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Theorem cidffn 13903
Description: The identity arrow construction is a function on categories. (Contributed by Mario Carneiro, 17-Jan-2017.)
Assertion
Ref Expression
cidffn  |-  Id  Fn  Cat

Proof of Theorem cidffn
Dummy variables  b 
c  f  g  h  o  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fvex 5742 . . 3  |-  ( Base `  c )  e.  _V
2 fvex 5742 . . . 4  |-  (  Hom  `  c )  e.  _V
3 fvex 5742 . . . . 5  |-  (comp `  c )  e.  _V
4 vex 2959 . . . . . 6  |-  b  e. 
_V
54mptex 5966 . . . . 5  |-  ( x  e.  b  |->  ( iota_ g  e.  ( x h x ) A. y  e.  b  ( A. f  e.  ( y
h x ) ( g ( <. y ,  x >. o x ) f )  =  f  /\  A. f  e.  ( x h y ) ( f (
<. x ,  x >. o y ) g )  =  f ) ) )  e.  _V
63, 5csbex 3262 . . . 4  |-  [_ (comp `  c )  /  o ]_ ( x  e.  b 
|->  ( iota_ g  e.  ( x h x ) A. y  e.  b  ( A. f  e.  ( y h x ) ( g (
<. y ,  x >. o x ) f )  =  f  /\  A. f  e.  ( x h y ) ( f ( <. x ,  x >. o y ) g )  =  f ) ) )  e. 
_V
72, 6csbex 3262 . . 3  |-  [_ (  Hom  `  c )  /  h ]_ [_ (comp `  c )  /  o ]_ ( x  e.  b 
|->  ( iota_ g  e.  ( x h x ) A. y  e.  b  ( A. f  e.  ( y h x ) ( g (
<. y ,  x >. o x ) f )  =  f  /\  A. f  e.  ( x h y ) ( f ( <. x ,  x >. o y ) g )  =  f ) ) )  e. 
_V
81, 7csbex 3262 . 2  |-  [_ ( Base `  c )  / 
b ]_ [_ (  Hom  `  c )  /  h ]_ [_ (comp `  c
)  /  o ]_ ( x  e.  b  |->  ( iota_ g  e.  ( x h x ) A. y  e.  b  ( A. f  e.  ( y h x ) ( g (
<. y ,  x >. o x ) f )  =  f  /\  A. f  e.  ( x h y ) ( f ( <. x ,  x >. o y ) g )  =  f ) ) )  e. 
_V
9 df-cid 13894 . 2  |-  Id  =  ( c  e.  Cat  |->  [_ ( Base `  c
)  /  b ]_ [_ (  Hom  `  c
)  /  h ]_ [_ (comp `  c )  /  o ]_ (
x  e.  b  |->  (
iota_ g  e.  (
x h x ) A. y  e.  b  ( A. f  e.  ( y h x ) ( g (
<. y ,  x >. o x ) f )  =  f  /\  A. f  e.  ( x h y ) ( f ( <. x ,  x >. o y ) g )  =  f ) ) ) )
108, 9fnmpti 5573 1  |-  Id  Fn  Cat
Colors of variables: wff set class
Syntax hints:    /\ wa 359    = wceq 1652   A.wral 2705   [_csb 3251   <.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:  cidpropd  13936
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-cid 13894
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