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Theorem cidfn 14609
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 6051 . . 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 2438 . . 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 5534 . 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 2438 . . . 4  |-  ( Hom  `  C )  =  ( Hom  `  C )
6 eqid 2438 . . . 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 14606 . . 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 5496 . 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 1369    e. wcel 1756   A.wral 2710   <.cop 3878    e. cmpt 4345    Fn wfn 5408   ` cfv 5413   iota_crio 6046  (class class class)co 6086   Basecbs 14166   Hom chom 14241  compcco 14242   Catccat 14594   Idccid 14595
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pr 4526
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-reu 2717  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-id 4631  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-cid 14599
This theorem is referenced by:  oppccatid  14650  fucidcl  14867  fucsect  14874  curfcl  15034  curf2ndf  15049
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