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Theorem cofurid 14043
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 2404 . . . . . 6  |-  ( Base `  C )  =  (
Base `  C )
3 cofulid.g . . . . . . . 8  |-  ( ph  ->  F  e.  ( C 
Func  D ) )
4 funcrcl 14015 . . . . . . . 8  |-  ( F  e.  ( C  Func  D )  ->  ( C  e.  Cat  /\  D  e. 
Cat ) )
53, 4syl 16 . . . . . . 7  |-  ( ph  ->  ( C  e.  Cat  /\  D  e.  Cat )
)
65simpld 446 . . . . . 6  |-  ( ph  ->  C  e.  Cat )
71, 2, 6idfu1st 14031 . . . . 5  |-  ( ph  ->  ( 1st `  I
)  =  (  _I  |`  ( Base `  C
) ) )
87coeq2d 4994 . . . 4  |-  ( ph  ->  ( ( 1st `  F
)  o.  ( 1st `  I ) )  =  ( ( 1st `  F
)  o.  (  _I  |`  ( Base `  C
) ) ) )
9 eqid 2404 . . . . . 6  |-  ( Base `  D )  =  (
Base `  D )
10 relfunc 14014 . . . . . . 7  |-  Rel  ( C  Func  D )
11 1st2ndbr 6355 . . . . . . 7  |-  ( ( Rel  ( C  Func  D )  /\  F  e.  ( C  Func  D
) )  ->  ( 1st `  F ) ( C  Func  D )
( 2nd `  F
) )
1210, 3, 11sylancr 645 . . . . . 6  |-  ( ph  ->  ( 1st `  F
) ( C  Func  D ) ( 2nd `  F
) )
132, 9, 12funcf1 14018 . . . . 5  |-  ( ph  ->  ( 1st `  F
) : ( Base `  C ) --> ( Base `  D ) )
14 fcoi1 5576 . . . . 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 2436 . . 3  |-  ( ph  ->  ( ( 1st `  F
)  o.  ( 1st `  I ) )  =  ( 1st `  F
) )
1773ad2ant1 978 . . . . . . . . . 10  |-  ( (
ph  /\  x  e.  ( Base `  C )  /\  y  e.  ( Base `  C ) )  ->  ( 1st `  I
)  =  (  _I  |`  ( Base `  C
) ) )
1817fveq1d 5689 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( Base `  C )  /\  y  e.  ( Base `  C ) )  ->  ( ( 1st `  I ) `  x
)  =  ( (  _I  |`  ( Base `  C ) ) `  x ) )
19 fvresi 5883 . . . . . . . . . 10  |-  ( x  e.  ( Base `  C
)  ->  ( (  _I  |`  ( Base `  C
) ) `  x
)  =  x )
20193ad2ant2 979 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( Base `  C )  /\  y  e.  ( Base `  C ) )  ->  ( (  _I  |`  ( Base `  C
) ) `  x
)  =  x )
2118, 20eqtrd 2436 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( Base `  C )  /\  y  e.  ( Base `  C ) )  ->  ( ( 1st `  I ) `  x
)  =  x )
2217fveq1d 5689 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( Base `  C )  /\  y  e.  ( Base `  C ) )  ->  ( ( 1st `  I ) `  y
)  =  ( (  _I  |`  ( Base `  C ) ) `  y ) )
23 fvresi 5883 . . . . . . . . . 10  |-  ( y  e.  ( Base `  C
)  ->  ( (  _I  |`  ( Base `  C
) ) `  y
)  =  y )
24233ad2ant3 980 . . . . . . . . 9  |-  ( (
ph  /\  x  e.  ( Base `  C )  /\  y  e.  ( Base `  C ) )  ->  ( (  _I  |`  ( Base `  C
) ) `  y
)  =  y )
2522, 24eqtrd 2436 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( Base `  C )  /\  y  e.  ( Base `  C ) )  ->  ( ( 1st `  I ) `  y
)  =  y )
2621, 25oveq12d 6058 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( Base `  C )  /\  y  e.  ( Base `  C ) )  ->  ( ( ( 1st `  I ) `
 x ) ( 2nd `  F ) ( ( 1st `  I
) `  y )
)  =  ( x ( 2nd `  F
) y ) )
2763ad2ant1 978 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( Base `  C )  /\  y  e.  ( Base `  C ) )  ->  C  e.  Cat )
28 eqid 2404 . . . . . . . 8  |-  (  Hom  `  C )  =  (  Hom  `  C )
29 simp2 958 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( Base `  C )  /\  y  e.  ( Base `  C ) )  ->  x  e.  (
Base `  C )
)
30 simp3 959 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( Base `  C )  /\  y  e.  ( Base `  C ) )  ->  y  e.  (
Base `  C )
)
311, 2, 27, 28, 29, 30idfu2nd 14029 . . . . . . 7  |-  ( (
ph  /\  x  e.  ( Base `  C )  /\  y  e.  ( Base `  C ) )  ->  ( x ( 2nd `  I ) y )  =  (  _I  |`  ( x
(  Hom  `  C ) y ) ) )
3226, 31coeq12d 4996 . . . . . 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 2404 . . . . . . . 8  |-  (  Hom  `  D )  =  (  Hom  `  D )
34123ad2ant1 978 . . . . . . . 8  |-  ( (
ph  /\  x  e.  ( Base `  C )  /\  y  e.  ( Base `  C ) )  ->  ( 1st `  F
) ( C  Func  D ) ( 2nd `  F
) )
352, 28, 33, 34, 29, 30funcf2 14020 . . . . . . 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 5576 . . . . . . 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 2436 . . . . 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 6097 . . . 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 14021 . . . . 5  |-  ( ph  ->  ( 2nd `  F
)  Fn  ( (
Base `  C )  X.  ( Base `  C
) ) )
41 fnov 6137 . . . . 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 189 . . . 4  |-  ( ph  ->  ( 2nd `  F
)  =  ( x  e.  ( Base `  C
) ,  y  e.  ( Base `  C
)  |->  ( x ( 2nd `  F ) y ) ) )
4339, 42eqtr4d 2439 . . 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 3952 . 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 14033 . . . 4  |-  ( C  e.  Cat  ->  I  e.  ( C  Func  C
) )
466, 45syl 16 . . 3  |-  ( ph  ->  I  e.  ( C 
Func  C ) )
472, 46, 3cofuval 14034 . 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 6352 . . 3  |-  ( ( Rel  ( C  Func  D )  /\  F  e.  ( C  Func  D
) )  ->  F  =  <. ( 1st `  F
) ,  ( 2nd `  F ) >. )
4910, 3, 48sylancr 645 . 2  |-  ( ph  ->  F  =  <. ( 1st `  F ) ,  ( 2nd `  F
) >. )
5044, 47, 493eqtr4d 2446 1  |-  ( ph  ->  ( F  o.func  I )  =  F )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1721   <.cop 3777   class class class wbr 4172    _I cid 4453    X. cxp 4835    |` cres 4839    o. ccom 4841   Rel wrel 4842    Fn wfn 5408   -->wf 5409   ` cfv 5413  (class class class)co 6040    e. cmpt2 6042   1stc1st 6306   2ndc2nd 6307   Basecbs 13424    Hom chom 13495   Catccat 13844    Func cfunc 14006  idfunccidfu 14007    o.func ccofu 14008
This theorem is referenced by:  catccatid  14212
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-map 6979  df-ixp 7023  df-cat 13848  df-cid 13849  df-func 14010  df-idfu 14011  df-cofu 14012
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