  =  (
Base `  C )
10 relfunc 13752 . . . . . . 7  |-  Rel  ( C  Func  D )
11 1st2ndbr 6185 . . . . . . 7  |-  ( ( Rel  ( C  Func  D )  /\  F  e.  ( C  Func  D
) )  ->  ( 1st `  F ) ( C  Func  D )
( 2nd `  F
) )
1210, 3, 11sylancr 644 . . . . . 6  |-  ( ph  ->  ( 1st `  F
) ( C  Func  D ) ( 2nd `  F
) )
139, 2, 12funcf1 13756 . . . . 5  |-  ( ph  ->  ( 1st `  F
) : ( Base `  C ) --> ( Base `  D ) )
14 fcoi2 5432 . . . . 5  |-  ( ( 1st `  F ) : ( Base `  C
) --> ( Base `  D
)  ->  ( (  _I  |`  ( Base `  D
) )  o.  ( 1st `  F ) )  =  ( 1st `  F
) )
1513, 14syl 15 . . . 4  |-  ( ph  ->  ( (  _I  |`  ( Base `  D ) )  o.  ( 1st `  F
) )  =  ( 1st `  F ) )
168, 15eqtrd 2328 . . 3  |-  ( ph  ->  ( ( 1st `  I
)  o.  ( 1st `  F ) )  =  ( 1st `  F
) )
1763ad2ant1 976 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( Base `  C )  /\  y  e.  ( Base `  C ) )  ->  D  e.  Cat )
18 eqid 2296 . . . . . . . 8  |-  (  Hom  `  D )  =  (  Hom  `  D )
1913ffvelrnda 5681 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( ( 1st `  F ) `  x )  e.  (
Base `  D )
)
20193adant3 975 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( Base `  C )  /\  y  e.  ( Base `  C ) )  ->  ( ( 1st `  F ) `  x
)  e.  ( Base `  D ) )
2113ffvelrnda 5681 . . . . . . . . 9  |-  ( (
ph  /\  y  e.  ( Base `  C )
)  ->  ( ( 1st `  F ) `  y )  e.  (
Base `  D )
)
22213adant2 974 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( Base `  C )  /\  y  e.  ( Base `  C ) )  ->  ( ( 1st `  F ) `  y
)  e.  ( Base `  D ) )
231, 2, 17, 18, 20, 22idfu2nd 13767 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( Base `  C )  /\  y  e.  ( Base `  C ) )  ->  ( ( ( 1st `  F ) `
 x ) ( 2nd `  I ) ( ( 1st `  F
) `  y )
)  =  (  _I  |`  ( ( ( 1st `  F ) `  x
) (  Hom  `  D
) ( ( 1st `  F ) `  y
) ) ) )
2423coeq1d 4861 . . . . . 6  |-  ( (
ph  /\  x  e.  ( Base `  C )  /\  y  e.  ( Base `  C ) )  ->  ( ( ( ( 1st `  F
) `  x )
( 2nd `  I
) ( ( 1st `  F ) `  y
) )  o.  (
x ( 2nd `  F
) y ) )  =  ( (  _I  |`  ( ( ( 1st `  F ) `  x
) (  Hom  `  D
) ( ( 1st `  F ) `  y
) ) )  o.  ( x ( 2nd `  F ) y ) ) )
25 eqid 2296 . . . . . . . 8  |-  (  Hom  `  C )  =  (  Hom  `  C )
26123ad2ant1 976 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( Base `  C )  /\  y  e.  ( Base `  C ) )  ->  ( 1st `  F
) ( C  Func  D ) ( 2nd `  F
) )
27 simp2 956 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( Base `  C )  /\  y  e.  ( Base `  C ) )  ->  x  e.  (
Base `  C )
)
28 simp3 957 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( Base `  C )  /\  y  e.  ( Base `  C ) )  ->  y  e.  (
Base `  C )
)
299, 25, 18, 26, 27, 28funcf2 13758 . . . . . . 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 )
) )
30 fcoi2 5432 . . . . . . 7  |-  ( ( x ( 2nd `  F
) y ) : ( x (  Hom  `  C ) y ) --> ( ( ( 1st `  F ) `  x
) (  Hom  `  D
) ( ( 1st `  F ) `  y
) )  ->  (
(  _I  |`  (
( ( 1st `  F
) `  x )
(  Hom  `  D ) ( ( 1st `  F
) `  y )
) )  o.  (
