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Theorem cidffn 13596
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 5555 . . 3  |-  ( Base `  c )  e.  _V
2 fvex 5555 . . . 4  |-  (  Hom  `  c )  e.  _V
3 fvex 5555 . . . . 5  |-  (comp `  c )  e.  _V
4 vex 2804 . . . . . 6  |-  b  e. 
_V
54mptex 5762 . . . . 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 3105 . . . 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 3105 . . 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 3105 . 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 13587 . 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 5388 1  |-  Id  Fn  Cat
Colors of variables: wff set class
Syntax hints:    /\ wa 358    = wceq 1632   A.wral 2556   [_csb 3094   <.cop 3656    e. cmpt 4093    Fn wfn 5266   ` cfv 5271  (class class class)co 5874   iota_crio 6313   Basecbs 13164    Hom chom 13235  compcco 13236   Catccat 13582   Idccid 13583
This theorem is referenced by:  cidpropd  13629
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-cid 13587
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