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Theorem cidfn 14953
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 6246 . . 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 2443 . . 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 5699 . 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 2443 . . . 4  |-  ( Hom  `  C )  =  ( Hom  `  C )
6 eqid 2443 . . . 4  |-  (comp `  C )  =  (comp `  C )
7 id 22 . . . 4  |-  ( C  e.  Cat  ->  C  e.  Cat )
8 cidfn.i . . . 4  |-  .1.  =  ( Id `  C )
94, 5, 6, 7, 8cidfval 14950 . . 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 5661 . 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 233 1  |-  ( C  e.  Cat  ->  .1.  Fn  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1383    e. wcel 1804   A.wral 2793   <.cop 4020    |-> cmpt 4495    Fn wfn 5573   ` cfv 5578   iota_crio 6241  (class class class)co 6281   Basecbs 14509   Hom chom 14585  compcco 14586   Catccat 14938   Idccid 14939
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-reu 2800  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-cid 14943
This theorem is referenced by:  oppccatid  14991  fucidcl  15208  fucsect  15215  curfcl  15375  curf2ndf  15390
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