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

Theorem cidfn 13904
Description: The identity arrow operator is a function from objects to arrows. (Contributed by Mario Carneiro, 4-Jan-2017.)
Hypotheses
Ref Expression
cidfn.b  |-  B  =  ( Base `  C
)
cidfn.i  |-  .1.  =  ( Id `  C )
Assertion
Ref Expression
cidfn  |-  ( C  e.  Cat  ->  .1.  Fn  B )

Proof of Theorem cidfn
Dummy variables  f 
g  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 riotaex 6553 . . 3  |-  ( iota_ g  e.  ( x (  Hom  `  C )
x ) A. y  e.  B  ( A. f  e.  ( y
(  Hom  `  C ) x ) ( g ( <. y ,  x >. (comp `  C )
x ) f )  =  f  /\  A. f  e.  ( x
(  Hom  `  C ) y ) ( f ( <. x ,  x >. (comp `  C )
y ) g )  =  f ) )  e.  _V
2 eqid 2436 . . 3  |-  ( x  e.  B  |->  ( iota_ g  e.  ( x (  Hom  `  C )
x ) A. y  e.  B  ( A. f  e.  ( y
(  Hom  `  C ) x ) ( g ( <. y ,  x >. (comp `  C )
x ) f )  =  f  /\  A. f  e.  ( x
(  Hom  `  C ) y ) ( f ( <. x ,  x >. (comp `  C )
y ) g )  =  f ) ) )  =  ( x  e.  B  |->  ( iota_ g  e.  ( x (  Hom  `  C )
x ) A. y  e.  B  ( A. f  e.  ( y
(  Hom  `  C ) x ) ( g ( <. y ,  x >. (comp `  C )
x ) f )  =  f  /\  A. f  e.  ( x
(  Hom  `  C ) y ) ( f ( <. x ,  x >. (comp `  C )
y ) g )  =  f ) ) )
31, 2fnmpti 5573 . 2  |-  ( x  e.  B  |->  ( iota_ g  e.  ( x (  Hom  `  C )
x ) A. y  e.  B  ( A. f  e.  ( y
(  Hom  `  C ) x ) ( g ( <. y ,  x >. (comp `  C )
x ) f )  =  f  /\  A. f  e.  ( x
(  Hom  `  C ) y ) ( f ( <. x ,  x >. (comp `  C )
y ) g )  =  f ) ) )  Fn  B
4 cidfn.b . . . 4  |-  B  =  ( Base `  C
)
5 eqid 2436 . . . 4  |-  (  Hom  `  C )  =  (  Hom  `  C )
6 eqid 2436 . . . 4  |-  (comp `  C )  =  (comp `  C )
7 id 20 . . . 4  |-  ( C  e.  Cat  ->  C  e.  Cat )
8 cidfn.i . . . 4  |-  .1.  =  ( Id `  C )
94, 5, 6, 7, 8cidfval 13901 . . 3  |-  ( C  e.  Cat  ->  .1.  =  ( x  e.  B  |->  ( iota_ g  e.  ( x (  Hom  `  C ) x ) A. y  e.  B  ( A. f  e.  ( y (  Hom  `  C
) x ) ( g ( <. y ,  x >. (comp `  C
) x ) f )  =  f  /\  A. f  e.  ( x (  Hom  `  C
) y ) ( f ( <. x ,  x >. (comp `  C
) y ) g )  =  f ) ) ) )
109fneq1d 5536 . 2  |-  ( C  e.  Cat  ->  (  .1.  Fn  B  <->  ( x  e.  B  |->  ( iota_ g  e.  ( x (  Hom  `  C )
x ) A. y  e.  B  ( A. f  e.  ( y
(  Hom  `  C ) x ) ( g ( <. y ,  x >. (comp `  C )
x ) f )  =  f  /\  A. f  e.  ( x
(  Hom  `  C ) y ) ( f ( <. x ,  x >. (comp `  C )
y ) g )  =  f ) ) )  Fn  B ) )
113, 10mpbiri 225 1  |-  ( C  e.  Cat  ->  .1.  Fn  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2705   <.cop 3817    e. cmpt 4266    Fn wfn 5449   ` cfv 5454  (class class class)co 6081   iota_crio 6542   Basecbs 13469    Hom chom 13540  compcco 13541   Catccat 13889   Idccid 13890
This theorem is referenced by:  oppccatid  13945  fucidcl  14162  fucsect  14169  curfcl  14329  curf2ndf  14344
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-riota 6549  df-cid 13894
  Copyright terms: Public domain W3C validator