x ( 2nd `  F
) y ) )  =  ( x ( 2nd `  F ) y ) )
3129, 30syl 15 . . . . . 6  |-  ( (
ph  /\  x  e.  ( Base `  C )  /\  y  e.  ( Base `  C ) )  ->  ( (  _I  |`  ( ( ( 1st `  F ) `  x
) (  Hom  `  D
) ( ( 1st `  F ) `  y
) ) )  o.  ( x ( 2nd `  F ) y ) )  =  ( x ( 2nd `  F
) y ) )
3224, 31eqtrd 2328 . . . . 5  |-  ( (
ph  /\  x  e.  ( Base `  C )  /\  y  e.  ( Base `  C ) )  ->  ( ( ( ( 1st `  F
) `  x )
( 2nd `  I
) ( ( 1st `  F ) `  y
) )  o.  (
x ( 2nd `  F
) y ) )  =  ( x ( 2nd `  F ) y ) )
3332mpt2eq3dva 5928 . . . 4  |-  ( ph  ->  ( x  e.  (
Base `  C ) ,  y  e.  ( Base `  C )  |->  ( ( ( ( 1st `  F ) `  x
) ( 2nd `  I
) ( ( 1st `  F ) `  y
) )  o.  (
x ( 2nd `  F
) y ) ) )  =  ( x  e.  ( Base `  C
) ,  y  e.  ( Base `  C
)  |->  ( x ( 2nd `  F ) y ) ) )
349, 12funcfn2 13759 . . . . 5  |-  ( ph  ->  ( 2nd `  F
)  Fn  ( (
Base `  C )  X.  ( Base `  C
) ) )
35 fnov 5968 . . . . 5  |-  ( ( 2nd `  F )  Fn  ( ( Base `  C )  X.  ( Base `  C ) )  <-> 
( 2nd `  F
)  =  ( x  e.  ( Base `  C
) ,  y  e.  ( Base `  C
)  |->  ( x ( 2nd `  F ) y ) ) )
3634, 35sylib 188 . . . 4  |-  ( ph  ->  ( 2nd `  F
)  =  ( x  e.  ( Base `  C
) ,  y  e.  ( Base `  C
)  |->  ( x ( 2nd `  F ) y ) ) )
3733, 36eqtr4d 2331 . . 3  |-  ( ph  ->  ( x  e.  (
Base `  C ) ,  y  e.  ( Base `  C )  |->  ( ( ( ( 1st `  F ) `  x
) ( 2nd `  I
) ( ( 1st `  F ) `  y
) )  o.  (
x ( 2nd `  F
) y ) ) )  =  ( 2nd `  F ) )
3816, 37opeq12d 3820 . 2  |-  ( ph  -> 
<. ( ( 1st `  I
)  o.  ( 1st `  F ) ) ,  ( x  e.  (
Base `  C ) ,  y  e.  ( Base `  C )  |->  ( ( ( ( 1st `  F ) `  x
) ( 2nd `  I
) ( ( 1st `  F ) `  y
) )  o.  (
x ( 2nd `  F
) y ) ) ) >.  =  <. ( 1st `  F ) ,  ( 2nd `  F
) >. )
391idfucl 13771 . . . 4  |-  ( D  e.  Cat  ->  I  e.  ( D  Func  D
) )
406, 39syl 15 . . 3  |-  ( ph  ->  I  e.  ( D 
Func  D ) )
419, 3, 40cofuval 13772 . 2  |-  ( ph  ->  ( I  o.func  F )  =  <. ( ( 1st `  I )  o.  ( 1st `  F ) ) ,  ( x  e.  ( Base `  C
) ,  y  e.  ( Base `  C
)  |->  ( ( ( ( 1st `  F
) `  x )
( 2nd `  I
) ( ( 1st `  F ) `  y
) )  o.  (
x ( 2nd `  F
) y ) ) ) >. )
42 1st2nd 6182 . . 3  |-  ( ( Rel  ( C  Func  D )  /\  F  e.  ( C  Func  D
) )  ->  F  =  <. ( 1st `  F
) ,  ( 2nd `  F ) >. )
4310, 3, 42sylancr 644 . 2  |-  ( ph  ->  F  =  <. ( 1st `  F ) ,  ( 2nd `  F
) >. )
4438, 41, 433eqtr4d 2338 1  |-  ( ph  ->  ( I  o.func  F )  =  F )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   <.cop 3656   class class class wbr 4039    _I cid 4320    X. cxp 4703    |` cres 4707    o. ccom 4709   Rel wrel 4710    Fn wfn 5266   -->wf 5267   ` cfv 5271  (class class class)co 5874    e. cmpt2 5876   1stc1st 6136   2ndc2nd 6137   Basecbs 13164    Hom chom 13235   Catccat 13582    Func cfunc 13744  idfunccidfu 13745    o.func ccofu 13746
This theorem is referenced by:  catccatid  13950
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-func 13748  df-idfu 13749  df-cofu 13750
  Copyright terms: Public domain W3C validator