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

Theorem cidfn 13581
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 6308 . . 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 2283 . . 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 5372 . 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 2283 . . . 4  |-  (  Hom  `  C )  =  (  Hom  `  C )
6 eqid 2283 . . . 4  |-  (comp `  C )  =  (comp `  C )
7 id 19 . . . 4  |-  ( C  e.  Cat  ->  C  e.  Cat )
8 cidfn.i . . . 4  |-  .1.  =  ( Id `  C )
94, 5, 6, 7, 8cidfval 13578 . . 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 5335 . 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 224 1  |-  ( C  e.  Cat  ->  .1.  Fn  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   <.cop 3643    e. cmpt 4077    Fn wfn 5250   ` cfv 5255  (class class class)co 5858   iota_crio 6297   Basecbs 13148    Hom chom 13219  compcco 13220   Catccat 13566   Idccid 13567
This theorem is referenced by:  oppccatid  13622  fucidcl  13839  fucsect  13846  curfcl  14006  curf2ndf  14021
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-riota 6304  df-cid 13571
  Copyright terms: Public domain W3C validator