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

Theorem cidffn 15596
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 vex 3050 . . . . . 6  |-  b  e. 
_V
21mptex 6141 . . . . 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
32csbex 4541 . . . 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
43csbex 4541 . . 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
54csbex 4541 . 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
6 df-cid 15587 . 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 ) ) ) )
75, 6fnmpti 5711 1  |-  Id  Fn  Cat
Colors of variables: wff setvar class
Syntax hints:    /\ wa 371    = wceq 1446   A.wral 2739   [_csb 3365   <.cop 3976    |-> cmpt 4464    Fn wfn 5580   ` cfv 5585   iota_crio 6256  (class class class)co 6295   Basecbs 15133   Hom chom 15213  compcco 15214   Catccat 15582   Idccid 15583
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-rep 4518  ax-sep 4528  ax-nul 4537  ax-pr 4642
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 988  df-tru 1449  df-fal 1452  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-ral 2744  df-rex 2745  df-reu 2746  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3734  df-if 3884  df-sn 3971  df-pr 3973  df-op 3977  df-uni 4202  df-iun 4283  df-br 4406  df-opab 4465  df-mpt 4466  df-id 4752  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 5549  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-fv 5593  df-cid 15587
This theorem is referenced by:  cidpropd  15627
  Copyright terms: Public domain W3C validator