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Theorem cofulid 15120
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 2467 . . . . . 6  |-  ( Base `  D )  =  (
Base `  D )
3 cofulid.g . . . . . . . 8  |-  ( ph  ->  F  e.  ( C 
Func  D ) )
4 funcrcl 15093 . . . . . . . 8  |-  ( F  e.  ( C  Func  D )  ->  ( C  e.  Cat  /\  D  e. 
Cat ) )
53, 4syl 16 . . . . . . 7  |-  ( ph  ->  ( C  e.  Cat  /\  D  e.  Cat )
)
65simprd 463 . . . . . 6  |-  ( ph  ->  D  e.  Cat )
71, 2, 6idfu1st 15109 . . . . 5  |-  ( ph  ->  ( 1st `  I
)  =  (  _I  |`  ( Base `  D
) ) )
87coeq1d 5164 . . . 4  |-  ( ph  ->  ( ( 1st `  I
)  o.  ( 1st `  F ) )  =  ( (  _I  |`  ( Base `  D ) )  o.  ( 1st `  F
) ) )
9 eqid 2467 . . . . . 6  |-  ( Base `  C )  =  (
Base `  C )
10 relfunc 15092 . . . . . . 7  |-  Rel  ( C  Func  D )
11 1st2ndbr 6834 . . . . . . 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
) )
139, 2, 12funcf1 15096 . . . . 5  |-  ( ph  ->  ( 1st `  F
) : ( Base `  C ) --> ( Base `  D ) )
14 fcoi2 5760 . . . . 5  |-  ( ( 1st `  F ) : ( Base `  C
) --> ( Base `  D
)  ->  ( (  _I  |`  ( Base `  D
) )  o.  ( 1st `  F ) )  =  ( 1st `  F
) )
1513, 14syl 16 . . . 4  |-  ( ph  ->  ( (  _I  |`  ( Base `  D ) )  o.  ( 1st `  F
) )  =  ( 1st `  F ) )
168, 15eqtrd 2508 . . 3  |-  ( ph  ->  ( ( 1st `  I
)  o.  ( 1st `  F ) )  =  ( 1st `  F
) )
1763ad2ant1 1017 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( Base `  C )  /\  y  e.  ( Base `  C ) )  ->  D  e.  Cat )
18 eqid 2467 . . . . . . . 8  |-  ( Hom  `  D )  =  ( Hom  `  D )
1913ffvelrnda 6022 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( ( 1st `  F ) `  x )  e.  (
Base `  D )
)
20193adant3 1016 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( Base `  C )  /\  y  e.  ( Base `  C ) )  ->  ( ( 1st `  F ) `  x
)  e.  ( Base `  D ) )
2113ffvelrnda 6022 . . . . . . . . 9  |-  ( (
ph  /\  y  e.  ( Base `  C )
)  ->  ( ( 1st `  F ) `  y )  e.  (
Base `  D )
)
22213adant2 1015 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( Base `  C )  /\  y  e.  ( Base `  C ) )  ->  ( ( 1st `  F ) `  y
)  e.  ( Base `  D ) )
231, 2, 17, 18, 20, 22idfu2nd 15107 . . . . . . 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 5164 . . . . . 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 2467 . . . . . . . 8  |-  ( Hom  `  C )  =  ( Hom  `  C )
26123ad2ant1 1017 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( Base `  C )  /\  y  e.  ( Base `  C ) )  ->  ( 1st `  F
) ( C  Func  D ) ( 2nd `  F
) )
27 simp2 997 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( Base `  C )  /\  y  e.  ( Base `  C ) )  ->  x  e.  (
Base `  C )
)
28 simp3 998 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( Base `  C )  /\  y  e.  ( Base `  C ) )  ->  y  e.  (
Base `  C )
)
299, 25, 18, 26, 27, 28funcf2 15098 . . . . . . 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 5760 . . . . . . 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 16 . . . . . 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 2508 . . . . 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 6346 . . . 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 15099 . . . . 5  |-  ( ph  ->  ( 2nd `  F
)  Fn  ( (
Base `  C )  X.  ( Base `  C
) ) )
35 fnov 6395 . . . . 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 196 . . . 4  |-  ( ph  ->  ( 2nd `  F
)  =  ( x  e.  ( Base `  C
) ,  y  e.  ( Base `  C
)  |->  ( x ( 2nd `  F ) y ) ) )
3733, 36eqtr4d 2511 . . 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 4221 . 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 15111 . . . 4  |-  ( D  e.  Cat  ->  I  e.  ( D  Func  D
) )
406, 39syl 16 . . 3  |-  ( ph  ->  I  e.  ( D 
Func  D ) )
419, 3, 40cofuval 15112 . 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 6831 . . 3  |-  ( ( Rel  ( C  Func  D )  /\  F  e.  ( C  Func  D
) )  ->  F  =  <. ( 1st `  F
) ,  ( 2nd `  F ) >. )
4310, 3, 42sylancr 663 . 2  |-  ( ph  ->  F  =  <. ( 1st `  F ) ,  ( 2nd `  F
) >. )
4438, 41, 433eqtr4d 2518 1  |-  ( ph  ->  ( I  o.func  F )  =  F )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   <.cop 4033   class class class wbr 4447    _I cid 4790    X. cxp 4997    |` cres 5001    o. ccom 5003   Rel wrel 5004    Fn wfn 5583   -->wf 5584   ` cfv 5588  (class class class)co 6285    |-> cmpt2 6287   1stc1st 6783   2ndc2nd 6784   Basecbs 14493   Hom chom 14569   Catccat 14922    Func cfunc 15084  idfunccidfu 15085    o.func ccofu 15086
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6246  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-1st 6785  df-2nd 6786  df-map 7423  df-ixp 7471  df-cat 14926  df-cid 14927  df-func 15088  df-idfu 15089  df-cofu 15090
This theorem is referenced by:  catccatid  15290
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