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Theorem cofulid 15378
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 2454 . . . . . 6  |-  ( Base `  D )  =  (
Base `  D )
3 cofulid.g . . . . . . . 8  |-  ( ph  ->  F  e.  ( C 
Func  D ) )
4 funcrcl 15351 . . . . . . . 8  |-  ( F  e.  ( C  Func  D )  ->  ( C  e.  Cat  /\  D  e. 
Cat ) )
53, 4syl 16 . . . . . . 7  |-  ( ph  ->  ( C  e.  Cat  /\  D  e.  Cat )
)
65simprd 461 . . . . . 6  |-  ( ph  ->  D  e.  Cat )
71, 2, 6idfu1st 15367 . . . . 5  |-  ( ph  ->  ( 1st `  I
)  =  (  _I  |`  ( Base `  D
) ) )
87coeq1d 5153 . . . 4  |-  ( ph  ->  ( ( 1st `  I
)  o.  ( 1st `  F ) )  =  ( (  _I  |`  ( Base `  D ) )  o.  ( 1st `  F
) ) )
9 eqid 2454 . . . . . 6  |-  ( Base `  C )  =  (
Base `  C )
10 relfunc 15350 . . . . . . 7  |-  Rel  ( C  Func  D )
11 1st2ndbr 6822 . . . . . . 7  |-  ( ( Rel  ( C  Func  D )  /\  F  e.  ( C  Func  D
) )  ->  ( 1st `  F ) ( C  Func  D )
( 2nd `  F
) )
1210, 3, 11sylancr 661 . . . . . 6  |-  ( ph  ->  ( 1st `  F
) ( C  Func  D ) ( 2nd `  F
) )
139, 2, 12funcf1 15354 . . . . 5  |-  ( ph  ->  ( 1st `  F
) : ( Base `  C ) --> ( Base `  D ) )
14 fcoi2 5742 . . . . 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 2495 . . 3  |-  ( ph  ->  ( ( 1st `  I
)  o.  ( 1st `  F ) )  =  ( 1st `  F
) )
1763ad2ant1 1015 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( Base `  C )  /\  y  e.  ( Base `  C ) )  ->  D  e.  Cat )
18 eqid 2454 . . . . . . . 8  |-  ( Hom  `  D )  =  ( Hom  `  D )
1913ffvelrnda 6007 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( Base `  C )
)  ->  ( ( 1st `  F ) `  x )  e.  (
Base `  D )
)
20193adant3 1014 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( Base `  C )  /\  y  e.  ( Base `  C ) )  ->  ( ( 1st `  F ) `  x
)  e.  ( Base `  D ) )
2113ffvelrnda 6007 . . . . . . . . 9  |-  ( (
ph  /\  y  e.  ( Base `  C )
)  ->  ( ( 1st `  F ) `  y )  e.  (
Base `  D )
)
22213adant2 1013 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( Base `  C )  /\  y  e.  ( Base `  C ) )  ->  ( ( 1st `  F ) `  y
)  e.  ( Base `  D ) )
231, 2, 17, 18, 20, 22idfu2nd 15365 . . . . . . 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 5153 . . . . . 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 2454 . . . . . . . 8  |-  ( Hom  `  C )  =  ( Hom  `  C )
26123ad2ant1 1015 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( Base `  C )  /\  y  e.  ( Base `  C ) )  ->  ( 1st `  F
) ( C  Func  D ) ( 2nd `  F
) )
27 simp2 995 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( Base `  C )  /\  y  e.  ( Base `  C ) )  ->  x  e.  (
Base `  C )
)
28 simp3 996 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( Base `  C )  /\  y  e.  ( Base `  C ) )  ->  y  e.  (
Base `  C )
)
299, 25, 18, 26, 27, 28funcf2 15356 . . . . . . 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 5742 . . . . . . 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 2495 . . . . 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 6334 . . . 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 15357 . . . . 5  |-  ( ph  ->  ( 2nd `  F
)  Fn  ( (
Base `  C )  X.  ( Base `  C
) ) )
35 fnov 6383 . . . . 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 2498 . . 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 4211 . 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 15369 . . . 4  |-  ( D  e.  Cat  ->  I  e.  ( D  Func  D
) )
406, 39syl 16 . . 3  |-  ( ph  ->  I  e.  ( D 
Func  D ) )
419, 3, 40cofuval 15370 . 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 6819 . . 3  |-  ( ( Rel  ( C  Func  D )  /\  F  e.  ( C  Func  D
) )  ->  F  =  <. ( 1st `  F
) ,  ( 2nd `  F ) >. )
4310, 3, 42sylancr 661 . 2  |-  ( ph  ->  F  =  <. ( 1st `  F ) ,  ( 2nd `  F
) >. )
4438, 41, 433eqtr4d 2505 1  |-  ( ph  ->  ( I  o.func  F )  =  F )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823   <.cop 4022   class class class wbr 4439    _I cid 4779    X. cxp 4986    |` cres 4990    o. ccom 4992   Rel wrel 4993    Fn wfn 5565   -->wf 5566   ` cfv 5570  (class class class)co 6270    |-> cmpt2 6272   1stc1st 6771   2ndc2nd 6772   Basecbs 14716   Hom chom 14795   Catccat 15153    Func cfunc 15342  idfunccidfu 15343    o.func ccofu 15344
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-1st 6773  df-2nd 6774  df-map 7414  df-ixp 7463  df-cat 15157  df-cid 15158  df-func 15346  df-idfu 15347  df-cofu 15348
This theorem is referenced by:  catccatid  15580
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